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http://math.stackexchange.com/questions/tagged/differential-geometry+analysis
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https://ask.sagemath.org/question/34001/orderedsetpartitions-variant/
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# OrderedSetPartitions (variant)
I would like to have the list of partitions of [1,2,...N] of finite size k but with using the NULL set
for example, if looking for the partitions of [1,2,3,4,5] of size 3, i would like [ [],[],[1,2,3,4,5] ]
and [ [],[1,2,3],[4,5] ] to be in this list
is it possible to do natively in Sage, or should i write a custom function ?
OrderedSetPartitions(5,3).list()
returns
[[{1, 2, 3}, {4}, {5}],
[{1, 2, 3}, {5}, {4}],
[{1, 2, 4}, {3}, {5}],
[{1, 2, 5}, {3}, {4}],...
EDIT: here is a (non optimized at all) attempt to solve naively this problem
def OrderedSetPartitions_0(A,k):
cols={i for i in range(k)}
# returns the list of k-OrderedSetPartitions of A, allowing for the empty set
s=Subsets(cols).list()
res=[]
count=0
P=[OrderedSetPartitions(A,i) for i in range(k+1)]
for sub in s:
print("sub=")
print(sub)
tmp=[ {} for i in range(k)]
c=sub.cardinality()
for part in P[c]:
print("part=")
print(part)
for i in range(c):
tmp[sub[i]]=part[i]
print("tmp=")
print(tmp)
res=res.append([tmp])
# res=res.append([tmp]) # tried this too
print("res=")
print(res)
count=count+1
return(res)
# print(count)
A=range(3)
k=2
A
P=[OrderedSetPartitions(A,i) for i in range(k+1)]
# note that P[2].list is a list of list !
P[2].list()
[[{0, 1}, {2}],
[{0, 2}, {1}],
[{1, 2}, {0}],
[{0}, {1, 2}],
[{1}, {0, 2}],
[{2}, {0, 1}]]
myset=OrderedSetPartitions_0(A,k)
I get this error message, and I admit I don't get it at all, because it looks fine when coding, but somehow res seems to be "None" instead of []
sub=
{}
sub=
{0}
part=
[{0, 1, 2}]
tmp=
[{0, 1, 2}, {}]
res=
None
sub=
{1}
part=
[{0, 1, 2}]
tmp=
[{}, {0, 1, 2}]
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_21.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("#
-- coding: utf-8 --\n" + _support_.preparse_worksheet_cell(base64.b64decode("bXlzZXQ9T3JkZXJlZFNldFBhcnRpdGlvbnNfMChBLGsp"),globals())+"\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module>
File "/private/var/folders/gm/z065gk616xg6g0xgn4c7_bvc0000gn/T/tmpryfYOj/___code___.py", line 2, in <module>
exec compile(u'myset=OrderedSetPartitions_0(A,k)
File "", line 1, in <module>
File "/private/var/folders/gm/z065gk616xg6g0xgn4c7_bvc0000gn/T/tmpSH_9LF/___code___.py", line 27, in OrderedSetPartitions_0
res=res.append([tmp])
AttributeError: 'NoneType' object has no attribute 'append'
The problem is about res if i omit the line res=res.append(tmp), i will get the enumeration ok i think
EDIT: i changed res=res.append(tmp) by res.append(tmp) i also changed the line initializing tmp into tmp=[ set() for i in range(k)] now as a result of print(tmp) i get the correct enumeration
[{0, 1, 2}, set([]), set([])]
[set([]), {0, 1, 2}, set([])]
[set([]), set([]), {0, 1, 2}]
[{0, 1}, {2}, set([])]
[{0, 2}, {1}, set([])]
[{1 ...
edit retag close merge delete
Could you please give us what complete output you expect for k=3, N=3 ? It is not clear to me if the empty sets shoud happen first or not.
( 2016-07-04 08:22:52 -0600 )edit
[{0, 1, 2}, {}, {}] [{}, {0, 1, 2}, {}] [{}, {}, {0, 1, 2}] [{0, 1}, {2}, {}] [{0, 2}, {1}, {}] [{1, 2}, {0}, {}] [{0}, {1, 2}, {}] [{1}, {0, 2}, {}] [{2}, {0, 1}, {}] [{0, 1}, {}, {2}] [{0, 2}, {}, {1}] [{1, 2}, {}, {0}] [{0}, {}, {1, 2}] [{1}, {}, {0, 2}] [{2}, {}, {0, 1}] [{}, {0, 1}, {2}] [{}, {0, 2}, {1}] [{}, {1, 2}, {0}] [{}, {0}, {1, 2}] [{}, {1}, {0, 2}] [{}, {2}, {0, 1}] [{0}, {1}, {2}] [{0}, {2}, {1}] [{1}, {0}, {2}] [{2}, {0}, {1}] [{1}, {2}, {0}] [{2}, {1}, {0}]
( 2016-07-04 08:31:10 -0600 )edit
Thanks, i posted an answser.
( 2016-07-04 09:20:09 -0600 )edit
Sort by เธขเธ oldest newest most voted
What is missing from the OrderedSetPartitions are the positions of the additional empty sets. Chosing a subset of size i in a list of size k is done via Combinations. So, here is a possibility mixing those two tools:
def OrderedSetPartitions_0(n,k):
my_list = []
for i in range(1,k+1):
for empty_spots in Combinations(k,k-i):
for part in OrderedSetPartitions(range(n),i):
L = list(part)
LL = []
for j in range(k):
if j in empty_spots:
LL.append(set())
else:
LL.append(L.pop())
my_list.append(LL)
return my_list
For example you get as expected:
sage: OrderedSetPartitions_0(3,3)
[[set(), set(), {0, 1, 2}],
[set(), {0, 1, 2}, set()],
[{0, 1, 2}, set(), set()],
[set(), {2}, {0, 1}],
[set(), {1}, {0, 2}],
[set(), {0}, {1, 2}],
[set(), {1, 2}, {0}],
[set(), {0, 2}, {1}],
[set(), {0, 1}, {2}],
[{2}, set(), {0, 1}],
[{1}, set(), {0, 2}],
[{0}, set(), {1, 2}],
[{1, 2}, set(), {0}],
[{0, 2}, set(), {1}],
[{0, 1}, set(), {2}],
[{2}, {0, 1}, set()],
[{1}, {0, 2}, set()],
[{0}, {1, 2}, set()],
[{1, 2}, {0}, set()],
[{0, 2}, {1}, set()],
[{0, 1}, {2}, set()],
[{2}, {1}, {0}],
[{1}, {2}, {0}],
[{2}, {0}, {1}],
[{1}, {0}, {2}],
[{0}, {2}, {1}],
[{0}, {1}, {2}]]
more
is set() different from {} ?
your solution works but i still wonder why i have this bug. I had all the elements right but when aggregating into a list with res.append() or the + operator i had strange side-effects...
( 2016-07-04 09:54:08 -0600 )edit
This is not a bug, in Python set() denotes the empty set, while {} denotes the empty dictionary:
sage: type({})
<type 'dict'>
sage: type(set())
<type 'set'>
What kind of side effect do you have ?
( 2016-07-04 10:02:55 -0600 )edit
thanks for this fact that i didn't know. i edited the question to reflect better the remaining problem that I have. Thanks
( 2016-07-04 10:38:30 -0600 )edit
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https://socratic.org/questions/how-do-you-simplify-the-expression-sin-arctan-x-arccos-x#292024
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# How do you simplify the expression Sin(arctan(x)+arccos(x))?
Jul 26, 2016
sin 2x
#### Explanation:
arctan x --> x
arccos x --> x
sin (arctan x + arccos x) = sin (x + x) = sin 2x
Jul 26, 2016
.$= \left(\frac{1}{\sqrt{1 + {x}^{2}}}\right) \left(\sqrt{1 - {x}^{2}} \pm {x}^{2}\right)$, x in [-1, 1].
The negative sign is used, when $x \in \left[- 1 , 0\right]$.
#### Explanation:
Let a = arc tan (x). The principal value of $a \in \left[- \frac{\pi}{2} , \frac{\pi}{2}\right]$
Then x = tan a. sin a = $\pm \frac{x}{\sqrt{1 + {x}^{2}}}$ and cos a = $\frac{1}{\sqrt{1 + {x}^{2}}}$.
Let b = arc cos x. The principal value of $b \in \left[0 , \pi\right]$
Then, x = cos b and sin b = $\sqrt{1 - {x}^{2}}$. Also, $x \in \left[- 1 , 1\right]$.
The given expression =
$\sin \left(a + b\right)$
$= \sin a \cos b + \cos a \sin b$
$= \left(\pm \frac{x}{\sqrt{1 + {x}^{2}}}\right) \left(x\right) + \left(\frac{1}{\sqrt{1 + {x}^{2}}}\right) \sqrt{1 - {x}^{2}}$
$= \left(\frac{1}{\sqrt{1 + {x}^{2}}}\right) \left(\sqrt{1 - {x}^{2}} \pm {x}^{2}\right)$, x in [-1, 1].
The negative sign is used when $x \in \left[- 1 , 0\right]$.
| 488
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|
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| 4.65625
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CC-MAIN-2024-38
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https://jutge.org/problems/P66413_en
| 1,718,721,896,000,000,000
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# Recursive traversal of a general tree P66413
Statement
thehtml
Write a program that reads the description of a general tree of natural numbers and prints its postorder traversal.
In this exercise as well as in the rest of exercises of this section, unless the contrary is said the description of a general tree consists of the number of nodes n followed by the pre-order traversal, in which the value of each node is followed by its number of children. This traversal has 2n elements.
(To see an instance with the tree corresponding to the input-output instance, consult the pdf or ps version of this wording.)
To solve this exercixe as well as most of the exercises of this section, you will need to store the tree in a vector. Do it using this code (slightly modified if it is necessary):
typedef vector<int> VE; struct Node { int value; VE children; }; // Reads a tree and stores a part of the vector v starting at the position j. // Modifies the variable j in order to indicates the following free position of v. // Returns the position in c of the root of the read (sub)tree. int tree(int& j, vector<Node>& v) { int a = j; ++j; int f; cin >> v[a].value >> f; v[a].children = VE(f); for (int i = 0; i < f; ++i) v[a].children[i] = tree(j, v); return a; } ... int main() { int n; cin >> n; vector<Node> v(n); int j = 0; tree(j, v); ... }
Each position of the vector stores the value of a node, and the vector with the positions of all its children from the left to the right. The position of the tree root is always 0.
Input
Input consists of the description of a general tree of natural numbers.
Output
Your program must print a line with the postorder traversal of the general tree. Each element must be preceded by a space.
Public test cases
โข Input
```12
7 3 8 0 4 2 3 1 0 1 6 0
5 0 2 4 9 0 1 0 8 0 5 0
```
Output
``` 8 6 0 3 5 4 9 1 8 5 2 7
```
โข Information
Author
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https://www.mainmoonwarren.com/beef/what-does-4-oz-of-beef-cubes-look-like.html
| 1,656,621,309,000,000,000
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What Does 4 Oz Of Beef Cubes Look Like?
In terms of both volume and weight, a deck of playing cards has approximately the same dimensions as a four-ounce slice of raw, lean, or medium-cooked beef.
What does 3 oz of meat look like?
A chunk of 3 ounces is about the same size as a standard deck of playing cards. The size of three dice is about equivalent to one ounce of cooked beef. One ounce of beef is equal to approximately one inch of meatball. After cooking, a four-ounce serving of lean, raw beef is about equivalent to a three-ounce serving. The weight of a checkbook is equivalent to three ounces of grilled fish.
What does 4 oz of meat mean?
As a point of clarification, the majority of measurements of meat and other goods are taken uncooked. Therefore, although the serving size on the box reads 4 ounces, it refers to raw, uncooked meat; once the meat is cooked, it will appear to be far less than that. One of these little Dixie cups holds exactly 4 ounces of liquid.
How do you measure 4 oz of ground beef?
Cups used for measuring liquid ingredients in baking are called measuring cups. The weight of the ground beef is intended to be 4 ounces. Iโm willing to wager that four ounces of meat is far less than a half cup of ground beef.
You might be interested:ย What Dish Is Served With Raw Ground Beef And Chopped Onion?
What is the size of 1 oz cooked meat?
The weight of a checkbook is equivalent to three ounces of grilled fish. 1 ounce of cooked beef is equivalent to around the size of a tennis ball when it comes to fruits and vegetables. A medium apple, peach, or orange is about the same size. 3 dice
How do you visualize 4 oz of meat?
A portion that is three ounces is about the same size as a standard deck of playing cards, while one ounce of cooked meat is about the same size as three dice.About one ounce of meat may be included in a meatball that is one inch in diameter.After cooking, a four-ounce serving of lean, raw beef will equal approximately three ounces.The weight of a checkbook is equivalent to three ounces of grilled fish.
How many cups is 4 oz of ground beef?
The last time I looked it up on Google, it indicated that one cup is equal to four ounces of ground beef. The unit of measurement for ounces of liquid is the cup (oz as in volume). You will need 4 ounces of hamburger that has been cooked (oz as in weight). 1/4 of a cup is equivalent to 1 ounce of cooked and drained hamburger that is lean.
How can I measure 4 ounces of ground beef without a scale?
IF YOU DONโT HAVE A SCALE, YOU CAN JUST WEIGH THE FOOD WITH YOUR HANDS Measuring Meat and Fish โ Approximately three ounces of protein, such as chicken, fish, or cattle, may fit into the palm of your hand. A word to the wise: the typical portion size for an adult consists of three ounces of meat.
How much is 3 oz of meat visually?
Both Fish and Meat Three ounces constitutes one serving of any kind of meat or fish, regardless of the kind. Instead of weighing this out, the best visual representation of this amount is about the size of the palm of your hand or the size of a regular deck of playing cards. This may be done by comparing the dimensions of the two.
You might be interested:ย FAQ: Why Can Beef Be Eaten Raw?
How do you visualize meat portions?
How to conceptualize the various serving amounts of meat
1. A chunk of beef weighing 3 ounces is around the same size as a deck of cards
2. The size of three dice is about equivalent to one ounce of cooked meat
3. A meatball that is one inch in diameter weighs around one ounce
4. About three ounces of cooked lean meat is equivalent to four ounces of raw meat
5. A checkbook is equivalent in size to 3 ounces of grilled fish.
How can I measure ground beef without a scale?
Simply using the palm of your hand as a measuring tool for grinding beef is one of the fastest, most straightforward, and hassle-free ways available.There are a lot of recipes that require two portions of ground beef.This is comparable to a total of 6 ounces, with each dish containing 3 ounces.One serving of ground beef is equal to approximately 3 ounces, which can be held in an adultโs palm at one time.
What is 4 oz of raw meat cooked?
As can be seen, the number of calories in 4 ounces raw is same to the number of calories in 3 ounces cooked.
How many oz is a serving of ground beef?
Beef is a good source of protein, as well as vitamins and minerals. The standard portion size for beef and other types of meat is three ounces, which is about equivalent to the size of a standard deck of playing cards. Approximately 180 calories, 10 grams of fat, and 15 percent of the daily requirement for iron may be found in a serving size of three ounces of lean ground beef.
You might be interested:ย Readers ask: What Percent Ground Beef For Lasagna?
Does 2 tablespoons equal 1 oz?
Because there are two tablespoons in a single fluid ounce, we utilize this amount in the calculation that we just presented above. Volume may be measured using a variety of different units, including fluid ounces and teaspoons.
How do you measure ounces of meat?
Due to the fact that peopleโs hand sizes differ, you should evaluate your fist size using a real measuring cup. Include two servings, or six ounces, of lean meat (poultry, fish, shellfish, or beef) in your daily diet. This is the recommended amount. Use your hand to get an accurate reading of the quantity. One serving is equivalent to three ounces, or one palm-sized piece.
How can I measure an ounce without a measuring cup?
Employ the use of a kitchen scale. One ounce of weight is equal to one fluid ounce, hence one fluid ounce of water is equal to one ounce. Because one cup is equal to eight ounces, the weight of one cup of water, or any other liquid having a density nearly equivalent to water, will approximately equal eight ounces of fluid.
What size is 4 oz?
It takes 0.375 cups to equal 3 US ounces. It takes 0.5 cups to equal 4 ounces in the US. It takes 0.625 cups to equal 5 US ounces.
How many tablespoons is 4 ounces?
8 Tbsp = 4 Fluid Oz. 10 Tbsp = 5 Fluid Oz. 12 Tbsp = 6 Fluid Oz. 16 Tbsp = 8 Fluid Oz.
What dies 3 oz of ground beef look like?
One ounce of cooked beef is about equivalent in size to three dice, while a three-ounce piece is about the same size as a standard deck of playing cards. A meatball that is one inch in diameter and weighs around one ounce. When cooked, four ounces of raw, lean beef is about equivalent to three ounces.
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# 2. Pseudo-Mathematics
## โ Numerologies โ
[Continued from here.]
However, we were not basically looking for the meaning of numerator per se, we were looking for recurrence, and our quick side-glance towards the blind blank and blackenโd speculum-punctum of n, โN.N.โ et consortes provides an example of โrecursive discourseโ, which is the course set out for our curriculum in this excursion.
MetaphorโโโMirror of Meaning
What n and N.N. are in the context of finding a formula for recursion could fall under the general headline of metaphor. Etymologically metaphor comes from words giving the idea of ferrying across (transfer/phore, across/meta). A ferry is a bridge over troubled water, the metaphor carries meaning from one shore to another, contains the meaning as a vessel contains a passenger. It is a Scylla and Charybdis-challenge to guide such a ship over the open ocean of understandings and ununderstandings.
No Nobody
To succeed, the captain of metaphor must navigate successfully between presenting too opaque a metaphor which might crash the vessel of meaning against an impenetrable mass of words, but just as much the navigator needs to abstain from getting carried away on overly wordy waves of inspiration, no matter how clear the waters. Navigare necesse est, there is no escaping that. Too long time upon the sea and the crew will finish all apples and oranges before the final port, get scurvy, die off and leave the captain to navigate all on his own even should he be lucky to survive, meaning time is of essence or the recipient the meaning is meant for will lose their patience, no patient Penelope awaiting Odysseus when he returns home, the reader having already let their glance glide off the page, the listener already lost the red thread that should have guided them through the Labyrinth. A metaphor must carry meaning, must not let it sink like a rock into a whirlpool. If it does, the captain is a no-good nobody, the captain being the meaning that must steer firmly the ship. The medium is the message, the metaphor the meaning.
*
Metaphor clarifies something unknown by what it is not, but which we know. It basically says that the unknown, X, which we want to explain, this unknown X is pretty much like Y. By equating the unknown with something known, we intimate the unknown. Itโs like asking Numerius to present us to their attractive younger sibling Nomen, whom we have not been introduced to but would very much like to get to know.
X or Y orโฆ?
The metaphor is incomplete, though, as they all are, by definition, as every metaphor explains one thing by another, meaning these two are unidentical. It is like building a bridge from nowhere to nowhere else. You have a ferry going from Scylla to Charybdis? No ferryman would get passengers for such a ferryboat.
This mirror X-sample of metaphor is an extreme case of incomplete metaphor, as we explain one unknown, โXโ, by another unknown, โYโ. Even the confused souls who might want to metaphorize the seas with such a ferryman could find no way to get onboard.
*
You just cannot ask Numerius to introduce you to attractive Nomen if you do not even know or recognize Numerius. You also do not know if Nomen would reject or accept no men or women, which though un-universal is a consideration we know is quite often brought into the equation in these pretty please present me to your pretty friend/sibling-schemes. A bridge from Scylla to Charybdis cannot be built.
And attempting to explain X by Y, we discover X may be XX as well as XY.
Universal and Un-Universal Unknown Unknowns
In our previous mirror metaphor, we found out that we, ourselves, have a line of sight intersecting with the line of sight of โviewerโ, and that neither we (w), nor Viewer (v), can see the point of intersecting lines. Which means that we do not see Numerius, we do not see Nomen, and we do not even see ourselves or our imago, being as hidden away as they are. Any and all siblings of our Nomen Nescio and our Numerius Negidius are now accompanied by an A.N. Other, yet another unknown N-name. Now we only need to find Mr. X, and we can set up a barbershop quartet.
As noted previously, any Mr. X may turn out to in reality be Ms. X, and for that matter in a situation where we do not know even who we are ourselves, we know nothing about any participant. We are all X in this together in this particularly peculiar barbershop quartet, meaning some X may be Y.
However, besides changing what we call the numerator and metaphorizing about mirrors, there does not seem to be much we can do about n, so let us leave it aside for now.
Let us move on and look at what is below the vinculum, in denominator land.
Below the Vinculum
First term
Under the vinculum bar first item is t divided by t. Originally one, in the first form given above, for the purpose of having one term only, we removed the numerical one and replaced it with t divided by t, which expresses the same but in a non-numerical manner. The point of doing so is that we now have a thoroughly self-similar and self-referencing expression. As this particular t describes division, we shall call it d. As we might as well have chosen to translate 1 into t times t it could also be represented by, say, m (multiplication).
However, we must choose, it can be represented by either, but logically not both at the same time. To divide is not to multiply and to multiply is not to divide. The first term thus is an illustration of the logical expression EITHER/OR. This, in turn, means that the specific numerical value of 1 is an expression of a particular logical operation. In everyday terms, EITHER/OR means choice. Choice is inevitable where one is.
However, as 1 could express either d or m it also means that it is an exception. A particular and strange case of its own rule, itโโโ1 that isโโโis the value which does allow equally for either variation of the identity to come forth. All numbers multiplied with one become themselves. All numbers self-divided by themselves become one too. The value 1 thus contains not only EITHER/OR but also the logical idea of AND. If one was a verb, it would mean you could both multiply and divide at the same time; โmultivideโ.
Consequently, announcing 1 as fundamental among integers, every numerical value being both a multiplication of one with itself and a division by one, every integer differing from every other by one, by definition if nothing else, we conclude that every integer may be seen as its own reciprocal as well.
The numerological series continues here.
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# How to Use Decimals in Real Life
by ยท
As a middle school math teacher, I always love showing my students how the math they learn relates to real life. It's an effective way to capture their attention, and it helps them retain the information they're learning.
When we were working on decimals in class, I wanted my students to understand that decimals pop up every dayโand that's true! Here are a few situations where your students might find decimal math in daily life.
## Want free resources?
Get my free resource library with digital & print activitiesโplus tips over email.
## Example #1: Managing Hotel Rooms
Let your students imagine they're the manager of a large hotel and need to keep track of occupancy, revenue and profit.
### The Math
Here's how to use decimals in this situation:
โข Occupancy Rate. First, calculate your hotel's occupancy rate by dividing the number of occupied rooms by the total number of rooms. For example, if your hotel has 100 total rooms and 30 are occupied, 30 รท 100 = 0.3 x 100 = 30% occupancy rate. Every day at noon, the front desk clerk will give you a report on these numbers. Using these numbers, you can make decisions on pricing and availability that will help to maximize occupancy while maintaining an optimal price point for customers.
โข Daily Room Revenue. Then calculate your daily room revenue by multiplying the average room rate by the number of occupied rooms. Average room rate is found by dividing your total daily room sales (the price you charge guests) by your total number of available rooms (100). For example: \$3,000 รท 100 = \$30 per night average rate x 30 occupied rooms = \$900 daily room revenue.
### Making The Math Engaging
Now, the math discussed above might feel a little dry. I've developed a kid friendly activity inspired by this called Hotel Math.
It has 3 scenarios, covering percents, decimals and fractions:
โข Percents. Apply discounts and understand revenue from booking rooms. Students will be asked to find things like 25% of 50, etc.
โข Decimals. Understand potentia
l revenue from different restaurants that want to rent space in the hotel. Select the best applications. Students will be asked to do basic operations with decimals (e.g. multiplication, addition).
โข Fractions. Understand employee
satisfaction survey results. Students will be asked to solve for things like 2/5 of 50, etc.
Here's what the Percents activity looks like:
The best part? In addition to no-prep PDF and TpT Easel, there's an auto-grading Google Form version of the activity. You can assign it, and students get instant feedback on how they're doing:
Give it a try in your classroom:
#### Fractions, Decimals, Percents Real-Life Math Project
\$3.99
Looking for a real world application of percents, decimals, and fractions? Do your students love simulation games? This activity has students play the role of a hotel manager, and use their math skills to book hotel rooms, lease restaurant spaces and improve employee satisfaction. Three formats are available in PDF, Google Form, EASEL.
## Example #2: Calculating Your Wedding Budget
Weddings are expensive. Even at the most affordable venues, youโre looking at a price tag of about \$5,000 for a wedding of 100 people. For this reason, you need to be able to calculate your budget and stick to it!
### The Math
Your wedding budget is the total amount of money that you have available to spend on your wedding. To calculate your budget, add together all of the money that you and your partner have available to spend on the wedding. This might include your own savings or contributions from family members if theyโre contributing towards your big day.
If you stay within your budget then you can feel confident that there wonโt be any unexpected costs for which you donโt have enough money. If you donโt stay within your budget, then there may be an option for borrowing extra money through an event planner or venue planner if necessary.
### Making the Math Engaging
I've developed a kid-friendly activity called Wedding Math that puts students in the shoes of an event planner, where they have to guide a couple through deciding on aspects of a wedding, and staying in budget:
โข Task 3: The Photography & Videography
โข Task 4: The Wedding Cake
Here's a preview of what Task 4 looks like:
Similar to Hotel Math, Wedding Math is available as a print-and-go PDF as well as on TpT's Easel. It's not yet available in a Google Form. Give it a try:
#### Decimal Operations Real-Life Math Project | Wedding Math
\$3.99
Mastering decimal operations just got a lot more engaging with our Wedding Math project! Designed for fifth and sixth graders, this real-life math project tackles budgeting through the lens of wedding costs, making it a perfect way to integrate decimal operations with relatable scenarios. Students play the role of a wedding coordinator, tasked with using decimals in a practical setting. Aligned with CCSS standards 5.NBT.B.7 and 6.NS.B.3, it features word problems that will make decimal operations fun and memorable for your students.
## Example #3: Calculate Restaurant Profit
Calculating profit is an important skill for business owners, especially restaurant owners.
### The Math
If you own a restaurant, you need to know how much of your revenue is profit.
To do that, you need to calculate your revenue and expenses:
โข Revenue = Number of orders x average order price
โข Expenses = Salary + food costs + rent + utilities + other expenses
Profit = Revenue - Expenses
### Making the Math Engaging
I haven't yet had a chance to develop a dedicated activity for this. But if you love freebies, and want to be the first to hear about future activities, subscribe to my newsletter.
## Hi, I'm Ping!
I spent 7 years in the classroom working to make math fun and relevant in middle school, by integrating math, art, and technology. I started Congruent Math to share this all with you.
## Want more ideas and freebies?
Get my free resource library with digital & print activitiesโplus tips over email.
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# n repeating digit of decimal number
โข Dec 22nd 2011, 04:43 AM
Bokas
n repeating digit of decimal number
I have a problem how to convert number 0.3333....3 where 3 repeats N times to fraction.
for N=1 0.3
N=2 0.33 and so on.
I know how to convert number where digits repeats forever, but how can I convert this to fraction to get correct result for any N.
โข Dec 22nd 2011, 04:47 AM
Prove It
Re: n repeating digit of decimal number
Quote:
Originally Posted by Bokas
I have a problem how to convert number 0.3333....3 where 3 repeats N times to fraction.
for N=1 0.3
N=2 0.33 and so on.
I know how to convert number where digits repeats forever, but how can I convert this to fraction to get correct result for any N.
\displaystyle \begin{align*} x &= 0.\dot{3} \\ 10x &= 3.\dot{3} \\ 10x - x &= 3.\dot{3} - 0.\dot{3} \\ 9x &= 3 \\ x &= \frac{3}{9} \\ x &= \frac{1}{3} \\ 0.\dot{3} &= \frac{1}{3} \end{align*}
โข Dec 22nd 2011, 06:24 AM
HallsofIvy
Re: n repeating digit of decimal number
Quote:
Originally Posted by Bokas
I have a problem how to convert number 0.3333....3 where 3 repeats N times to fraction.
for N=1 0.3
N=2 0.33 and so on.
I know how to convert number where digits repeats forever, but how can I convert this to fraction to get correct result for any N.
A terminating decimal is easier than a repeating decimal (which is what ProveIt did). If x= 0.33333..3 where the "3" repeats N times, then $10^Nx= 3333...3$ so that $x= \frac{3333...3}{10^N}$. Because of that " $10^N$" in the denominator, a terminating decimal, converted to a fraction, has only powers of 2 and 5 in its denominator.
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# A mystifying grid
Puzzle Name: TAKE TWOs
Text Only:
L R D N S D Q
D B L B T E U
N T F Y P K U
A L M C D O N
Z C F D B S I
E I H N T U F
L O R G C E L
K H I E A K U
T S L O U N L
Solve this puzzle, and you'll discover a location that could be associated with the phrase "Take Two".
Hint 1:
There's a reason why the puzzle's name ISN'T 'Take Two'.
Hint 2:
The relative positions of the letters is a red herring (i.e. This puzzle is still solvable even if the entire grid is completely scrambled).
โข Most certainly knowledge tag is applicable for this. Nov 21 '17 at 12:54
โข I don't agree. To my knowledge, you can solve this without googling, and I have tested it out with people who were able to solve it without external knowledge. The final sentence is more of a teaser to the solution - you can safely ignore it and still figure the answer. Nov 21 '17 at 12:57
โข I can't help but to see "BL", "BS" and think of BioShock and Borderlands from Take-Two. Seems to require knowledge though, and I can't seem to make anything else work with it.
โย Tas
Nov 22 '17 at 1:01
After taking the letters appear an even number of times we end up with
AEEHRTT (we count the letters appearing 4 times as 2 * 2 times)
which gives the word
THEATER
that we can associate with the phrase "Take two" which give
โข you got it, although the location isn't specific. The association is with the place being somewhere where you can watch movies, and movie directors yelling that phrase when shooting movies. As mentioned, the association is meant to be loose, and not needed to solve the puzzle. Nov 22 '17 at 10:25
Seems like
answer is FILM STUDIO (as we can see that pairs of letters are available in the grid as FI, LM, ST, UD, IO) and it is that place where generally for a re-take that word - Take two is used.
โข I can see where you're coming from, but that is not the solution I have in mind. Nov 21 '17 at 13:19
It can be
EARTH / THRAE (taken letters in pairs i.e. occur 2 or 4 times in grid)
OR
E.T. (Extra Terrestrial) taken letters those occur four times
Take Two:
may be only two letters
โข you are on the right track, but are you sure you took the right letters? Nov 22 '17 at 4:44
โข ok edited but not sure yet Nov 22 '17 at 5:48
โข I think you could get a better word if you took the letters that occur four times twice. Nov 22 '17 at 6:12
โข @M Oehm i also think about it that it could be extra-terrestrial but no idea about take two. I have no knowledge:( Nov 22 '17 at 6:19
โข M Oehm's got the gist! Just follow his thought process and you'll probably get the answer. You don't need external knowledge. Nov 22 '17 at 8:38
After counting the repeated letters:
L, U, N, D where the most repeated
The place is
LUND
It is a city in the province of Scania, southern Sweden.
Another place might be:
Dunet
Dunet is a commune in the Indre department in central France
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## How do you calculate CP of nitrogen?
Isobaric specific heat (Cp) is used for substances in a constant pressure (ฮP = 0) systemโฆ.Specific heat of Nitrogen Gas โ N2 โ at temperatures ranging 175 โ 6000 K.
Nitrogen Gas โ N2
Temperature โ T โ (K) Specific Heat โ cp โ (kJ/(kg K))
275 1.039
300 1.040
325 1.040
What is r of N2?
0.2968
The Individual Gas Constant โ R
Gas Individual Gas Constant โ R
Name Formula [kJ/kg K]
Neon Ne 0.4120
Nitrogen N2 0.2968
Nitrogen dioxide NO2 0.1807
What is CP to CV ratio?
The Cp/Cv ratio is also called the heat capacity ratio. In thermodynamics, the heat capacity ratio is known as the adiabatic index. (i.e.) Heat Capacity ratio = Cp/Cv = Heat capacity at constant pressure/ Heat capacity at constant volume.
### What is CP and CV formula?
Cp= Cv + R. Cp-Cv= R. The above equation shows the correlation between the molar heat capacity at constant pressure and molar heat capacity at constant volume. Thus, the difference between the two parameters Cp and Cv for a substance is equal to the universal gas constant.
How do you calculate CP of a gas?
cp = cv + R The specific heat constants for constant pressure and constant volume processes are related to the gas constant for a given gas.
What is CP of a gas?
Specific heat constant pressure: Cp. Cp is the amount of heat energy released or absorbed by the unit mass of the substance with the change in temperature at constant pressure. In another word, it is the heat energy transfer at a constant pressure between system and surroundings. Specific heat constant volume: Cv.
## What is CP for co2?
37.35 J/mol K
Chemical, physical and thermal properties of carbon dioxide. Phase diagram included.
Property Value Unit
Specific heat capacity, Cp (isobaric) 37.35 J/mol K
Specific heat capacity, Cv (isochoric) 28.96 J/mol K
Ionization potential 13.77 eV
Molecular Weight 44.0095 g/mol
Is CP a CV nR?
Cp-Cv =R is used when we are asked to solve numericals which are based on calculating the values on the basis of 1 mole of a given gas. But if more than 1 mole of a gas is taken, then we use Cp-Cv=nR.
How do you calculate CP CV for gas mixture?
Also you can use, (Cp/Cv) โ 1 = (R/Cv). Therefore, (Cp/Cv) = 1 + (R/Cv) = 1 + (6/11) = 17/11.
### How do I find my CV and CP?
How do you find CP on CV?
What is the name of the compound with the formula n2?
Nitrogen, also known as N2 or N#N, belongs to the class of inorganic compounds known as other non-metal nitrides.These are inorganic compounds of nitrogen where nitrogen has a formal oxidation state of -3, and the heaviest atom bonded to it belongs to the class of โother non-metalsโ.
## What is the meaning of CP/CV in chemistry?
Cp/Cv is an indicator of how much gas in adiabatic conditions with dQ=0 can extract heat internally to do work. Cp/Cv is an indicator of how much gas in adiabatic conditions with dQ=0 can extract heat internally to do work.
What does QP/CV mean?
QP = Cp โT = โU + W = โU + P โV. Cp/Cv is an indicator of how much gas in adiabatic conditions with dQ=0 can extract heat internally to do work. Cp/Cv is an indicator of how much gas in adiabatic conditions with dQ=0 can extract heat internally to do work. This can be further explained like this.
What is the CP/CV ratio for monoatomic and diatomic molecules?
Cp/Cv ratio for monoatomic, diatomic, triatomic is 1.67,1.4,1.33 respectively Therefore, Cp-Cv only shows that Cp exceeds Cv by an amount equivalent to R. But if individually Cp and Cv are probed lots of information emerges.
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Vectors - Motion and Forces in Two Dimensions - Lesson 3 - Forces in Two Dimensions
# Equilibrium and Statics
When all the forces that act upon an object are balanced, then the object is said to be in a state of equilibrium. The forces are considered to be balanced if the rightward forces are balanced by the leftward forces and the upward forces are balanced by the downward forces. This however does not necessarily mean that all the forces are equal to each other. Consider the two objects pictured in the force diagram shown below. Note that the two objects are at equilibrium because the forces that act upon them are balanced; however, the individual forces are not equal to each other. The 50 N force is not equal to the 30 N force.
If an object is at equilibrium, then the forces are balanced. Balanced is the key word that is used to describe equilibrium situations. Thus, the net force is zero and the acceleration is 0 m/s/s. Objects at equilibrium must have an acceleration of 0 m/s/s. This extends from Newton's first law of motion. But having an acceleration of 0 m/s/s does not mean the object is at rest. An object at equilibrium is either ...
โข at rest and staying at rest, or
โข in motion and continuing in motion with the same speed and direction.
This too extends from Newton's first law of motion.
### Analyzing a Static Equilibrium Situation
If an object is at rest and is in a state of equilibrium, then we would say that the object is at "static equilibrium." "Static" means stationary or at rest. A common physics lab is to hang an object by two or more strings and to measure the forces that are exerted at angles upon the object to support its weight. The state of the object is analyzed in terms of the forces acting upon the object. The object is a point on a string upon which three forces were acting. See diagram at right. If the object is at equilibrium, then the net force acting upon the object should be 0 Newton. Thus, if all the forces are added together as vectors, then the resultant force (the vector sum) should be 0 Newton. (Recall that the net force is "the vector sum of all the forces" or the resultant of adding all the individual forces head-to-tail.) Thus, an accurately drawn vector addition diagram can be constructed to determine the resultant. Sample data for such a lab are shown below.
Force A Force B Force C Magnitude 3.4 N 9.2 N 9.8 N Direction 161 deg. 70 deg. 270 deg
For most students, the resultant was 0 Newton (or at least very close to 0 N). This is what we expected - since the object was at equilibrium, the net force (vector sum of all the forces) should be 0 N.
Another way of determining the net force (vector sum of all the forces) involves using the trigonometric functions to resolve each force into its horizontal and vertical components. Once the components are known, they can be compared to see if the vertical forces are balanced and if the horizontal forces are balanced. The diagram below shows vectors A, B, and C and their respective components. For vectors A and B, the vertical components can be determined using the sine of the angle and the horizontal components can be analyzed using the cosine of the angle. The magnitude and direction of each component for the sample data are shown in the table below the diagram.
The data in the table above show that the forces nearly balance. An analysis of the horizontal components shows that the leftward component of A nearly balances the rightward component of B. An analysis of the vertical components show that the sum of the upward components of A + B nearly balance the downward component of C. The vector sum of all the forces is (nearly) equal to 0 Newton. But what about the 0.1 N difference between rightward and leftward forces and the 0.2 N difference between the upward and downward forces? Why do the components of force only nearly balance? The sample data used in this analysis are the result of measured data from an actual experimental setup. The difference between the actual results and the expected results is due to the error incurred when measuring force A and force B. We would have to conclude that this low margin of experimental error reflects an experiment with excellent results. We could say it's "close enough for government work."
### Analyzing a Hanging Sign
The above analysis of the forces acting upon an object in equilibrium is commonly used to analyze situations involving objects at static equilibrium. The most common application involves the analysis of the forces acting upon a sign that is at rest. For example, consider the picture at the right that hangs on a wall. The picture is in a state of equilibrium, and thus all the forces acting upon the picture must be balanced. That is, all horizontal components must add to 0 Newton and all vertical components must add to 0 Newton. The leftward pull of cable A must balance the rightward pull of cable B and the sum of the upward pull of cable A and cable B must balance the weight of the sign.
Suppose the tension in both of the cables is measured to be 50 N and that the angle that each cable makes with the horizontal is known to be 30 degrees. What is the weight of the sign? This question can be answered by conducting a force analysis using trigonometric functions. The weight of the sign is equal to the sum of the upward components of the tension in the two cables. Thus, a trigonometric function can be used to determine this vertical component. A diagram and accompanying work is shown below.
Since each cable pulls upwards with a force of 25 N, the total upward pull of the sign is 50 N. Therefore, the force of gravity (also known as weight) is 50 N, down. The sign weighs 50 N.
In the above problem, the tension in the cable and the angle that the cable makes with the horizontal are used to determine the weight of the sign. The idea is that the tension, the angle, and the weight are related. If the any two of these three are known, then the third quantity can be determined using trigonometric functions.
As another example that illustrates this idea, consider the symmetrical hanging of a sign as shown at the right. If the sign is known to have a mass of 5 kg and if the angle between the two cables is 100 degrees, then the tension in the cable can be determined. Assuming that the sign is at equilibrium (a good assumption if it is remaining at rest), the two cables must supply enough upward force to balance the downward force of gravity. The force of gravity (also known as weight) is 49 N (Fgrav = m*g), so each of the two cables must pull upwards with 24.5 N of force. Since the angle between the cables is 100 degrees, then each cable must make a 50-degree angle with the vertical and a 40-degree angle with the horizontal. A sketch of this situation (see diagram below) reveals that the tension in the cable can be found using the sine function. The triangle below illustrates these relationships.
### Thinking Conceptually
There is an important principle that emanates from some of the trigonometric calculations performed above. The principle is that as the angle with the horizontal increases, the amount of tensional force required to hold the sign at equilibrium decreases. To illustrate this, consider a 10-Newton picture held by three different wire orientations as shown in the diagrams below. In each case, two wires are used to support the picture; each wire must support one-half of the sign's weight (5 N). The angle that the wires make with the horizontal is varied from 60 degrees to 15 degrees. Use this information and the diagram below to determine the tension in the wire for each orientation. When finished, click the button to view the answers.
In conclusion, equilibrium is the state of an object in which all the forces acting upon it are balanced. In such cases, the net force is 0 Newton. Knowing the forces acting upon an object, trigonometric functions can be utilized to determine the horizontal and vertical components of each force. If at equilibrium, then all the vertical components must balance and all the horizontal components must balance.
### We Would Like to Suggest ...
Sometimes it isn't enough to just read about it. You have to interact with it! And that's exactly what you do when you use one of The Physics Classroom's Interactives. We would like to suggest that you combine the reading of this page with the use of ourย Name That Vector Interactive, ourย Vector Addition Interactive, or ourย Vector Guessing Game Interactive. All three Interactives can be found in the Physics Interactive section of our website and provide an interactive experience with the skill of adding vectors.
The following questions are meant to test your understanding of equilibrium situations. Click the button to view the answers to these questions.
1. The following picture is hanging on a wall. Use trigonometric functions to determine the weight of the picture.
2. The sign below hangs outside the physics classroom, advertising the most important truth to be found inside. The sign is supported by a diagonal cable and a rigid horizontal bar. If the sign has a mass of 50 kg, then determine the tension in the diagonal cable that supports its weight.
3. The following sign can be found in Glenview. The sign has a mass of 50 kg. Determine the tension in the cables.
4. After its most recent delivery, the infamous stork announces the good news. If the sign has a mass of 10 kg, then what is the tensional force in each cable? Use trigonometric functions and a sketch to assist in the solution.
5. Suppose that a student pulls with two large forces (F1 and F2) in order to lift a 1-kg book by two cables. If the cables make a 1-degree angle with the horizontal, then what is the tension in the cable?
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# Thread: exansion of mathematical terms used by Erwin Kreyszig in Differential Geometry text
1. ## exansion of mathematical terms used by Erwin Kreyszig in Differential Geometry text
On pages 128,129 in the Principal Curvature section of his text Kreyszig states the series of mathematical steps copied in the graphic attached. Beginning with and beyond the differentiation step I cannot follow his mathematics. For example, where does the minus sign come from between the parentheses? Each step condenses many intermediate steps, particularly because of the indices. His expression incorporating the kronecker delta is particularly puzzling. Anyone who can expand -or deconstruct one might say- the mathematical steps would be greatly appreciated.
2. ## Re: exansion of mathematical terms used by Erwin Kreyszig in Differential Geometry te
Hey cyberbaffled.
Consider that when you differentiate something then everything that isn't relative to the independent variable is treated as a constant.
Lets say you have w = f(x0,x1,x2) and you only want to differentiate against x, then if you only want df/dx^0 you can use tensor notation to mask this out by using d_(m,0)*df/dx^m where m is a dummy variable and d_(m,0) is the kronecker delta function where d_(x,y) = 1 if x=y and 0 otherwise.
3. ## Re: exansion of mathematical terms used by Erwin Kreyszig in Differential Geometry te
Thanks for your quick reply, Chiro. I'm only familiar with the asterisk designating the star operator and the carot designating the wedge product. How are you using them in your expression? Also, I've worked three quarters of the way through Robert Wrede's text Vector and Tensor Analysis without encountering the procedures used by Kreyszig. Let me jump ahead of the curve a bit and ask if you know of a good text that thoroughly exercises the procedures I'm inquiring about and that you've explained in a summary fashion?
4. ## Re: exansion of mathematical terms used by Erwin Kreyszig in Differential Geometry te
Well as a quick example, we can have df/dx^m in Einstein Summation Form to be for m = 1 to 3 to be equal to df/dx^1 + df/dx^2 + df/dx^3 where dx^1 is not dx to the power 1 but instead a particular variable given by that index.
Similarly if we have say d_(m,1)*df/dx^m using Einstein summation, for m = 1 to 3 we get 1*df/dx^1 + 0*df/dx^2 + 0*df/dx^3 = df/dx^1 wher d_(x,y) is the kronecker delta function. If you were summing over two arbitrary indices for kronecker delta (like d_(i,j)) then d_(i,j) would be an identity matrix.
5. ## Re: exansion of mathematical terms used by Erwin Kreyszig in Differential Geometry te
Sorry I went away there for awhile. Got preoccupied with a presidential election. Tomorrow I'll focus on your example. Then reply again.
6. ## Re: exansion of mathematical terms used by Erwin Kreyszig in Differential Geometry te
Please find recent attachment regarding origin of minus sign.
7. ## Re: exansion of mathematical terms used by Erwin Kreyszig in Differential Geometry te
How do I describe the process of making a tessellation using mathematical terms?
Social media recruitment
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UnitConverter.netV1.2
Home / Speed Converter / Mph To Kph
# mph to kph
Enter the value that you want to convert mph to kph or kph to mph.
From: mile/hour To: kilometer / hour
Value:
### Description
#### Mph (mile/hour):
The MPH stands for miles per hour & It is measuring unit of speed in both the imperial and United States customary systems. It is denoted as mph or mi/h. One mph is equal to 1.609344 kilometers per hour (km/h).
#### Kph (kilometer/hour):
The KPH stands for kilometer per hour and it is measuring unit of speed. It is denoted as kph or km/h. One kph is equal to 0.6213711922 mi/h.
#### Miles per hour to kilometers per hour (mph to kph):
It is a free online Miles per hour to kilometers per hour (mph to kmh) speed converter. The MPH stands for miles per hour & It is measuring unit of speed in both the imperial and United States customary systems. It is denoted as mph or mi/h. One mph is equal to 1.609344 kilometers per hour (km/h). The KPH stands for kilometer per hour and it is measuring unit of speed. It is denoted as kph or km/h. One kph is equal to 0.6213711922 mi/h.
1 mph = 1.6093 kmh 2 mph = 3.218 kmh 3 mph = 4.828 kmh 4 mph = 6.4374 kmh 5 mph = 8.0467 kmh 6 mph = 9.6560 kmh 7 mph = 11.2654 kmh 8 mph = 12.8747 kmh 9 mph = 14.4840 kmh 10 mph = 16.0934 kmh
Mph to kph metric conversion table
11 mph = 17.702 kmh 30 mph = 48.2803 kmh
12 mph = 19.3121 kmh 40 mph = 64.3737 kmh
13 mph = 20.9214 kmh 50 mph = 80.4672 kmh
14 mph = 22.5308 kmh 60 mph = 96.560 kmh
15 mph = 24.140 kmh 70 mph = 112.65 kmh
16 mph = 25.7495 kmh 80 mph = 128.74 kmh
17 mph = 27.358 kmh 90 mph = 144.84 kmh
18 mph = 28.968 kmh 100 mph = 160.93 kmh
19 mph = 30.577 kmh 500 mph = 804.67 kmh
20 mph = 32.1869 kmh 1000 mph = 1609.34 kmh
### How to Convert Mile/hour to Kilometer/hour?
For converting mph to kph we go with an example.
Example:
Convert 70 mph to kph?
We know 1 mph = 1.609344 kph; 1 kph = 0.6213711922 mph.
70 mph = __kph
70x1.609344 = 112.65408 kph
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Info
# Filter of sin component of sound signal
1 view (last 30 days)
Even Wee on 18 Apr 2015
Closed: MATLAB Answer Bot on 20 Aug 2021
Hey
This is a project that i am working on, there we are suppose to filter out a sin-component from a small sound signal.
I have the signal "x" already, this "x" contains the sound signal og the energy-spectral-density.
And I have done all the calculations for the filter and signal, and ended up with this differential equation:
y = 2*a*cos(W_s)*y(n-1) - a.^2*y(n-2) - b0*x(n) - b0*2*cos(W_s)*x(n-1) + b0*x(n-2);
So my next step is to get this signal y(n) and check and se if the sin-component is gone. I know that I should be loop like this:
for n = 3:length(x)
y(n) = 2*a*cos(W_s)*y(n-1) - a.^2*y(n-2) - b0*x(n) - b0*2*cos(W_s)*x(n-1) + b0*x(n-2);
end
But i am getting this error: Attempted to access y(2); index out of bounds because numel(y)=1.
Youssef Khmou on 18 Apr 2015
Edited: Youssef Khmou on 18 Apr 2015
Concerning the equation of y, can you explain its origin. As for the error, initial condition must be set :
y=zeros(size(x));
y(1)=x(1);
y(2)=x(2);
% for loop
Even Wee on 18 Apr 2015
Sorry, i have the y(n). It is an array of data like x(n), that contains the sound signal with the sin-component that need to be filtered.
So the differential equation is the differential-equation of the filter. And it should be causal, therefore I need to begin on n = 3.
Even Wee on 18 Apr 2015
Edited: Even Wee on 18 Apr 2015
Okay.. now i got it to work, and I manage to also filter the sin-component. It was a sign error in my differential-equation, it should be like this
y = 2*a*cos(W_s)*y(n-1) - a.^2*y(n-2) + b0*x(n) - b0*2*cos(W_s)*x(n-1) + b0*x(n-2);
And i think "y" was not initialized when i got this error: "Attempted to access y(2); index out of bounds because numel(y)=1."
But now it works as it should :)
Thank you.
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# why use homogeneous coordinates?
This topic is 3936 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
hi, I'm positing this because I don't really understand the use of homogeneous coordinates in a 4x4 matrix. I understand that the extra dimension is just a dividing factor so...
[ 1 0 0 px ]
[ 0 1 0 py ]
[ 0 0 1 pz ]
[ 0 0 0 1 ]
makes sense as the x, y and z axis would extend to infinity and the point p would remain the same. When adding a vector though, if the w component of each vector in this matrix remain the same, will the resultant vectors w component not be the same as the point/vector to be translated? eg.
[ 1 0 0 0 ] [ 10 ] [ 10 ]
[ 0 1 0 0 ] [ 10 ] [ 10 ]
[ 0 0 1 0 ] X [ 10 ] = [ 10 ]
[ 0 0 0 1 ] [ 1 ] [ 1 ]
or do the values in the w components in the matrix change?
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Since nobody answered yet, I will try to add my 2 cents. 4x4 matrices are useful because they let you use linear and affine transformations in the 3x3 space. If you wanted to translate a point, with a 3x3 matrix (and 3-components vectors) you should perform it separately from rotation and scale. Using 4x4 matrices you can perform scale, translation and rotation using just one matrix. Using 4-components vectors lets you use the same classes for vectors and point, since they are semantically different: i.e you can't translate a direction, or get a point length. If you use w = 0 for directions and w = 1 for points, translation wont apply on directions (w = 0). In addition, point + vector = point (1+0=1), point - point = direction (1-1=0), point+direction = point (1+0=1) and so on. Of course, you cannot do point + point or direction - point directly (because these operations are not legal really).
4x4 matrices make possible to apply transformations such as translation with the same matrix you use for rotations and scaling.
Hope this help
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Homogeneous coordinates are used for projections
e.g.: perspective projections
set w=0 would translate the point infinitely far away after dehomogenization
here is a derivation of the perspective transformation matrix
http://www.cs.kuleuven.ac.be/cwis/research/graphics/INFOTEC/viewing-in-3d/node8.html
there are also some nice properties that you will likely learn when studying computer science
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I thought that perspective projection was achieved by dividing the x and y components of the translated coordinate by the z component?
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Quote:
4x4 matrices make possible to apply transformations such as translation with the same matrix you use for rotations and scaling.
Right.
Quote:
I thought that perspective projection was achieved by dividing the x and y components of the translated coordinate by the z component?
Kinda. The typical projection matrix is most real-time 3D graphics applications looks like
A 0 0 00 B 0 00 0 C D0 0 1 0
so when a vertex is multiplied by this vector, we have
x y z w * A 0 0 0 = xA yB zC + wD zE 0 B 0 0 0 0 C E 0 0 D 0
(for row vectors, like D3D; transpose everything for column vectors), and typically E and w are both 1, so we end up with a clip-space vector (xA, yB, zC + D, z). Once clipping is performed, the vector is component-wise divided by it's w component, which is proportional to the original z component of the vector.
So while technically the pipeline performs division by w, in practice, the value of w is usually proportional to the view-space z of the vertex.
Quote:
When adding a vector though, if the w component of each vector in this matrix remain the same, will the resultant vectors w component not be the same as the point/vector to be translated?
How the w component is handled depends on the context in which you are handling it; frequently most CPU-side manipulation uses 3-vectors, and the API understands that 3-vectors should be extended to 4-vectors by setting w to one.
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in the last matrix diagram - would the result vector not be:
xA
yB
zC + wE
zD
?
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Quote:
Original post by staticVoid2in the last matrix diagram - would the result vector not be:xAyBzC + wEzD?
In his example, [x y z w] is a row on the left. In your examples, the vector is a column on the right.
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Quote:
in the last matrix diagram - would the result vector not be:
Nope.
Looking at the multiplication visually and isolated to this specific scenario, we calculate element i of the result matrix as the dot product of the left vector and column i from the right matrix, for example:
X: (x y z w) <dot> (A 0 0 0) = (xA + 0 + 0 + 0) = xAY: (x y z w) <dot> (0 B 0 0) = (0 + yB + 0 + 0) = xBZ: (x y z w) <dot> (0 0 C D) = (0 + 0 + zC + wD) = zC + wDW: (x y z w) <dot> (0 0 E 0) = (0 + 0 + zE + 0) = zE
The columns, above, are written as rows to save space.
You can achive the answer you thought would be correct, as JohnBolton says. However, in doing so you change the result of the operation. As I said before, if you want to use columns on the right (and still want to have the same result, but obtained through a different process), you should transpose the matrix as well (consider a translation matrix).
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so if was to code a camera class that stored a 3x4 matrix:
a b c d
1 0 0 0
0 1 0 0
0 0 1 0
where a was the x axis vector, b - the y axis vector, c - the z axis vector and d the camera location. then I had a function to transform a 3d point.
eg.
1 0 0 10 500 1 0 10 x 1000 0 1 10 50 -1 40= 90 40
then with the new coordinate (40, 90, 40), divide the x and y components by z
(x/z, y/z) = (1, 2) to give the 2d screen coord.
why do you need an extra component(w) for each vector?
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Quote:
divide the x and y components by z(x/z, y/z) = (1, 2) to give the 2d screen coord.
The actual pipeline is more complex than this; the projection matrix which is applied after the view matrix brings the view-space vertices into clip space, where clipping is performed. The vertices that survive are divided by their w coordinates to bring them into normalized-device-coordinate space, which is then offset and scaled by further matrices to bring them into actual pixel coordinates.
Quote:
why do you need an extra component(w) for each vector?
Because otherwise you cannot use a 4x4 matrix. If you don't use a 4x4 matrix, you can't use a single matrix to represent rotational, scaling, and translation.
You can't actually multiply a 3x4 matrix and a 3x1 vector. It's mathematically impossible for the accepted definition of matrix multiplication.
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# A complex limit cycle not intersecting the real plane
Edit: This is a real coefficient version of the current post.
Is there a polynomial vector field $$X$$ with complex coefficients on $$\mathbb{C}^2$$ with the property quoted bellow?
There is a regular leaf $$L$$ whose holonomy, along at least one closed curve on it, is not trivial but $$L$$ does not intersect the real part $$im (z)=im(w)=0,\;(z,w) \in \mathbb{C}^2$$.
# Note:
A leaf with non trivial holonomy is called a complex limit cycle, according to the terminology used in the video lecture by Ilyashenko described in the following answer:
The error in Petrovski and Landis' proof of the 16th Hilbert problem
โข When I saw your question I did the bet that you would eventually, at some point, edit its tags just to add "limitcycle". It's too predictable :) โย YCor Feb 18 '18 at 0:30
โข @YCor Yes. As you have predicted, I could not resist myself from adding this tag:) โย Ali Taghavi Feb 18 '18 at 5:32
โข I think your question stands even without asking about a non-trivial holonomy. What may come into play is the fact that a non-constant entire function omits at most 2 values. Of course solutions to polynomial ODE are not always entire, but they can be extended almost everywhere, hence my guess is that the answer is no. โย Loรฏc Teyssier Mar 18 '18 at 20:02
โข @LoรฏcTeyssier Very interesting point. It is a very good idea. โย Ali Taghavi Mar 23 '18 at 11:42
โข @YCor I will just add that (limitcycle) seems to be renamed to (limit-cycles). I suppose that was your suggestion on meta - although I no longer can see the post, since it was deleted. โย Martin Sleziak Jan 20 at 11:29
The answer to this question is yes. There is a complex polynomial vector field on $$\mathbb{C}^2$$ with a complex limit cycle which does not intersect the real plane $$im(z)=im(w)=0$$.
Consider the differential equation $$\begin{cases}z'=w+(z^2+w^2-4i)\\ w'=-z+(z^2+w^2-4i) \end{cases}$$
The regular leaf $$L: z^2+w^2=4i$$ of this singular foliation does not intersect the real part of $$\mathbb{C}^2$$. This leaf, which is topologically a cylinder, has a non trivial holonomy. In fact we have more: there is a closed curve on this leaf whose corresponding holonomy map is a hyperbolic map: namely the holonomy is not tangent to the identity map. Here is the argument:
The hyperbolicity, hence non triviality, of the holonomy of this leaf is a consequence of Theorem 3.2 Page 333 of the paper: First Variation of Holomorphic forms and some applications.
Elaboration: The foliation is defined by $$\omega= (w+(z^2+w^2-4i))dw-(-z+(z^2+w^2-4i))dz=0$$
To apply the theorem 3.2 in the above paper we find a $$1-$$ form $$\alpha$$ which satisfies $$d\omega=\alpha \wedge \omega$$, locally around an appropriate closed curve $$\gamma$$ in $$L$$.
Represent the above $$1$$- form $$\omega$$ in the form $$\omega=Pdw-Qdz$$. Then for $$\alpha=(P_z+Q_w)/(P^2+Q^2)(Pdz+Qdw)$$ we have $$d\omega=\alpha \wedge \omega$$. Note that $$P^2+Q^2$$ does not vanish on $$L$$. Now we have to compute $$\int_{\gamma} \alpha$$, along an appropriate closed curve $$\gamma \subset L$$, and show that this integral is non zero.
To compute this integral we parametrize the cylinder $$L$$ with
$$\phi(t)= \begin{cases} z(t)=t+i/t\\w(t)=t/i-1/t \end{cases}$$ where $$\phi:\mathbb{C}\setminus \{0\}\to \mathbb{C}^2$$ is the global parametrization of $$L$$. We will see that the desired appropriate curve $$\gamma$$ is $$\phi(S^1)$$.
We denote by $$\phi^*(\alpha)$$, the pull back of $$\alpha$$ under embedding $$\phi$$. Now a very simple computation shows that $$\int_{S^1} \phi^* \alpha$$ is non zero since we obtain a pole of order 1 at the origin. In fact the later integral is $$\int_{S^1} 2(z(t)+w(t))(wdz-zdw)$$. An straightforward and short computation shows that we have a non degenerate pole, namely a pole of order 1. so the integral does not vanish. So the multiplier $$e^{\int _{S^1} \alpha}$$ is different from $$1$$. Then the leaf $$L$$ is a hyperbolic leaf. $$\square$$
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# 13939 (number)
13,939 (thirteen thousand nine hundred thirty-nine) is an odd five-digits composite number following 13938 and preceding 13940. In scientific notation, it is written as 1.3939 ร 104. The sum of its digits is 25. It has a total of 2 prime factors and 4 positive divisors. There are 13,624 positive integers (up to 13939) that are relatively prime to 13939.
## Basic properties
โข Is Prime? No
โข Number parity Odd
โข Number length 5
โข Sum of Digits 25
โข Digital Root 7
## Name
Short name 13 thousand 939 thirteen thousand nine hundred thirty-nine
## Notation
Scientific notation 1.3939 ร 104 13.939 ร 103
## Prime Factorization of 13939
Prime Factorization 53 ร 263
Composite number
Distinct Factors Total Factors Radical ฯ(n) 2 Total number of distinct prime factors ฮฉ(n) 2 Total number of prime factors rad(n) 13939 Product of the distinct prime numbers ฮป(n) 1 Returns the parity of ฮฉ(n), such that ฮป(n) = (-1)ฮฉ(n) ฮผ(n) 1 Returns: 1, if n has an even number of prime factors (and is square free) โ1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor ฮ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0
The prime factorization of 13,939 is 53 ร 263. Since it has a total of 2 prime factors, 13,939 is a composite number.
## Divisors of 13939
1, 53, 263, 13939
4 divisors
Even divisors 0 4 2 2
Total Divisors Sum of Divisors Aliquot Sum ฯ(n) 4 Total number of the positive divisors of n ฯ(n) 14256 Sum of all the positive divisors of n s(n) 317 Sum of the proper positive divisors of n A(n) 3564 Returns the sum of divisors (ฯ(n)) divided by the total number of divisors (ฯ(n)) G(n) 118.064 Returns the nth root of the product of n divisors H(n) 3.91105 Returns the total number of divisors (ฯ(n)) divided by the sum of the reciprocal of each divisors
The number 13,939 can be divided by 4 positive divisors (out of which 0 are even, and 4 are odd). The sum of these divisors (counting 13,939) is 14,256, the average is 3,564.
## Other Arithmetic Functions (n = 13939)
1 ฯ(n) n
Euler Totient Carmichael Lambda Prime Pi ฯ(n) 13624 Total number of positive integers not greater than n that are coprime to n ฮป(n) 6812 Smallest positive number such that aฮป(n) โก 1 (mod n) for all a coprime to n ฯ(n) โ 1647 Total number of primes less than or equal to n r2(n) 0 The number of ways n can be represented as the sum of 2 squares
There are 13,624 positive integers (less than 13,939) that are coprime with 13,939. And there are approximately 1,647 prime numbers less than or equal to 13,939.
## Divisibility of 13939
m n mod m 2 3 4 5 6 7 8 9 1 1 3 4 1 2 3 7
13,939 is not divisible by any number less than or equal to 9.
## Classification of 13939
โข Arithmetic
โข Semiprime
โข Deficient
โข Polite
โข Square Free
### Other numbers
โข LucasCarmichael
## Base conversion (13939)
Base System Value
2 Binary 11011001110011
3 Ternary 201010021
4 Quaternary 3121303
5 Quinary 421224
6 Senary 144311
8 Octal 33163
10 Decimal 13939
12 Duodecimal 8097
20 Vigesimal 1egj
36 Base36 ar7
## Basic calculations (n = 13939)
### Multiplication
nรy
nร2 27878 41817 55756 69695
### Division
nรทy
nรท2 6969.5 4646.33 3484.75 2787.8
### Exponentiation
ny
n2 194295721 2708288055019 37750827198909841 526208780325604273699
### Nth Root
yโn
2โn 118.064 24.0664 10.8657 6.74289
## 13939 as geometric shapes
### Circle
Diameter 27878 87581.3 6.10398e+08
### Sphere
Volume 1.13445e+13 2.44159e+09 87581.3
### Square
Length = n
Perimeter 55756 1.94296e+08 19712.7
### Cube
Length = n
Surface area 1.16577e+09 2.70829e+12 24143.1
### Equilateral Triangle
Length = n
Perimeter 41817 8.41325e+07 12071.5
### Triangular Pyramid
Length = n
Surface area 3.3653e+08 3.19175e+11 11381.1
## Cryptographic Hash Functions
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[Photo by Betsssssy.]
Do you ever take your kidsโ math tests? It helps me remember what it is like to be a student. I push myself to work quickly, trying to finish in about 1/3 the allotted time, to mimic the pressure students feel. And whenever I do this, I find myself prone to the same stupid mistakes that students make.
Even teachers are human.
In this case, it was a multi-step word problem, a barrage of information to stumble through. In the middle of it all sat this statement:
โฆand there were 3/4 as many dragons as gryphonsโฆ
My eyes saw the words, but my mind heard it this way:
โฆand 3/4 of them were dragonsโฆ
What do you think โ did I get the answer right? Of course not! Every little word in a math problem is important, and misreading even the smallest word can lead a student astray. My mental glitch encompassed several words, and my final tally of mythological creatures was correspondingly screwy.
But here is the more important question: Can you explain the difference between these two statements?
If Johnny Canโt Read, Then He Canโt Do Math
To solve word problems, students must be able to read and understand what is written, and they must be able to follow directions. They need to comprehend what they read โ to paraphrase it, concentrating on the relevant facts โ and then to translate that information into a mathematical expression. Many times, they must be able to โread between the linesโ and understand something that is implied, not explicitly stated.
When students struggle with word problems, more often than not it is a language issue that confuses them.
Paraphrasing is one of the most important skills we can teach junior high and high school students. Often they want to rush into interpreting and reacting to a text even before they know what it means. We teachers sometimes suffer from the delusion that since a student can read the words on the page, he or she understands whatโs been read. But thatโs not always true.
That quote is from an article at Teen Literacy Tips blog. Does a literature teacher have anything useful to say about solving math problems? Well, the fact that word problems are also called story problems should clue us in to a significant connection.
As important as mathematics is, it is a distant second to the need for good reading comprehension. We teachers so often hear students summarize a course by saying, โI could do everything except the word problems.โ
Sadly, in the textbook of life, there are only word problems.
โ Herb Gross,
quoted by Jerome Dancis in
Reading Instruction for Arithmetic Word Problems
[The entire article by Dancis is worth reading, and you may want to explore the rest of his webpage as well. I will be using Supposedly Difficult Arithmetic Word Problems as ratio practice with my MathCounts students later this semester.]
For a simple (yet often confusing) example, consider these two statements. Can you explain the difference?
โข Eight divided in half is four.
and
โข Eight divided by one-half is sixteen.
If your students keep a Math Journal, this would be a great writing prompt. An answer is given at bottom of this post.
Now, Letโs Analyze My Mistake
In my word problem, it turned out there were 56 creatures in all. I got that part of the answer just fine, but then I needed to know how many of those creatures were gryphons.
This is how I did it:
โฆand 3/4 of them were dragonsโฆ
4 units = 56
1 unit = 56 รท 4 = 14 gryphons
But that was not at all what the problem said. There should have been several more gryphons than dragons. If I had been paying better attention to what I read, this is how I should have solved the problem:
โฆand there were 3/4 as many dragons as gryphonsโฆ
7 units = 56
1 unit = 56 รท 7 = 8
4 units = 8 x 4 = 32 gryphons
Just to make the language issue more difficult, consider this: All of the following statements are equivalent. Compare each statement to the second drawing above (the correct one). Can you see each relationship?
โข There are 3/4 as many dragons as gryphons.
โข For every 4 gryphons, there are 3 dragons.
โข The ratio of dragons to gryphons is 3:4.
โข 4 out of every 7 creatures is a gryphon.
โข There are 1/3 more gryphons than dragons.
โข There are 25% fewer dragons than gryphons.
โข If you tag a creature at random from the group, the probability of choosing a dragon is 3/7.
Can you think of any other ways to say it? This would be another good math journal writing prompt.
Ratio problems like this are some of the most confusing word problems our pre-algebra students will face. The more we can work with them on reading, paraphrasing, and translating these problems into mathematical expressions, the better prepared our students will be to face the word problems they meet in โthe textbook of life.โ
[Edited to add: This problem follows students beyond middle school. Jackie is struggling to get her high school math students to read carefully. See her post Mis-Reading in Mathematics (and the comments section).]
When you divide a number in half, you split it into two equal parts. But if you divide a number by 1/2, you are finding how many halves it takes to make that number โ that is, you are cutting it into half-size pieces and counting how many there are. And in that case, because each whole thing is two halves, there will be double as many pieces as the number you started with.
12 thoughts on โReading to Learnย Mathโ
1. Di says:
How timely! My students are doing word problems right now and converting from prose to mathematical expressions is challenging. Iโll watch out for tricky wording like you pointed out.
2. These things are hard to read. Even for good readers. I slow kids down, it probably helps.
3. I was just working on this (again) with my pre-algebra students. They admitted they just skip the word problems โ especially on standardized tests. Our new goal is not to be โtrickedโ by them anymore.
I love your quotes โ especially the one by Herb Gross.
sorry โ somehow your comment was marked as spam โ not anymore!
4. I started out tutoring algebra, then switched to remedial reading using phonics. Once you get into higher math where you need to be able to read the explanations and the word problems, reading is important to math. Without the ability to read well, youโll never excel in your other subjects.
5. disconnect says:
Eight divided in half is four.
Eight divided by (one-)half is sixteen.
Okay.
Eight divided in two is four.
Eight divided by two is four.
(brain explodes)
6. LOL!
I did it again this week. My son gets a laugh out of my mistakes on his homework, especially when he got the problem right. Last time, I ignored the word โmoreโ in a MathCounts problem, and this week I missed the word โadditional.โ Youโd think I would have learned by nowโฆ
As for comments, Jackie, I have given up on rescuing them from the spam folder. I have been getting way over 100 spam a day, and I just donโt have that much time to sort them. But I did fix your blog link for you!
7. I teach a lot of literacy in my class, even when it seems itโs weird to do so in a math class. It helps when kids actually read with me the problems they have to do. Thatโll be especially important for a year in which I have a class full of ELLs. In any case, good post.
8. Very good post. This is where we had many fun discussion in college about the validity of tests and knowing what you are really testing.
But the problem is, if we want to prepare our children for life, they need this kind of reasoning as well.
9. You have some good examples here about the critical importance of knowing how to translate from words to math. I also liked the quote about the paraphrasing.
Of course, I like those because my book, โSolving Word Problems,โ explains exactly these things (and more)! ๐
The dragons-gryphons sentence is actually a very difficult one. Your mistake was a simple one but the sentence structure โ3/4 as many dragons as gryphonsโ requires either an ability to manipulate the two objects, dragons and gryphons, in oneโs head and understand that dragons = 3/4 grypohns, or to know how to rephrase it to an easy to understand sentence. The first is EXTREMELY hard to do and most students will write 3/4dragons = gryphons. The second can be easily taught! [See my book of course LOL].
To the one with the exploding brain: you canโt divide eight in two. You can only use this grammatical structure for fractions. It took me a minute to understand that this is the issue, but maybe I noticed it because English is not my first language so Iโm more sensitive to the translation issue ๐
10. Hi.
This is Herb Gross and I am now putting toglether all of my arithmetic and algebra materials (including textbooks, videos and slide shows) on my website for anyone ot use free of charge. The website is just temporary and in a short time it will be made more user-friendly in terms of being able to access items quickly.
Please feel free tot use the material on my site (www.adjectivenounmath.com) in any ways that you wish. You may email me at hgross3@comcast.net
PS
Please excuse any typos. At age 81 the small prnt is my nemesis.
This site uses Akismet to reduce spam. Learn how your comment data is processed.
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## Monday, July 11, 2005
### Separating hyperplanes, witnesses for unitary graphs
A separating vector is a vector which belongs to the dependence cone.
A witness is a special form of separating vectors: One that induces only +ve cycles in the given graph.
--
To Do: Read DRV or DV-KMW and draw the dual polyhedron for the examples from KMW.
To do: DRV has a PRDG in chapter 5. work on it.
--
The hyperplane/cycle algorithm doesnot seem to work when all the weights are (0,0,0). Is this a case that can be handled easily?
--
Example from DRV-EX-213: The input graph is already in unitarized form. DRV say that all the vectors that are orthogonal to (1,0,0) vanish. Our algorithm identifies the plane i=0 (the one orthogonal to (1,0,0)) to be, what CM-Z would call, a separating hyperplane . The edges from S2 to the bottom node and a two more edges disappear.
A few questions: What about the edge from S1 to the bottom node? Why does it not disappear? What about the vectors (0,0,0)?
--
Prove/disprove the following (important) : The unariness of the vectors (q,v).
Soln(of vector (q,v)): The vector q need not be unitary. A graph with three edges having the vectors e1=(1,0), e2=(-1,+), e3=(-1,-1) would have the solution 2q1 = q2 + q3. A circulation would be (q1,q2,q3) = (2,-1,-1), which is not unitary.
--
Possible choices for the Level sets: (1) The norm, (2) square of the norm (3) number of non-zero components in the dependence vectors
One more advantage of unitarization: The level sets of the vectors are from a finite set. If the dependences are not unitarized, the level sets are from a possibly infinite set.
--
Here may be the tentative algorithm:
(1) Unitarize the graph
(2) Find the set of edges with the largest level sets. This can be done by
(a) finding the largest level set
(b) selecting the edges with norm == that particular level set
(3) Amon the edges with the greatest level set, find the plane which is parallel to the maximal number of edges. This can be done by ........
--
The Unitarization Problem: Unitarization for ZC-D, multi-dim scheduling, MDS followed by optimal memory allocation, simultaneous MDS and OMA, tiling for parallelism, tiling for locality.
--
To do: How does the dual polyhedron look like? What is special about it when it has been unitarized? How do its level sets look like?
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# Tag Info
Accepted
### Why do rockets have multiple stages?
The basic reason: tossing an extra stage can be far, far, more of a mass-savings than trying to make one stage that can do everything. There's a handful of reasons for this: Engines weigh much less ...
โข 10.7k
### Why did the space shuttle's altitude go down after reaching 108,000m?
The drop in acceleration around 40s into the flight is the shuttle throttling down to reduce the aerodynamic load on the vehicle. It then accelerates when past this point. The drop in acceleration at ...
โข 4,625
Accepted
### Did any spacecraft ever use the Sun's gravity for acceleration?
It doesn't really work that way. We can use the Sun to change direction, but we need rocket thrust to increase speed with the maneuver. To begin with, the closest stars (apart from the Sun) are not ...
โข 8,525
### Did any spacecraft ever use the Sun's gravity for acceleration?
The "gravitational" (slingshot) maneuvers space probes are performing are actually not so much about gravity. The gravity is method to "tie" temporarily these two bodies, but you could (purely ...
โข 1,831
Accepted
### How to account for burned fuel mass when calculating spacecraft acceleration?
The quantity you ultimately need when planning your manoeuvre is change in velocity, which in spaceflight terminology is called delta-v, $\Delta v$ (searching this term would give you a lot of ...
### Why do rockets have multiple stages?
Then why not use only 1 stage? Because we don't know how to do that. That we don't know how to do make a single stage to orbit is a consequence Tsiolkovsky rocket equation and of the fact that some ...
โข 75k
Accepted
### How do vibrational isolators reduce the g-forces of a payload on a launch vehicle?
The g-forces from the rocket's acceleration remain unaffected, of course. What the vibrational isolators do is isolate the payload from vibrations. Without them, the payload would experience both the ...
Accepted
### SpaceX rocket strange velocity
Two possibilities: The Earth rotates at about 465โm/s at the equator. This video's measurements could include that speed, though I don't believe so as it starts from 0โm/s. Most likely it's because ...
โข 2,900
Accepted
### How much acceleration g-force can prevent pilot from moving their arms?
According to the book Human Engineering guide for Equipment Designers, authored by Wesley E. Woodson and Donald W. Conover and published by the University of California Press, suggests the following ...
โข 2,137
### Did any spacecraft ever use the Sun's gravity for acceleration?
I think the question is based on a misconception about how gravity assists work. If you just let yourself get pulled to a distant object then continue out the other side, the same gravity that ...
### Why do rockets have multiple stages?
Other answers address the core construct of the rocket equation with words and equations, but here it is visually: Where the Y-axis is the $\Delta V$ and the X-axis is the propellant mass. $b$ is a ...
โข 9,805
### Are we actually that close to techniques of accelerating probes to speeds like a quarter $c$?
Given that the interstellar medium (ISM) has a density of about 1 atom per cubic centimeter and given that laser propulsion could, in theory, accelerate a spacecraft to 30% of the speed of light in ...
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### Could you survive very high G's if your whole body was accelerated uniformly?
Of course. (if you replace that pesky "infinite" with "arbitrarily large", physics really really hates infinite forces) Who said you (and everything around you), is not currently ...
### Orbital Railgun for launching deep space probes
Let's look at some numbers. For the sake of argument let's target a delta-V from the railgun of 4 km/s enough to get to low lunar orbit or Mars transfer orbit. If the railgun is 1km long, that would ...
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### Orbital Railgun for launching deep space probes
Concepts for mass-drivers or railguns situated on the moon exist. What you're asking about is a mass driver space station in low earth orbit that a payload or ship launched from earth can dock with, ...
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# Apรฉry's constant
Apรฉry's constant is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Apรฉry's constant is the sum of the reciprocals of the positive cubes.
That is, it is defined as the number ${\displaystyle \zeta (3)=\sum _{k=1}^{\infty }{\frac {1}{k^{3}}},}$ where ฮถ is the Riemann zeta function.
Approximately equal to:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
This constant was known and studied by many early mathematicians, but was not named until 1978, after Roger Apรฉry, who was first to prove it was an irrational number.
${\displaystyle \zeta (3)}$ by summing reciprocal cubes is easy to calculate, but converges very slowly. The first 1000 terms are only accurate to 6 decimal places.
There have been many fast convergence representations developed / discovered that generate correct digits much more quickly.
One of the earliest, discovered in the late 1800s by A. Markov and later widely published by Apรฉry is:
${\displaystyle \zeta (3)={\frac {5}{2}}\sum _{k=1}^{\infty }(-1)^{k-1}{\frac {k!^{2}}{(2k)!k^{3}}}.}$
Much better than direct calculation of ${\displaystyle \zeta (3)}$, but still only yielding about .63 correct digits per iteration.
Several even faster converging representions are available. The fastest known to date, yielding about 5.04 correct digits per term, is by Sebastian Wedeniwski.
${\displaystyle \zeta (3)={\frac {1}{24}}\sum _{k=0}^{\infty }(-1)^{k}{\frac {(2k+1)!^{3}(2k)!^{3}k!^{3}(126392k^{5}+412708k^{4}+531578k^{3}+336367k^{2}+104000k+12463)}{(3k+2)!(4k+3)!^{3}}}.}$
Task
โข Show the value of Apรฉry's constant calculated at least three different ways.
1. Show the value of at least the first 1000 terms of ${\displaystyle \zeta (3)}$ by direct summing of reciprocal cubes, truncated to 100 decimal digits.
2. Show the value of the first 158 terms of Markov / Apรฉry representation truncated to 100 decimal digits.
3. Show the value of the first 20 terms of Wedeniwski representation truncated to 100 decimal digits.
See also
## ALGOL 68
Works with: ALGOL 68G version Any - tested with release 2.8.3.win32
Uses Algol 68G's LONG LONG INT and LONG LONG REAL which have programmer specified precision. The factorials can get quuite large so more than 101 digits are needed.
BEGIN # find Apรฉry's constant: the sum of the positive cubes' reciprocals #
# (this is the value of the Riemann zeta function applied to 3) #
PR precision 1000 PR # set precision of LONG LONG REAL #
# returns a string representation of z, truncated to 100 decimals #
PROC truncate100 = ( LONG LONG REAL z )STRING:
BEGIN
STRING result = fixed( z, 0, 101 ); # format with 101 decimals #
result[ย : UPB result - 1 ] # remove the final digit #
END # truncate100 #ย ;
# methind 1 - sum the reciprocols of the cubes - 1000 terms from 1 #
BEGIN
LONG LONG REAL zeta3ย := 0;
FOR k TO 1000 DO
LONG LONG INT llk = LENG LENG k;
zeta3 +:= 1 / ( llk * llk * llk )
OD;
print( ( truncate100( zeta3 ), newline ) )
END;
# method 2 - Markov's alternative representaton, 158 terms from 1 #
# 5/2 * sum [ (-1)^(k-1)( k!^2 / (2k!)k^2 ) ] from 1 #
BEGIN
LONG LONG INT fkย := 1, f2kย := 1;
LONG LONG REAL zeta3ย := 0;
FOR k TO 158 DO
LONG LONG INT llk = k;
LONG LONG INT ll2k = llk * 2;
fk *:= llk;
f2k *:= ( ll2k - 1 ) *:= ll2k;
LONG LONG REAL term = ( fk * fk ) / ( f2k * llk * llk * llk );
IF ODD k THEN
zeta3 +:= term
ELSE
zeta3 -:= term
FI
OD;
zeta3 *:= 5 / 2;
print( ( truncate100( zeta3 ), newline ) )
END;
# method 3 - Wedeniwski representation - 20 terms from 0 #
# 1/24 * sum [ (-1)^k (2k+1)!^3(2k)!^3(k!)^3 #
# * ( 126392k^5 + 412708k^4 + 531578k^3 #
# + 336367k^2 + 104000k + 12463 #
# ) / (3k+2)! (4k + 3)!^3 #
# ] from 0 #
BEGIN
[]INT w coefficients = ( 126392, 412708, 531578, 336367, 104000, 12463 );
LONG LONG INT fkย := 1, f2kย := 1, f3kย := 1, f4kย := 1;
# ensure the divisor is a LONG LONG INT so the LHS is calculated as a #
# LONG LONG REAL value and not a REAL which is then widened #
LONG LONG REAL zeta3ย := w coefficients[ UPB w coefficients ]
/ LENG LENG ( 2 * 6 * 6 * 6 );
FOR k TO 19 DO
LONG LONG INT llk = k;
LONG LONG INT ll2k = llk + llk;
LONG LONG INT ll3k = ll2k + llk;
LONG LONG INT ll4k = ll3k + llk;
fk *:= llk;
f2k *:= ( ll2k - 1 ) * ll2k;
f3k *:= ( ll3k - 2 ) * ( ll3k - 1 ) * ll3k;
f4k *:= ( ll4k - 3 ) * ( ll4k - 2 ) * ( ll4k - 1 ) * ll4k;
LONG LONG INT f2k1 = f2k * ( ll2k + 1 );
LONG LONG INT f3k2 = f3k * ( ll3k + 1 ) * ( ll3k + 2 );
LONG LONG INT f4k3 = f4k * ( ll4k + 1 ) * ( ll4k + 2 ) * ( ll4k + 3 );
LONG LONG REAL termย := 0;
FOR c TO UPB w coefficients DO
term *:= llk +:= w coefficients[ c ]
OD;
LONG LONG INT fp = f2k1 * f2k * fk;
term *:= fp * fp * fp /:= f3k2 * f4k3 * f4k3 * f4k3;
IF ODD k THEN
zeta3 -:= term
ELSE
zeta3 +:= term
FI
OD;
zeta3 /:= 24;
print( ( truncate100( zeta3 ), newline ) )
END
END
Output:
1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
## F#
// Apรฉry's constant. Nigel Galloway: March 3rd., 2023
open MathNet.Numerics
let fact=let g=Seq.unfold(fun(n,g)->Some(n,(n*g,g+1N)))(1N,2N)|>Seq.cache in (fun n->Seq.item (n-1) g)
let fN g=let g=BigRational.FromInt g in 126392N*g**5+412708N*g**4+531578N*g**3+336367N*g**2+104000N*g+12463N
let fG n g l=let i=n/g in (int i,Seq.unfold(fun(n,i)->if i=0 then None else let l=n/g in Some(int l,(10I*(n-l*g),i-1)))(10I*(n-i*g),l))
let r3=Seq.initInfinite(fun g->BigRational.PowN(((+)1>>BigRational.FromInt>>BigRational.Reciprocal)g,3))|>Seq.take 1000|>Seq.sum
let ma=(5N/2N)*(Seq.unfold(fun(n,g,l)->Some(n*g,(-n,(fact l)*(fact l)/(fact(2*l)*BigRational.FromInt(pown l 3)),l+1)))(1N,1N/2N,2)|>Seq.take 158|>Seq.sum)
let sw=(1N/24N)*(Seq.unfold(fun(n,g,l)->Some(n*g,(-n,(fact(2*l+1)**3*fact(2*l)**3*(fact l)**3*(fN l))/(fact(3*l+2)*(fact(4*l+3)**3)),l+1)))(1N,12463N/432N,1)|>Seq.take 20|>Seq.sum)
[r3;ma;sw]|>List.iter(fun n->let n,g=fG (n.Numerator) (n.Denominator) 100 in printf $"%d{n}."; g|>Seq.iter(printf "%d"); printfn "") Output: 1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 ### Bonus 4th. way as a continued fraction I extend Continued fraction#Apรฉry's_constant to provide the 101 digits required here. This method requires 31 iterations to provide the 101 digits, not quite as good as Wedeniwski but much better than the other 2. This method has historic interest as Apรฉry used it to prove that this number is irrational. let cf2br ฮฑ ฮฒ=let n0,g1,n1,g2=ฮฒ(),ฮฑ(),ฮฒ(),ฮฒ() seq{let (ฮ :BigRational)=g1/n1 in yield n0+ฮ ; yield! Seq.unfold(fun(n,g,ฮ )->let a,b=ฮฑ(),ฮฒ() in let ฮ =ฮ *g/n in Some(n0+ฮ ,(b+a/n,b+a/g,ฮ )))(g2+ฮฑ()/n1,g2,ฮ )} let aฯ()=let mutable n=0N in (fun ()->n<-n+1N; -(BigRational.Pow(n,6))) let bฯ()=let mutable n=0N in (fun ()->n<-n+1N; (2N*n-1N)*(17N*n*n-17N*n+5N)) cf2br (aฯ()) (bฯ())|>Seq.skip 31|>Seq.take 1|>Seq.iter(fun n->let n=6N/n in let n,g=fG (n.Numerator) (n.Denominator) 100 in printf$"%d{n}."; g|>Seq.iter(printf "%d"); printfn "")
Output:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
## J
app3=: {{ +/_3x^~1+i.y }}
app3m=: {{ _5r2 * +/ {{ (_1^y)*(2^~!y)%(!2*y)*y^3}} 1x+i.y }}
app3sm=: {{ 1r24* +/ {{
(_1^y)*(3^~*/!y,0 1+/2*y)*(12463 104000 336367 531578 412708 126392 p. y)%(!2 3 p.y)*(!3 4 p.y)^3
}} x:i.y
}}
0j100 ": app3 1000
1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473112
0j100 ": app3m 158
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
0j100 ": app3sm 20
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
## Julia
using SpecialFunctions
setprecision(120, base=10)
println("Apรฉry's constant via Julia's zeta:\n$(string(zeta(big"3"))[1:102])") """ zeta(3) via Riemann summation of 1/(k cubed) """ Apรฉry_r(nterms = 1_000_000) = sum(big"1" / k^big"3" for k in 1:nterms) println("\nApรฉry's constant via reciprocal cubes:\n$(string(Apรฉry_r())[1:102])")
""" zeta(3) via Markov's summation """
function Apรฉry_m(nterms = 158)
return big"2.5" * sum((isodd(k) ? 1 : -1) * factorial(big(k))^2 /
(factorial(big"2" * k) * k^big"3") for k in 1:nterms)
end
println("\nApรฉry's constant via Markov's summation:\n$(string(Apรฉry_m())[1:102])") """ zeta(3) via Wedeniwski's summation """ function Apรฉry_w(nterms = 20) return big"1"/24 * sum((iseven(k) ? 1 : -1) * factorial(big"2" * k + 1)^3 * factorial(big"2" * k)^3 * factorial(big(k))^3 * (126392 * k^big"5" + 412708 * k^big"4" + 531578 * k^big"3" + 336367 * k^big"2" + big"104000" * k + 12463) / (factorial(big"3" * k + 2) * factorial(big"4" * k+3)^3) for k in 0:nterms) end println("\nApรฉry's constant via Wedeniwski's summation:\n$(string(Apรฉry_w())[1:102])")
Output:
Apรฉry's constant via Julia's zeta:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
Apรฉry's constant via reciprocal cubes:
1.2020569031590942858997379115114499908483196256737488817922717053418382053696464235214344450378979367
Apรฉry's constant via Markov's summation:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
Apรฉry's constant via Wedeniwski's summation:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
## Mathematica/Wolfram Language
ClearAll["Global*"];
TruancateTo100DecimalDigits = N[#, 100 + 1] &;
MyShowApรฉryConstant[expr_, caption_String] :=
Print[caption <>
ToString@Activate@TruancateTo100DecimalDigits[expr]];
MyShowApรฉryConstant[
Zeta[3], "Apรฉry's constant via Mathematica's Zeta:\n"]
MyShowApรฉryConstant[
Sum[1/(k^3), {k, 1,
1000}], "Apรฉry's constant via reciprocal cubes:\n"]
MyShowApรฉryConstant[(5/2*
Sum[(-1)^(k - 1)*(k!)^2/((2 k)!*k^3), {k, 1,
158}]), "Apรฉry's constant via Markov's summation:\n"]
MyShowApรฉryConstant[
1/24*Sum[(-1)^
k*((2 k + 1)!)^3*((2 k)!)^3*(k!)^3*(126392 k^5 + 412708 k^4 +
531578 k^3 + 336367 k^2 + 104000 k +
12463)/(((3 k + 2)!)*((4 k + 3)!)^3), {k, 0,
19}], "Apรฉry's constant via Wedeniwski's summation:\n"]
Output:
Apรฉry's constant via Mathematica's Zeta:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
Apรฉry's constant via reciprocal cubes (accurate to 6 decimal places):
1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473112
Apรฉry's constant via Markov's summation: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
Apรฉry's constant via Wedeniwski's summation: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
## Nim
Translation of: Wren
Library: bignum
import std/strformat
import bignum
func toDecimal100(r: Rat): string =
## Return the representation of a rational up to 100 decimals.
r *= newInt(10)^100
result.setLen(102)
result = ($r.toInt)[0..100] result.insert(".", 1) proc apery(n: Positive) = var sum = newRat() for k in 1..n: sum += newRat(1, k^3) echo &"First {n} terms of ฮถ(3) truncated to 100 decimal places (accurate to 6 decimal places):" echo sum.toDecimal100 echo() proc markov(n: Positive) = var neg = true var fact1, fact2 = newInt(1) var sum = newRat() for k in 1..n: neg = not neg fact1 *= k var num = fact1 * fact1 if neg: num = -num fact2 *= 2 * k * (2 * k - 1) let denom = fact2 * k^3 sum += newRat(num, denom) sum *= newRat(5, 2) echo &"First {n} terms of Markov / Apรฉry representation truncated to 100 decimal places:" echo sum.toDecimal100 echo() proc wedeniwski(n: Positive) = var fact1, fact2 = newInt(1) var neg = true var sum = newRat() for k in 0..<n: neg = not neg if k > 0: fact1 *= k fact2 *= 2 * k * (2 * k - 1) let fact3 = fact2 * (2 * k + 1) var num = (fact1 * fact2 * fact3)^3 num *= ((((126392 * k + 412708) * k + 531578) * k + 336367) * k + 104000) * k + 12463 if neg: num = -num let denom = fac(4 * k + 3)^3 * fac(3 * k + 2) sum += newRat(num, denom) sum /= 24 echo &"First {n} terms of Wedeniwski representation truncated to 100 decimal places:" echo sum.toDecimal100 echo() echo "Actual value to 100 decimal places:" echo "1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581" echo() apery(1000) markov(158) wedeniwski(20) Output: Actual value to 100 decimal places: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 First 1000 terms of ฮถ(3) truncated to 100 decimal places (accurate to 6 decimal places): 1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111 First 158 terms of Markov / Apรฉry representation truncated to 100 decimal places: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 First 20 terms of Wedeniwski representation truncated to 100 decimal places: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 ## PARI/GP \\ Set the precision to, say, 110 digits default(realprecision, 110); \\ Function to display Apรฉry's constant my_show_apery_constant(expr, caption) = printf("%s %.101g\n\n", caption, expr); \\ Apรฉry's constant via PARI/GP's zeta function my_show_apery_constant(zeta(3), "Apรฉry's constant via PARI/GP's Zeta:\n"); \\ Apรฉry's constant via reciprocal cubes my_show_apery_constant(sum(k = 1, 1000, 1/k^3), "Apรฉry's constant via reciprocal cubes:\n"); \\ Apรฉry's constant via Markov's summation my_show_apery_constant((5/2) * sum(k = 1, 158, (-1)^(k - 1) * (k!)^2 / ((2*k)! * k^3)), "Apรฉry's constant via Markov's summation:\n"); \\ Apรฉry's constant via Wedeniwski's summation my_show_apery_constant(1/24 * sum(k = 0, 19, (-1)^k * ((2*k + 1)!)^3 * ((2*k)!)^3 * (k!)^3 * (126392*k^5 + 412708*k^4 + 531578*k^3 + 336367*k^2 + 104000*k + 12463) / (((3*k + 2)!) * ((4*k + 3)!)^3)), "Apรฉry's constant via Wedeniwski's summation:\n"); Output: Apรฉry's constant via PARI/GP's Zeta: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 Apรฉry's constant via reciprocal cubes: 1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473112 Apรฉry's constant via Markov's summation: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 Apรฉry's constant via Wedeniwski's summation: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 ## Perl use v5.36; use bigrat try => 'GMP'; sub f { my$r = 1; $r *=$_ for 1..shift; $r } say 'Actual value to 100 decimal places:'; say '1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581'; say "\nFirst 1000 terms of ฮถ(3) truncated to 100 decimal places. (accurate to 6 decimal places):"; my$z3;
$z3 += 1/$_**3 for 1..1000;
say $z3->as_float(101); say "\nFirst 158 terms of Markov / Apรฉry representation truncated to 100 decimal places:";$z3 = 0;
$z3 += (-1)**($_-1) * (f($_)**2 / (f(2*$_) * $_**3)) for 1..158;$z3 *= 5/2;
say $z3->as_float(101); say "\nFirst 20 terms of Wedeniwski representation truncated to 100 decimal places:";$z3 = 0;
$z3 += (-1)**$_ * f(2*$_+1)**3 * f(2*$_)**3 * f($_)**3 * (126392*$_**5 + 412708*$_**4 + 531578*$_**3 + 336367*$_**2 + 104000*$_ + 12463)
/ ( f(3*$_+2) * f(4*$_+3)**3 )
for 0..19;
$z3 *= 1/24; say$z3->as_float(101);
Output:
Actual value to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
First 1000 terms of ฮถ(3) truncated to 100 decimal places. (accurate to 6 decimal places):
1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473112
First 158 terms of Markov / Apรฉry representation truncated to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
First 20 terms of Wedeniwski representation truncated to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
## Phix
Ugh. If you ran this on the James Webb, you might just be able to pick out a faint small print outline of the word "elegant".
Still, at least it is not like you do this sort of thing every day... and I got to fix a couple of bugs in my mpfr.js code.
with javascript_semantics
requires("1.0.2") -- (missing mpfr_ui_pow_ui() and bug in mpfr_mul_d(), both in mpfr.js)
include mpfr.e
mpfr_set_default_precision(-100)
mpfr {d,a,w,t} = mpfr_inits(4)
mpz {z,pk} = mpz_inits(2)
for k=1 to 1000 do
mpfr_ui_pow_ui(t,k,3)
mpfr_si_div(t,1,t)
mpfr_add(d,d,t)
end for
for k=1 to 158 do
mpz_fac_ui(z,k)
mpz_mul(z,z,z)
mpfr_set_z(t,z)
mpz_fac_ui(z,2*k)
mpfr_div_z(t,t,z)
mpz_ui_pow_ui(z,k,3)
mpfr_div_z(t,t,z)
if even(k) then
mpfr_sub(a,a,t)
else
mpfr_add(a,a,t)
end if
end for
mpfr_mul_d(a,a,5/2)
for k=0 to 19 do
mpz_ui_pow_ui(z,k,5)
mpz_mul_si(z,z,126392)
mpz_ui_pow_ui(pk,k,4)
mpz_mul_si(pk,pk,412708)
mpz_add(z,z,pk)
mpz_ui_pow_ui(pk,k,3)
mpz_mul_si(pk,pk,531578)
mpz_add(z,z,pk)
mpz_add_si(z,z,k*k*336367)
mpz_add_si(z,z,k*104000)
mpz_add_si(z,z,12463)
mpfr_set_z(t,z)
mpz_fac_ui(z,2*k+1)
mpz_pow_ui(z,z,3)
mpfr_mul_z(t,t,z)
mpz_fac_ui(z,2*k)
mpz_pow_ui(z,z,3)
mpfr_mul_z(t,t,z)
mpz_fac_ui(z,k)
mpz_pow_ui(z,z,3)
mpfr_mul_z(t,t,z)
mpz_fac_ui(z,3*k+2)
mpfr_div_z(t,t,z)
mpz_fac_ui(z,4*k+3)
mpz_pow_ui(z,z,3)
mpfr_div_z(t,t,z)
if odd(k) then
mpfr_sub(w,w,t)
else
mpfr_add(w,w,t)
end if
end for
mpfr_div_si(w,w,24)
constant fmt = """
Actual value to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
First 1000 terms of zeta(3) truncated to 100 decimal places. (accurate to 6 decimal places):
%s
First 158 terms of Markov / Apery representation truncated to 100 decimal places:
%s
First 20 terms of Wedeniwski representation truncated to 100 decimal places:
%s
"""
string direct = mpfr_get_fixed(d,100),
mapery = mpfr_get_fixed(a,100),
wdnski = mpfr_get_fixed(w,100)
printf(1,fmt,{direct,mapery,wdnski})
Output:
Actualโvalueโtoโ100โdecimalโplaces:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
Firstโ1000โtermsโofโzeta(3)โtruncatedโtoโ100โdecimalโplaces.โ(accurateโtoโ6โdecimalโplaces):
1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111
Firstโ158โtermsโofโMarkovโ/โAperyโrepresentationโtruncatedโtoโ100โdecimalโplaces:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
Firstโ20โtermsโofโWedeniwskiโrepresentationโtruncatedโtoโ100โdecimalโplaces:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
Last digit of the 1000 terms line is 2 under pwa/p2js...
As per Wren, you can verify or completely replace all this with mpfr_zeta_ui(w,3) [on desktop/Phix only, not supported under pwa/p2js]
## Python
from sympy import zeta, factorial
from decimal import Decimal, getcontext
# Set the desired precision
getcontext().prec = 120
def my_sympy_format_to_decimal(sympy_result):
return Decimal(str(sympy_result.evalf(getcontext().prec)))
def print_apery_constant(description, value):
print(f"{description}:\n{str(value)[:102]}")
# Apรฉry's constant via SymPy's zeta function
zeta_3_str = str(zeta(3).evalf(getcontext().prec))
zeta_3_decimal = Decimal(zeta_3_str)
print_apery_constant("Apรฉry's constant via SymPy's zeta", zeta_3_decimal)
# Apรฉry's constant via Riemann summation of 1/(k cubed)
def apery_r(nterms=1_000):
total = sum(Decimal('1') / Decimal(k) ** 3 for k in range(1, nterms + 1))
return total
print_apery_constant("Apรฉry's constant via reciprocal cubes", apery_r())
# Apรฉry's constant via Markov's summation
def apery_m(nterms=158):
total = Decimal(2.5) * sum(
(Decimal(1) if k % 2 != 0 else Decimal(-1)) *
my_sympy_format_to_decimal(factorial(k) ** 2) /
my_sympy_format_to_decimal(factorial(2*k) * (k ** 3) )
for k in range(1, nterms + 1)
)
return total
print_apery_constant("Apรฉry's constant via Markov's summation", apery_m())
# Apรฉry's constant via Wedeniwski's summation
def apery_w(nterms=20):
total = Decimal('1') / Decimal('24') * sum(
(Decimal('1') if k % 2 == 0 else Decimal('-1')) *
my_sympy_format_to_decimal(factorial(2 * k + 1)) ** 3 *
my_sympy_format_to_decimal(factorial(2 * k)) ** 3 *
my_sympy_format_to_decimal(factorial(k)) ** 3 *
(Decimal('126392') * Decimal(k) ** 5 +
Decimal('412708') * Decimal(k) ** 4 +
Decimal('531578') * Decimal(k) ** 3 +
Decimal('336367') * Decimal(k) ** 2 +
Decimal('104000') * Decimal(k) +
Decimal('12463')) /
(my_sympy_format_to_decimal(factorial(3 * k + 2)) * my_sympy_format_to_decimal(factorial(4 * k + 3)) ** 3)
for k in range(0, nterms + 1)
)
return total
print_apery_constant("Apรฉry's constant via Wedeniwski's summation", apery_w())
Output:
Apรฉry's constant via SymPy's zeta:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
Apรฉry's constant via reciprocal cubes:
1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111
Apรฉry's constant via Markov's summation:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
Apรฉry's constant via Wedeniwski's summation:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
## Raku
sub postfix:<!> (Int $n) { (constant f = 1, |[\ร] 1..*)[$n] }
say 'Actual value to 100 decimal places:';
say '1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581';
say "\nFirst 1000 terms of ฮถ(3) truncated to 100 decimal places. (accurate to 6 decimal places):";
say (1..1000).map({FatRat.new: 1, .ยณ}).sum.substr: 0, 102;
say "\nFirst 158 terms of Markov / Apรฉry representation truncated to 100 decimal places:";
say (5/2 ร (1..158).map( -> \k { (-1)**(k-1) ร FatRat.new: k!ยฒ, ((2รk)! ร kยณ) } ).sum).substr: 0, 102;
say "\nFirst 20 terms of Wedeniwski representation truncated to 100 decimal places:";
say (1/24 ร ((^20).map: -> \k {
(-1)**k ร FatRat.new: (2รk+1)!ยณ ร (2รk)!ยณ ร k!ยณ ร (126392รkโต + 412708รkโด + 531578รkยณ + 336367รkยฒ + 104000รk + 12463), (3รk+2)! ร (4รk+3)!ยณ
}).sum).substr: 0, 102;
Output:
Actual value to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
First 1000 terms of ฮถ(3) truncated to 100 decimal places. (accurate to 6 decimal places): 1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111
First 158 terms of Markov / Apรฉry representation truncated to 100 decimal places: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
First 20 terms of Wedeniwski representation truncated to 100 decimal places: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
## Sidef
local Num!PREC = 4*101
say "Actual value to 100 decimal places:\n#{zeta(3)}"
say "\nFirst 1000 terms of ฮถ(3) truncated to 100 decimal places. (accurate to 6 decimal places):";
say sum(1..1000, {|k| 1/k**3 }).as_float(101)
say "\nFirst 158 terms of Markov / Apรฉry representation truncated to 100 decimal places:";
say ((5/2)*sum(1..158, {|k|
(-1)**(k-1) * (k!**2 / ((2*k)! * k**3))
}) -> as_float(101))
say "\nFirst 20 terms of Wedeniwski representation truncated to 100 decimal places:";
say ((1/24)*sum(^20, {|k|
(-1)**k * (2*k + 1)!**3 * (2*k)!**3 * k!**3 * (
126392*k**5 + 412708*k**4 + 531578*k**3 + 336367*k**2 + 104000*k + 12463
) / ((3*k + 2)! * (4*k + 3)!**3)
}) -> as_float(101))
Output:
Actual value to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
First 1000 terms of ฮถ(3) truncated to 100 decimal places. (accurate to 6 decimal places):
1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473112
First 158 terms of Markov / Apรฉry representation truncated to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
First 20 terms of Wedeniwski representation truncated to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
## Wren
Library: Wren-big
import "./big" for BigInt, BigRat
var apery = Fn.new { |n|
var sum = BigRat.zero
for (k in 1..n) sum = sum + BigRat.new(1, k*k*k)
System.print("First %(n) terms of ฮถ(3) truncated to 100 decimal places (accurate to 6 decimal places):")
System.print(sum.toDecimal(100, false))
System.print()
}
var markov = Fn.new { |n|
var fact = BigInt.one
var fact2 = BigInt.one
var sign = BigInt.minusOne
var sum = BigRat.zero
for (k in 1..n) {
sign = sign * BigInt.minusOne
fact = fact * k
var num = fact.square * sign
var mult = 2 * k * (2*k - 1)
fact2 = fact2 * mult
var cube = k * k * k
var den = fact2 * cube
sum = sum + BigRat.new(num, den)
}
sum = sum * BigRat.new(5, 2)
System.print("First %(n) terms of Markov / Apรฉry representation truncated to 100 decimal places:")
System.print(sum.toDecimal(100, false))
System.print()
}
var wedeniwski = Fn.new { |n|
var fact = BigInt.one
var fact2 = BigInt.one
var sign = BigInt.minusOne
var sum = BigRat.zero
var mult = 1
for (k in 0..n-1) {
sign = sign * BigInt.minusOne
if (k > 0) {
fact = fact * k
mult = 2 * k * (2*k - 1)
fact2 = fact2 * mult
}
var fact3 = fact2 * (2*k + 1)
var num = sign * fact3.cube * fact2.cube * fact.cube
var cube = k * k * k
var quad = cube * k
var pent = quad * k
var tmp = 126392*pent + 412708*quad + 531578*cube + 336367*k*k + 104000*k + 12463
num = num * tmp
var den = BigInt.factorial(3*k + 2) * BigInt.factorial(4*k + 3).cube
sum = sum + BigRat.new(num, den)
}
sum = sum / 24
System.print("First %(n) terms of Wedeniwski representation truncated to 100 decimal places:")
System.print(sum.toDecimal(100, false))
System.print()
}
System.print("Actual value to 100 decimal places:")
System.print("1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581")
System.print()
apery.call(1000)
markov.call(158)
wedeniwski.call(20)
Output:
Actual value to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
First 1000 terms of ฮถ(3) truncated to 100 decimal places (accurate to 6 decimal places):
1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111
First 158 terms of Markov / Apรฉry representation truncated to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
First 20 terms of Wedeniwski representation truncated to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
We can also verify the actual value of Apรฉry's constant to 100 decimal places using MPFR which has a zeta function built in. A precision of 324 bits is needed.
Library: Wren-gmp
import "./gmp" for Mpf
var x = Mpf.new(324)
var zeta = x.zetaUi(3)
System.print(zeta.toString(101))
Output:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
`
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# Minimum possible value T such that at most D Partitions of the Array having at most sum T is possible
โข Difficulty Level : Easy
โข Last Updated : 07 Sep, 2021
Given an array arr[] consisting of N integers and an integer D, the task is to find the least integer T such that the entire array can be partitioned into at most D subarrays from the given array with sum atmost T.
Examples:
Input: D = 5, arr[] = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Output: 15
Explanation:
If T = 15, then 5 subarrays {{1, 2, 3, 4, 5}, {6, 7}, {8}, {9}, {10}}
Input: D = 2, arr[] = {1, 1, 1, 1, 1}
Output:
Explanation:
If T = 3, then the 2 partitions are {{1, 1, 1}, {1, 1}}
Naive Approach: The idea is to check for all possible values of T in the range [max(element), sum(element)] whether it is possible to have at most D partition.
Time Complexity: O( N*R )
Auxiliary Space: O(1)
Efficient Approach: The idea is to use Binary search to optimize the above approach. Follow the steps below to solve the problem:
โข Consider T in the range R = [ max(element), sum(element) ].
โข If median T can generate at most D partitions, then check for a median lesser than T.
โข Otherwise, check for a median greater than the current median T.
โข Return the possible value of T at the end.
Below is the implementation of the above approach:
## C++
`// C++ Program for the above approach``#include ``using` `namespace` `std;`ย `// Function to check if the array``// can be partitioned into atmost d``// subarray with sum atmost T``bool` `possible(``int` `T, ``int` `arr[], ``int` `n, ``int` `d)``{``ย ย ย ย ``// Initial partition``ย ย ย ย ``int` `partition = 1;`ย `ย ย ย ย ``// Current sum``ย ย ย ย ``int` `total = 0;`ย `ย ย ย ย ``for` `(``int` `i = 0; i < n; i++) {`ย `ย ย ย ย ย ย ย ย ``total = total + arr[i];`ย `ย ย ย ย ย ย ย ย ``// If current sum exceeds T``ย ย ย ย ย ย ย ย ``if` `(total > T) {`ย `ย ย ย ย ย ย ย ย ย ย ย ย ``// Create a new partition``ย ย ย ย ย ย ย ย ย ย ย ย ``partition = partition + 1;``ย ย ย ย ย ย ย ย ย ย ย ย ``total = arr[i];`ย `ย ย ย ย ย ย ย ย ย ย ย ย ``// If count of partitions``ย ย ย ย ย ย ย ย ย ย ย ย ``// exceed d``ย ย ย ย ย ย ย ย ย ย ย ย ``if` `(partition > d) {``ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ``return` `false``;``ย ย ย ย ย ย ย ย ย ย ย ย ``}``ย ย ย ย ย ย ย ย ``}``ย ย ย ย ``}`ย `ย ย ย ย ``return` `true``;``}`ย `// Function to find the minimum``// possible value of T``void` `calcT(``int` `n, ``int` `d, ``int` `arr[])``{``ย ย ย ย ``// Stores the maximum and``ย ย ย ย ``// total sum of elements``ย ย ย ย ``int` `mx = -1, sum = 0;`ย `ย ย ย ย ``for` `(``int` `i = 0; i < n; i++) {`ย `ย ย ย ย ย ย ย ย ``// Maximum element``ย ย ย ย ย ย ย ย ``mx = max(mx, arr[i]);`ย `ย ย ย ย ย ย ย ย ``// Sum of all elements``ย ย ย ย ย ย ย ย ``sum = sum + arr[i];``ย ย ย ย ``}`ย `ย ย ย ย ``int` `lb = mx;``ย ย ย ย ``int` `ub = sum;`ย `ย ย ย ย ``while` `(lb < ub) {`ย `ย ย ย ย ย ย ย ย ``// Calculate medianย T``ย ย ย ย ย ย ย ย ``int` `T_mid = lb + (ub - lb) / 2;`ย `ย ย ย ย ย ย ย ย ``// If atmost D partitions possible``ย ย ย ย ย ย ย ย ``if` `(possible(T_mid, arr, n, d) == ``true``) {`ย `ย ย ย ย ย ย ย ย ย ย ย ย ``// Check for smaller T``ย ย ย ย ย ย ย ย ย ย ย ย ``ub = T_mid;``ย ย ย ย ย ย ย ย ``}`ย `ย ย ย ย ย ย ย ย ``// Otherwise``ย ย ย ย ย ย ย ย ``else` `{`ย `ย ย ย ย ย ย ย ย ย ย ย ย ``// Check for larger T``ย ย ย ย ย ย ย ย ย ย ย ย ``lb = T_mid + 1;``ย ย ย ย ย ย ย ย ``}``ย ย ย ย ``}`ย `ย ย ย ย ``// Print the minimum T required``ย ย ย ย ``cout << lb << endl;``}`ย `// Driver Code``int` `main()``{``ย ย ย ย ``int` `d = 2;``ย ย ย ย ``int` `arr[] = { 1, 1, 1, 1, 1 };`ย `ย ย ย ย ``int` `n = ``sizeof` `arr / ``sizeof` `arr[0];``ย ย ย ย ``// Function call``ย ย ย ย ``calcT(n, d, arr);`ย `ย ย ย ย ``return` `0;``}`
## Java
`// Java program for the above approach``import` `java.util.*;``import` `java.io.*;`ย `class` `GFG{``ย ย ย ย `ย `// Function to check if the array``// can be partitioned into atmost d``// subarray with sum atmost T``public` `static` `boolean` `possible(``int` `T, ``int` `arr[],``ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ``int` `n, ``int` `d)``{``ย ย ย ย `ย `ย ย ย ย ``// Initial partition``ย ย ย ย ``int` `partition = ``1``;`ย `ย ย ย ย ``// Current sum``ย ย ย ย ``int` `total = ``0``;`ย `ย ย ย ย ``for``(``int` `i = ``0``; i < n; i++)``ย ย ย ย ``{``ย ย ย ย ย ย ย ย ``total = total + arr[i];`ย `ย ย ย ย ย ย ย ย ``// If current sum exceeds T``ย ย ย ย ย ย ย ย ``if` `(total > T)``ย ย ย ย ย ย ย ย ``{``ย ย ย ย ย ย ย ย ย ย ย ย `ย `ย ย ย ย ย ย ย ย ย ย ย ย ``// Create a new partition``ย ย ย ย ย ย ย ย ย ย ย ย ``partition = partition + ``1``;``ย ย ย ย ย ย ย ย ย ย ย ย ``total = arr[i];`ย `ย ย ย ย ย ย ย ย ย ย ย ย ``// If count of partitions``ย ย ย ย ย ย ย ย ย ย ย ย ``// exceed d``ย ย ย ย ย ย ย ย ย ย ย ย ``if` `(partition > d)``ย ย ย ย ย ย ย ย ย ย ย ย ``{``ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ``return` `false``;``ย ย ย ย ย ย ย ย ย ย ย ย ``}``ย ย ย ย ย ย ย ย ``}``ย ย ย ย ``}``ย ย ย ย ``return` `true``;``}`ย `// Function to find the minimum``// possible value of T``public` `static` `void` `calcT(``int` `n, ``int` `d,``ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ``int` `arr[])``{``ย ย ย ย `ย `ย ย ย ย ``// Stores the maximum and``ย ย ย ย ``// total sum of elements``ย ย ย ย ``int` `mx = -``1``, sum = ``0``;`ย `ย ย ย ย ``for``(``int` `i = ``0``; i < n; i++)``ย ย ย ย ``{``ย ย ย ย ย ย ย ย `ย `ย ย ย ย ย ย ย ย ``// Maximum element``ย ย ย ย ย ย ย ย ``mx = Math.max(mx, arr[i]);`ย `ย ย ย ย ย ย ย ย ``// Sum of all elements``ย ย ย ย ย ย ย ย ``sum = sum + arr[i];``ย ย ย ย ``}`ย `ย ย ย ย ``int` `lb = mx;``ย ย ย ย ``int` `ub = sum;`ย `ย ย ย ย ``while` `(lb < ub)``ย ย ย ย ``{``ย ย ย ย ย ย ย ย `ย `ย ย ย ย ย ย ย ย ``// Calculate median T``ย ย ย ย ย ย ย ย ``int` `T_mid = lb + (ub - lb) / ``2``;`ย `ย ย ย ย ย ย ย ย ``// If atmost D partitions possible``ย ย ย ย ย ย ย ย ``if` `(possible(T_mid, arr, n, d) == ``true``)``ย ย ย ย ย ย ย ย ``{``ย ย ย ย ย ย ย ย ย ย ย ย `ย `ย ย ย ย ย ย ย ย ย ย ย ย ``// Check for smaller T``ย ย ย ย ย ย ย ย ย ย ย ย ``ub = T_mid;``ย ย ย ย ย ย ย ย ``}`ย `ย ย ย ย ย ย ย ย ``// Otherwise``ย ย ย ย ย ย ย ย ``else``ย ย ย ย ย ย ย ย ``{``ย ย ย ย ย ย ย ย ย ย ย ย `ย `ย ย ย ย ย ย ย ย ย ย ย ย ``// Check for larger T``ย ย ย ย ย ย ย ย ย ย ย ย ``lb = T_mid + ``1``;``ย ย ย ย ย ย ย ย ``}``ย ย ย ย ``}``ย ย ย ย `ย `ย ย ย ย ``// Print the minimum T required``ย ย ย ย ``System.out.println(lb);``}`ย `// Driver code``public` `static` `void` `main(String args[])``{``ย ย ย ย ``int` `d = ``2``;``ย ย ย ย ``int` `arr[] = { ``1``, ``1``, ``1``, ``1``, ``1` `};`ย `ย ย ย ย ``int` `n = arr.length;``ย ย ย ย `ย `ย ย ย ย ``// Function call``ย ย ย ย ``calcT(n, d, arr);``}``}`ย `// This code is contributed by decoding`
## Python3
`# Python3 program for the above approach`ย `# Function to check if the array``# can be partitioned into atmost d``# subarray with sum atmost T``def` `possible(T, arr, n, d):``ย ย ย ย `ย `ย ย ย ย ``# Initial partition``ย ย ย ย ``partition ``=` `1``;`ย `ย ย ย ย ``# Current sum``ย ย ย ย ``total ``=` `0``;`ย `ย ย ย ย ``for` `i ``in` `range``(n):``ย ย ย ย ย ย ย ย ``total ``=` `total ``+` `arr[i];`ย `ย ย ย ย ย ย ย ย ``# If current sum exceeds T``ย ย ย ย ย ย ย ย ``if` `(total > T):`ย `ย ย ย ย ย ย ย ย ย ย ย ย ``# Create a new partition``ย ย ย ย ย ย ย ย ย ย ย ย ``partition ``=` `partition ``+` `1``;``ย ย ย ย ย ย ย ย ย ย ย ย ``total ``=` `arr[i];`ย `ย ย ย ย ย ย ย ย ย ย ย ย ``# If count of partitions``ย ย ย ย ย ย ย ย ย ย ย ย ``# exceed d``ย ย ย ย ย ย ย ย ย ย ย ย ``if` `(partition > d):``ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ``return` `False``;`ย `ย ย ย ย ``return` `True``;`ย `# Function to find the minimum``# possible value of T``def` `calcT(n, d, arr):``ย ย ย ย `ย `ย ย ย ย ``# Stores the maximum and``ย ย ย ย ``# total sum of elements``ย ย ย ย ``mx ``=` `-``1``; ``sum` `=` `0``;`ย `ย ย ย ย ``for` `i ``in` `range``(n):``ย ย ย ย ย ย ย ย `ย `ย ย ย ย ย ย ย ย ``# Maximum element``ย ย ย ย ย ย ย ย ``mx ``=` `max``(mx, arr[i]);`ย `ย ย ย ย ย ย ย ย ``# Sum of all elements``ย ย ย ย ย ย ย ย ``sum` `=` `sum` `+` `arr[i];`ย `ย ย ย ย ``lb ``=` `mx;``ย ย ย ย ``ub ``=` `sum``;`ย `ย ย ย ย ``while` `(lb < ub):`ย `ย ย ย ย ย ย ย ย ``# Calculate median T``ย ย ย ย ย ย ย ย ``T_mid ``=` `lb ``+` `(ub ``-` `lb) ``/``/` `2``;`ย `ย ย ย ย ย ย ย ย ``# If atmost D partitions possible``ย ย ย ย ย ย ย ย ``if` `(possible(T_mid, arr, n, d) ``=``=` `True``):`ย `ย ย ย ย ย ย ย ย ย ย ย ย ``# Check for smaller T``ย ย ย ย ย ย ย ย ย ย ย ย ``ub ``=` `T_mid;`ย `ย ย ย ย ย ย ย ย ``# Otherwise``ย ย ย ย ย ย ย ย ``else``:`ย `ย ย ย ย ย ย ย ย ย ย ย ย ``# Check for larger T``ย ย ย ย ย ย ย ย ย ย ย ย ``lb ``=` `T_mid ``+` `1``;`ย `ย ย ย ย ``# Print the minimum T required``ย ย ย ย ``print``(lb);`ย `# Driver code``if` `__name__ ``=``=` `'__main__'``:``ย ย ย ย `ย `ย ย ย ย ``d ``=` `2``;``ย ย ย ย ``arr ``=` `[ ``1``, ``1``, ``1``, ``1``, ``1` `];`ย `ย ย ย ย ``n ``=` `len``(arr);`ย `ย ย ย ย ``# Function call``ย ย ย ย ``calcT(n, d, arr);`ย `# This code is contributed by Rajput-Ji`
## C#
`// C# program for the above approach``using` `System;`ย `class` `GFG{``ย ย ย ย `ย `// Function to check if the array``// can be partitioned into atmost d``// subarray with sum atmost T``public` `static` `bool` `possible(``int` `T, ``int` `[]arr,``ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ``int` `n, ``int` `d)``{``ย ย ย ย `ย `ย ย ย ย ``// Initial partition``ย ย ย ย ``int` `partition = 1;`ย `ย ย ย ย ``// Current sum``ย ย ย ย ``int` `total = 0;`ย `ย ย ย ย ``for``(``int` `i = 0; i < n; i++)``ย ย ย ย ``{``ย ย ย ย ย ย ย ย ``total = total + arr[i];`ย `ย ย ย ย ย ย ย ย ``// If current sum exceeds T``ย ย ย ย ย ย ย ย ``if` `(total > T)``ย ย ย ย ย ย ย ย ``{``ย ย ย ย ย ย ย ย ย ย ย ย `ย `ย ย ย ย ย ย ย ย ย ย ย ย ``// Create a new partition``ย ย ย ย ย ย ย ย ย ย ย ย ``partition = partition + 1;``ย ย ย ย ย ย ย ย ย ย ย ย ``total = arr[i];`ย `ย ย ย ย ย ย ย ย ย ย ย ย ``// If count of partitions``ย ย ย ย ย ย ย ย ย ย ย ย ``// exceed d``ย ย ย ย ย ย ย ย ย ย ย ย ``if` `(partition > d)``ย ย ย ย ย ย ย ย ย ย ย ย ``{``ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ``return` `false``;``ย ย ย ย ย ย ย ย ย ย ย ย ``}``ย ย ย ย ย ย ย ย ``}``ย ย ย ย ``}``ย ย ย ย ``return` `true``;``}`ย `// Function to find the minimum``// possible value of T``public` `static` `void` `calcT(``int` `n, ``int` `d,``ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ``int` `[]arr)``{``ย ย ย ย `ย `ย ย ย ย ``// Stores the maximum and``ย ย ย ย ``// total sum of elements``ย ย ย ย ``int` `mx = -1, sum = 0;`ย `ย ย ย ย ``for``(``int` `i = 0; i < n; i++)``ย ย ย ย ``{``ย ย ย ย ย ย ย ย `ย `ย ย ย ย ย ย ย ย ``// Maximum element``ย ย ย ย ย ย ย ย ``mx = Math.Max(mx, arr[i]);`ย `ย ย ย ย ย ย ย ย ``// Sum of all elements``ย ย ย ย ย ย ย ย ``sum = sum + arr[i];``ย ย ย ย ``}`ย `ย ย ย ย ``int` `lb = mx;``ย ย ย ย ``int` `ub = sum;`ย `ย ย ย ย ``while` `(lb < ub)``ย ย ย ย ``{``ย ย ย ย ย ย ย ย `ย `ย ย ย ย ย ย ย ย ``// Calculate median T``ย ย ย ย ย ย ย ย ``int` `T_mid = lb + (ub - lb) / 2;`ย `ย ย ย ย ย ย ย ย ``// If atmost D partitions possible``ย ย ย ย ย ย ย ย ``if` `(possible(T_mid, arr, n, d) == ``true``)``ย ย ย ย ย ย ย ย ``{``ย ย ย ย ย ย ย ย ย ย ย ย `ย `ย ย ย ย ย ย ย ย ย ย ย ย ``// Check for smaller T``ย ย ย ย ย ย ย ย ย ย ย ย ``ub = T_mid;``ย ย ย ย ย ย ย ย ``}`ย `ย ย ย ย ย ย ย ย ``// Otherwise``ย ย ย ย ย ย ย ย ``else``ย ย ย ย ย ย ย ย ``{``ย ย ย ย ย ย ย ย ย ย ย ย `ย `ย ย ย ย ย ย ย ย ย ย ย ย ``// Check for larger T``ย ย ย ย ย ย ย ย ย ย ย ย ``lb = T_mid + 1;``ย ย ย ย ย ย ย ย ``}``ย ย ย ย ``}``ย ย ย ย `ย `ย ย ย ย ``// Print the minimum T required``ย ย ย ย ``Console.WriteLine(lb);``}`ย `// Driver code``public` `static` `void` `Main(String []args)``{``ย ย ย ย ``int` `d = 2;``ย ย ย ย ``int` `[]arr = { 1, 1, 1, 1, 1 };`ย `ย ย ย ย ``int` `n = arr.Length;``ย ย ย ย `ย `ย ย ย ย ``// Function call``ย ย ย ย ``calcT(n, d, arr);``}``}`ย `// This code is contributed by 29AjayKumar`
## Javascript
``
Output
`3`
Time complexity: O( N*log(sum) )
Auxiliary Space: O(1)
My Personal Notes arrow_drop_up
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# Homework Help: How to calculate the rebound distance of the bungge jump
1. Oct 3, 2005
### superkui
Hie all, curently i working on a mini project which require us to design a bungee jumping game where we had to decide all the data by myself.
here i got a few question wanna ask n hope someone can help me a bit..
a) how to calcualte the rebound force and distance he will rebound??as i givin the k = 150 N/m, original cord length = 17m, elongation = 23m..n the distance between the platform and the ground = 45m.
b)if there is a obstacle is 5m from the platform, how to avoid it??
2. Oct 3, 2005
### Gokul43201
Staff Emeritus
1. What do you mean by "rebound force" ? Please define this clearly.
2. There is not enough info to determine the rebound height. That is a function of the damping coefficient of the bungee cord, and the answer in the absence of damping will be the original height.
3. Oct 4, 2005
### superkui
hie there..sorry for the unconvenient. I wanna find out the maximum rebound height since i wil have a obstacle 5m below the jumping platform and need to avoid it. i know after the cord reach the maximum length, the cord wil retract to it's original length, which mean 17m. But when the cord retract, there is a acceleration.. so there is a force when the cord is retract.
4. Oct 5, 2005
### Gokul43201
Staff Emeritus
It's hard to understand the question without a picture. Can you attach one ?
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Question
# The equation of common tangent to the curves y2=16x and xy= ฤยย4, is :
A
x2y+16=0
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B
x+y+4=0
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C
2xy+2=0
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D
xy+4=0
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Solution
## The correct option is D xโy+4=0Let the equation of the tangent to y2=16x be y=mx+4m This is also a tangent to the curve xy=โ4 โx(mx+4m)=โ4โm2x2+4x+4m=0 For tangent, D=0โ16โ16m3=0โm=1 So, equation of common tangent is y=x+4
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12 Replies Latest reply: Jan 30, 2012 3:30 AM by andrelam
# How to convert an exponential noย to decimal
Hi friends, I m facing a problem .Please suggest me the solution for the problem.
hey guys,how to convert an exponential number to decimal in
qlikview?
i tried mantissa*pow(10,exp) formula but mantissa that i
separate from the original field
,it is showing that it is neither a text nor a number.
Kamal Naithani
โข ###### How to convert an exponential noย to decimal
Hi,
Its very simple. Go to the Settings -> Document Properties -> Numberย Tab-> Click on the field which have Exponential value and then click on Number.
Regards,
Kaushik Solanki
โข ###### How to convert an exponential noย to decimal
That does not work for this format:
+1.961776E+03
+1.855197E+04
-1.304177E+07
-7.642861E+06
I need to read in these as number, can someone help me?
โข ###### How to convert an exponential noย to decimal
HI,
Try this.
Data:
NUMBERS
+1.961776E+03
+1.855197E+04
-1.304177E+07
-7.642861E+06
];
Dat1:
Load Num(num#(NUMBERS)) as Num Resident Data;
Regards,
Kaushik Solanki
โข ###### How to convert an exponential noย to decimal
Sorry, im a complete qlikview noob, can you modify this for me sow it reads in the exponetioal values: (meas, mean, lower, upper)
SET ThousandSep='.';
SET DecimalSep=',';
SET MoneyThousandSep='.';
SET MoneyDecimalSep=',';
SET MoneyFormat='โฌ #.##0,00;โฌ #.##0,00-';
SET TimeFormat='h:mm:ss';
SET DateFormat='DD-MM-YYYY';
SET TimestampFormat='DD-MM-YYYY h:mm:ss[.fff]';
SET MonthNames='jan;feb;mrt;apr;mei;jun;jul;aug;sep;okt;nov;dec';
SET DayNames='ma;di;wo;do;vr;za;zo';
@2 as name,
@3,
@4,
@5 as meas,
@6 as mean,
@7 as lower,
@8 as upper,
@9,
@10,
@11,
@12,
@13,
@14,
@15,
@16,
@17,
@18,
@19
FROM
"C:\Documents and Settings\a.lam\Desktop\temp\A000*"
(txt, codepage is 1252, no labels, delimiter is '|', msq);
โข ###### How to convert an exponential noย to decimal
Hi,
SET ThousandSep='.';
SET DecimalSep=',';
SET MoneyThousandSep='.';
SET MoneyDecimalSep=',';
SET MoneyFormat='โฌ #.##0,00;โฌ #.##0,00-';
SET TimeFormat='h:mm:ss';
SET DateFormat='DD-MM-YYYY';
SET TimestampFormat='DD-MM-YYYY h:mm:ss[.fff]';
SET MonthNames='jan;feb;mrt;apr;mei;jun;jul;aug;sep;okt;nov;dec';
SET DayNames='ma;di;wo;do;vr;za;zo';
@2 as name,
@3,
@4,
num(num#(@5)) as meas,
num(num#(@6)) as mean,
num(num#(@7)) as lower,
num(num#(@8)) as upper,
@9,
@10,
@11,
@12,
@13,
@14,
@15,
@16,
@17,
@18,
@19
FROM
"C:\Documents and Settings\a.lam\Desktop\temp\A000*"
(txt, codepage is 1252, no labels, delimiter is '|', msq);
Regards,
Kaushik Solanki
โข ###### How to convert an exponential noย to decimal
Thank you for your help, but.. now the values are gone, it shows "-" in my table object.
โข ###### How to convert an exponential noย to decimal
HI,
Can you send me the txt file.
Regards,
Kaushik Solanki
โข ###### Re: How to convert an exponential noย to decimal
Hello, thanks again for your efford!. The file is attached.
Best Regards,
Andrรฉ Lam
โข ###### How to convert an exponential noย to decimal
HI,
Its working my side.
Regards,
Kaushik Solanki
โข ###### How to convert an exponential noย to decimal
But what can i do?, i used the code from your post..
โข ###### Re: How to convert an exponential noย to decimal
what version are you on ?
โข ###### Re: How to convert an exponential noย to decimal
11.00.11154.0
I got it working:
Num(Left(@5, Index(@5, 'E') -1) * Pow(10, TextBetween(@5, 'E', ''))) as meas,
Num(Left(@6, Index(@6, 'E') -1) * Pow(10, TextBetween(@6, 'E', ''))) as mean,
Num(Left(@7, Index(@7, 'E') -1) * Pow(10, TextBetween(@7, 'E', ''))) as lower,
Num(Left(@8, Index(@8, 'E') -1) * Pow(10, TextBetween(@8, 'E', ''))) as upper,
For now i'm happy, although it would have been easier to just say "this is a number".....
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| 3,850
|
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| 2.9375
| 3
|
CC-MAIN-2018-39
|
latest
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en
| 0.653172
|
https://consultglp.com/2020/03/
| 1,695,697,017,000,000,000
|
text/html
|
crawl-data/CC-MAIN-2023-40/segments/1695233510130.53/warc/CC-MAIN-20230926011608-20230926041608-00554.warc.gz
| 210,423,581
| 23,799
|
## Training and consultancy for testing laboratories.
### Theory behind decision rule simply explained โ Part B
There are a few approaches for decision making leading to a conformity statement after testing.
Simple approaches for binary decision rule involving comments of pass/fail, compliant/non-compliant:
โข A result implies non-compliance with an upper limit if the measured value exceeds the limit by the expanded uncertainty. See Figure 1A.ย This is a clear-cut case.
โข A result equal to or above the upper limit implies non-compliance and a result below the limit implies compliance โ provided that uncertainty is below a specified value or assumed zero.ย This is normally used where the uncertainty is so small compared with the limit that the risk of making a wrong decision is acceptable. For example, the relative uncertainty of the measured value is 1-2% whilst the Type I error that you are prepared to take is 5%.ย See Figure 1B.
However, to use such a rule without specifying the maximum permitted value of the uncertainty would mean the risk (probability) of making a wrong decision would not be known.
More complicated approaches for decision rule by use of Guard Bands:
Many learned organizations like ILAC, Eurachem, etc. have suggested to consider incorporating some tolerance limits (or interval) or guard bands (say, +g) added to the nominal specification for risk decision. ย In this instance, a rejection zone can be defined as starting from the specification limit L plus or minus an amountg (the Guard Band).ย The purpose of establishing such an โexpandedโ or โconservativeโ error on the specification value is to draw โsafeโ conclusions concerning whether measurement errors are within acceptable limits with a calculated risk as agreed by both the customers and the laboratory concerned.
The value of g is chosen so that for a measurement result greater than or equal to L + g, the probability of false rejection is less than or equal to alpha (Type I error) which is the accepted risk level.
In general, g will be a multiple of the standard uncertainty of the test parameter, u. The multiplying factor can be 1.645 or 1.65 (95% confidence) or 3.3 (>99% confidence).ย That is to say that the amount of uncertainty in the measurement process and where the measurement result lies with respect to the tolerance limit set help to determine the probability of an incorrect decision.
A situation is, for example, when you set your guard band g to be the expanded uncertainty of the measurement, that is U = 2u above the upper limit of specification.ย In this case, your estimated critical measurement result plus 1.645u with 95% confidence is well inside the L + g zone, and hence, your risk of making a wrong decision is at 5%.ย This is shown graphically in Figure 2A below:
Often it is the customer who would specify such a tolerance limit as in Figure 2A, indicating that he would be happy to accept when such tolerance level or guard band is above the upper specification limit or below the lower specification limit in the rejection zones.ย Hence, the risk is at the customerโs side.ย It is also known as โrelax rejection zoneโ which covers the Type II (beta) error.
However, if the laboratory operator is to set his own risk limit, it is best for him to set the tolerance limit or guard band below the upper specification level or above the lower specification level to safeguard his own interest.ย It is known as โconservative or stringent acceptance zoneโ, leading to the Type I (alpha) error.
How to estimate the critical value for acceptance?
Letโs illustrate it via a worked example.
One of the toxic elements in soil is cadmium (Cd).ย Let the upper acceptable limit on the total Cd in soil required by the environmental consultant client as 2.0 mg/kg on dried matter.ย The measurand is therefore the total Cd content in soil by ICP-OES method.
Upon analysis, the average value of Cd content in soil samples, say, was found to be 1.81 mg/kg on dried basis, and the uncertainty of measurement U was 0.20 mg/kg with a coverage factor of 2 (95% confidence). Hence, the standard uncertainty of the measurement = 0.20 / 2 = 0.10 mg/kg. This standard uncertainty included both sampling and analytical uncertainties.
Our Decision ruleThe critical value or the decision limit was the Cd concentration where it could be decided with a confidence of approximately 95% (alpha=0.05) that the sample batch had a concentration below the set upper limit of 2 mg/kg.
The guard band g is then calculated as:
1.645 x u = 1.645 x 0.10 = 0.165 mg/kg
where k = z = 1.645 for one-tailed value of normal probability distribution at 95% confidence.
The decision (critical) limit therefore = 2.0 โ 0.165 = 1.84 mg/kg.
The client would then be duly informed and agreed that all reported values below this critical limit value of 1.84 mg/kg were in the acceptance zone.ย Hence, the test result of 1.81 mg/kg in this study was in compliance with the Cd specification limit of 2.0 mg/kg maximum.
Suggested types of guard bands
The guard band is often based on a multiple, r, of the expanded measurement uncertainty, U where g = rU
For a binary decision rule, a measurement result below the acceptance limit AL = (L-g) is accepted.
The above example of g = U is quite commonly used, but there may be cases where a multiplier other than 1 is more appropriate.ย ILAC Guide G08:09/2019 titled โGuidelines on decision rules and statements of conformityโ provides a table showing examples of different guard bands to achieve certain levels of specific risks, based on the customer application, as reproduced in Figure 3 Table 1 below.ย Note that probability of False Accept PFA refers to false positive or Type I error.
It may be noted that the multiplying factor of 0.83 in the guard band of 0.83U as given by ISO 14253-1:2017 is derived by calculation of 1.65/2, where 1.645 has been approximated to 1.65 and 2 is the coverage factor of 1.96 rounded up to the nearest integer, for 95% confidence interval.
### A worked example for decision rule on conformity statement
Consider a measurement value y = 2.70ppm with a standard uncertainty of u(y) = 0.20ppm.ย (Its expanded uncertainty = k x 0.20ppm = 2 x 0.20ppm = 0.40ppm where coverage factor k = 2 at 95% confidence). It is also given that the single tolerance or specification upper limit of Tu = 3.0ppm.
Assuming the normal probability distribution data and a type I error alpha = 0.05 (5%), we are to make a statement of specification conformity at probability of (1-alpha) or 0.95 (95%).
Our decision rule is that :ย โAcceptance if the hypothesis Ho: P(y< 3.0ppm) > 0.95โis true.
Use Microsoft Excel spreadsheet function: โ= 1-NORM.DIST(2.7,3.0,0.2,TRUE)โ to calculate P(y< 3.0ppm) to get 0.933 or 93.3%.ย Note that the function โ=NORM.DIST(2.7,3.0,0.2,TRUE)โ gives the cumulative area under the curve from far left to right for a value of 0.067 approximately.
Alternatively, we can also calculate a normalized z -value as (2.7 โ 3.0)/0.2 = โ 1.50, and look up the one-tailed normal distribution table for cumulative probability under the curve with z =|1.5| which gives 0.5000 + 0.4332 = 0.9332, as a normal distribution curve is symmetrical in shape. See Appendix A for the normal distribution cumulative table. In fact, we would get the same answer if we were to use the Excel function โ=1- NORM.DIST(-1.5,0,1,TRUE)โ as well.
Since 93.3% < 95.0%, the Ho is rejected, i.e. the sample result of 2.70ppm can be declared non-compliant with the specification limit, or put it more mildly, โnot possible to state complianceโ or โconditional passโ or some other qualification wordings!
If, for discussion sake, the measured value was 2.60ppm, instead. Would it be within the upper specification limit of 3.0ppm by the above evaluation?
Indeed, by following the above reasoning, we would find that the normalized z-value as (2.6-3.0)/0.2 = โ 2.0 and the cumulative area under the curve was 0.5000 + 0.4772 = 0.977 which is larger than 0.950.ย Therefore, the Ho is not rejected, i.e. the sample or test item is declared in compliant with the specification limit.
What is the critical acceptable value Xppm in order not to get Ho rejected?
The task will be simple if we know how to find the critical z -value in a normal distribution curve where the area under the curve on the right tail is 0.05 out of 1.00, or 5%, as we have fixed our Type I (alpha) risk as 5%.
Reading from the normal distribution cumulative table in Appendix A, we note that when z = 1.645, the area under the curve is 0.5000 + 0.4500 = 0.9500.ย Similarly, the absolute value of Excel function โ=NORM.INV(0.05,0,1)โ also gives a |z|-value 0f 1.645.
The critical acceptable value X is then calculated as below:
which gives X = 2.67ppm.
The conclusion therefore is that any test value found to be less than or equal to 2.67ppm will be declared as in compliance with the specification of 3.0ppm maximum with 95% confidence (or 5% error risk). ย Any value found larger than 2.67ppm will be assessed for compliant by considering the higher than 5% risk that the test laboratory is willing to undertake, probably based on some commercial reason.ย In other words, where a confidence level of less than 95% is acceptable to the laboratory, a compliance statement may be possible.ย Decision is entirely yours!
Appendix A
### Theory behind decision rule simply explained โ Part A
All testing and calibration laboratories accredited under ISO/IEC 17025:2017 are required to prepare and implement a set of decision rules when the customer requests for a statement of conformity in the test or calibration report issued.
As the word โconformityโ is defined as โcompliance with standards, rules and lawsโ, a statement of conformity is an expression that clearly describes the state of compliance or non-compliance to a specification, standard, regulatory limits or requirements, after calibration or testing.
Like any decision made, you have to assume a certain amount of risk as you might make a wrong decision. So, how much is a risk that you can comfortably undertake when you issue a statement of conformity in your test or calibration report?
Generally, decision rules give a prescription for the acceptance or rejection of a product based on:
โข the measurement result
โข its uncertainty due to inherent errors (random and/or systematic)
โข the specification (or regulatory) limit or limits, and,
โข the acceptable risk level based on the probability of making a wrong decision
Certainly, you want to minimize our risk in issuing a statement of conformity that is to be proven wrong by others.ย But, what is the type of risk you are answering when making such decision rule?ย In short, it ย is either
โข the supplierโs (laboratoryโs) risk (statistically speaking, false positive or Type I error, alpha) or
โข the consumerโs (customerโs) risk (false negative or Type II error, beta).
From the laboratory service point of view, you should be interested in the Type I (alpha) error to protect your own interest.
Before indulging further in the discussion, letโs take note of an important assumption, that is, the uncertainty of measurement is represented by a normal (Gaussian) probability distribution function, which is consistent with the typical measurement results (being assumed the applicability of the Central Limit Theorem).
After calibration or testing an item with its measurement uncertainty known, our subsequent statement of conformance with a specification or regulatory limits can lead us to 2 possible outcomes:
โข We are right
โข We are wrong
The decision rule made is related to statistical hypothesis testing where we propose a null hypothesis Ho for a situation and an alternative hypothesis H1 should Ho be rejected after some test statistics.ย ย In this case, we can make either a Type I (false POSITIVE or false ALARM, i.e. rejecting null hypothesis Ho when in fact Ho is true) or Type II (false NEGATIVE, i.e. not rejecting Ho when in fact Ho is actually false) errors.
It follows that the probabilities of making the correct decisions are (1 โ alpha) and (1 โ beta), respectively.ย Generally we would take a 5% Type I risk, hence we had alpha = 0.05 and would claim that we have 95% confidence in making this statement of conformity.
In laymanโs language:
โข Type I :ย Deciding that something is NOT OK when it actually is OK, ย given the probability (risk):ย alpha
โข Type II:ย Deciding something is OK when it really was NOT OK, given the probability (risk):ย beta
Figure 1 shows the matrix of such decision making and potential errors involved:
The statistical basis of the decision rules is to determine where the โAcceptance zoneโ and the โRejection zoneโ are, such that if the measurement result lies in the acceptance zone, the product is declared compliant, and, if in the rejection zone, it is declared non-compliant. ย Graphically, it can be shown as in Figure 2 below:
We should not have any issue in deciding the conformity in Case 1 and non-conformity in Case 4 due to a clear cut situation as shown in Figure 2 above, but we need to assess if Cases 2 and 3 are in conformity or not, as illustrated in Figure 3 below for an upper specification limit:
For the situations in Cases 2 and 3, we may include the following thoughts in the decision rule making before considering the amount of risk to be taken in deciding conformity:
โข Making a request for additional measurement(s)
โข Re-evaluating measurement uncertainty to narrow the range, if possible
โข A manufactured (and tested) product must be compared with an alternative specification to decide on possible sale at a discounted price, as a rejected goods
Part B of this article will discuss both simple and more complicated decision rules that can be made during issuing statement of conformance after testing or calibration. Before that, we shall study a practical worked example.
### R evaluation of Measurement uncertainty
At the recent Eurachem/PUC ISO 17025 training course in Nicosia, Cyprus on 20-21 February 2020, I had learnt something new from Dr Stephen Ellisonโs presentation.
There is a measurement uncertainty package in the R Language, named โmetRologyโ.ย You can download this library when you are in the R environment.
For example, if we were asked to evaluate the uncertainty of the following expression:
expr = A + 2xB + 3xC + D/2
where A = 1, B = 3, C=2, D=11.ย The sensitive coefficients, cโs, from the above expression are thus 1, 2, 3 and ยฝ for A, B, C and D, respectively.
Assuming the standard uncertainties of these parameters are constant at 1/10th of their values, the following steps demonstrate how the combined standard uncertainty can be evaluated.
> library(โmetRologyโ)
Attaching package: โmetRologyโ
The following objects are masked from โpackage:baseโ:
cbind, rbind
> expr<-expression(A+B*2+C*3+D/2)
> x=list(A=1,B=3,C=2,D=11)
> u=lapply(x,function(x) x/10)
> u
\$A
[1] 0.1
\$B
[1] 0.3
\$C
[1] 0.2
\$D
[1] 1.1
>
> u.expr<-uncert(expr,x,u,method=โNUMโ)
> u.expr
Uncertainty evaluation
Call:
uncert.expression(obj = expr, x = x, u = u, method = โNUMโ)
Expression: a + b * 2 + C * 3 + D/2
Evaluation method:ย NUM
Uncertainty budget:
x ย ย ย uย ย ย ย ย cย ย ย ย u.c
Aย ย 1 ย ย 0.1 ย ย 1.0 ย ย 0.10
Bย ย 3 ย ย 0.3 ย ย 2.0 ย ย 0.60
Cย ย 2 ย ย 0.2 ย ย 3.0 ย ย 0.60
D ย 11 ย 1.1 ย ย 0.5 ย 0.55
y:ย 18.5
u(y):ย 1.01612
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https://help.scilab.org/docs/5.3.2/ja_JP/m2sci_max.html
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| 276,588,586
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Change language to:
English - Franรงais - Portuguรชs
See the recommended documentation of this function
Scilab help >> Matlab to Scilab Conversion Tips > Matlab-Scilab equivalents > M > max (Matlab function)
# max (Matlab function)
Maximum
### Matlab/Scilab equivalent
Matlab Scilab `max` `max`
### Particular cases
Matlab max function can work with complexes, what Scilab max can not, so a emulation function called mtlb_max has been written.
Note that in Scilab, second input parameter can give the dimension to use to find the maximum values or another matrix (maximum of two matrices), in Matlab, dimension parameter is given in a third input parameter (in this case, second parameter must be []).
C=max(A)
If A is a matrix, max(A) is equivalent to max(A,[],1) in Matlab whereas in Scilab max(A) gives the maximum value found in A. Matlab max treats the values along the first non-singleton dimension.
### Examples
Matlab Scilab ```A = [1,2,3;4,5,6] C = max(A) C = [4,5,6] C = max(A,[],1) C = [4,5,6] B = [7,8,9;2,3,4] C = max(A,B) C = [7,8,9;4,5,6]``` ```A = [1,2,3;4,5,6] C = max(A) C = 6 C = max(A,"r") C = [4,5,6] B = [7,8,9;2,3,4] C = max(A,B) C = [7,8,9;4,5,6]```
<< M M min (Matlab function) >>
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http://www.solve-variable.com/math-variable/cramer%E2%80%99s-rule/rational-equation-solver.html
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โข order from least to greatest calculator decimals and fractions
โข gcf worksheets
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โข simplify polynomial expressions calculator
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# Maths
Today we are going to be learning how to use the 'part part whole method' in order to help us add two numbers together.
The 'part part whole method' or sometimes known as the 'part whole method' shows how numbers can be split in to parts. Using this method will help children be able to see the relationship between the 'whole' number and the two 'parts' that add together to make the whole. This method can be used to help children with addition as well as subtraction but today we are just going to be focusing on addition.
Watch the video below to help you understand how a part part whole model can help with addition as well number bonds.
## How to use a 'part-whole' model
More maths help for Year One and Reception
Challenge - Take a look at the example done below using the blue cubes.
We now can use our part part model above to help us write our number sentence. Using the example above with the blue cubes I now know that 7 + 3 = 10. I also know that 3 + 7 = 10.
Now have a go yourself trying to write the correct number sentence that matches the example done below with the red cubes.
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# In a trapezium ABCD, O is the point of intersection of AC and BD, AB || CD and AB = 2 ร CD. If the area of โAOB = 84 cm^2 . Find the area of โCOD - Mathematics
Sum
In a trapezium ABCD, O is the point of intersection of AC and BD, AB || CD and AB = 2 ร CD. If the area of โAOB = 84 cm2 . Find the area of โCOD
#### Solution
In โAOB and โCOD, we have
โ OAB = โ OCD (alt. int. โ s)
โ OBA = โ ODC (alt. int. โ s)
โด โAOB ~ โCOD [By AA-similarity]
\Rightarrow \frac{ar\ (\Delta AOB)}{ar\ (\Delta COD)}=(AB^2)/(CD^2)=(2CD)^2/(CD^2)ย [โต AB = 2 ร CD]
=>(4xxCD^2)/(CD^2)=4
โ ar (โCOD) = 1/4 ร ar (โAOB)
=>(1/4xx84)cm^2=21cm^2
Hence, the area of โCOD is 21 cm2 .
Concept: Areas of Similar Triangles
Is there an error in this question or solution?
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# Endurance calculator could generate new product line
Posting in Energy
### Amateur athletes can get hit the wall just like elite ones do. Ben Rapoport has a simple online calculator that can prevent it.
A lot of folks are real excited today about marathoner Ben I. Rapoport (right).
Rapoport, who is both a Harvard MD and MIT Ph.D candidate (the latter degree is in electrical engineering) produced a paper in PLoS Computational Biology with what he calls a formula for doing your best marathon time. (The picture is from Ben's MIT page.)
Many marathoners "hit a wall" or "bonk" in a race's latter stages because they don't have enough carbohydrates in their system to provide ready energy, he writes. Once you start trying to burn fat, you are no longer capable of the high energy output a difficult endurance event requires.
He put this online in an endurance calculator. From your age, your weight, and your resting pulse, the calculator tells you the energy you will need to run a marathon in your targeted time, how much you will have to eat beforehand, and even whether your target time is attainable.
A better estimate of your true capacity is your VO2Max -- your maximum ability to process oxygen -- which is calculated from a treadmill test, he writes.
The present system just estimates that figure. (It calculated mine at 27.)ย You can make that estimate yourself by comparing your resting heart rate to your maximum rate, which is usually 220 beats minus your age. Or see how far you can go on a treadmill in 12 minutes.
It's an important paper. And it wouldn't take much to turn this simple calculator into a valuable product, and its software into a basic component of athletic training technology.
I've done a half-marathon and once even completed a bicycle marathon of 112 miles. But my best endurance days are behind me. These days I'm happy with a 10 km Peachtree Road Race in the summer and maybe a 37 mile ride down the Silver Comet Trail in the fall.
But amateur athletes can get hit the wall just like elite ones do. I hit it a few years ago when I tried to go both ways on the Silver Comet, a total of 74 miles, on a 95 degree day. My dear wife had to come get me.
Proper training toward a goal, and hands-on data concerning game day nutrition, would have prevented that. Rapoport's calculator offers it. All that you need do is estimate the energy requirements of any endurance event, then plug it in.
Rapaport thinks that with proper training I could possibly run a 4:52 marathon if I put away 4,090 calories beforehand. That's a bit more than 4 regular servings of spaghetti with meatballs and tomato sauce.
Cool. But this can go much further.
I can easily see this turned into a hand held device, or a SaaS service, that will let you plug in the type of endurance contest you want to do, follow your training, and then give you your final meal instructions at the end of the process.
The people who run large events like the Peachtree or San Francisco's Bay to Breakers could offer this service to runners and save on medical costs. I once saw someone have a heart attack during a Peachtree run -- it was not pretty.
I know MIT and Harvard are expensive, but a little birdie tells me Ben's tuition bills can be paid for.
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# GATE (TF) Textile 2010 Question Paper Solution | GATE/2010/TF/37
###### Question 37 (Textile Engineering & Fibre Science)
The effective length of cotton with 21 mm lower quartile length and 25% dispersion percentage is
Given in the question-
Lower quartile length=21 mm
Dispersion %=25%
Effective length of cotton(L)=?
By formula-
Dispersion %=Inter quartile length/Effective length
In baer sorter graph-
Effective length=Inter quartile length+Lower quartile length
L=Inter quartile length+21
Inter quartile length=(L-21)
Dispersion %=Inter quartile length/Effective length
25%=(L-21)/L
0.25L=L-21
0.75L=21
L=21/0.75
L=28 mm
i.e.,
Effective length=28 mm (Ans)
### What is effective length in Baer sorter?
In the context of a Baer sorter, the effective length refers to the distance between the starting point of the sorting process and the last accepted defect in the fabric.
The Baer sorter is a machine that is used in the textile industry to detect and remove defects in fabric, such as knots, holes, and stains. As the fabric passes through the machine, it is inspected by a series of cameras or sensors that detect any defects in the material. The fabric is then cut or marked at the location of the defect and removed from the production line.
The effective length is an important parameter in the Baer sorter, as it determines the amount of fabric that is removed from the production line due to defects. A shorter effective length means that more of the fabric is rejected due to defects, which can result in higher production costs and lower yields. Conversely, a longer effective length means that fewer defects are removed from the fabric, which can result in lower production costs and higher yields.
The effective length is typically set by the operator of the Baer sorter and can be adjusted based on the specific requirements of the production process. It is an important factor to consider when using the Baer sorter to ensure that the right balance is struck between product quality and production efficiency.
### What is upper quartile length in textile?
Upper quartile length is a statistical parameter used in the textile industry to measure the length of the longest fibers in a sample of textile material. It is also known as the 75th percentile length or UQL.
To determine the upper quartile length, a representative sample of the textile material is taken and the length of each individual fiber is measured. The fibers are then sorted in ascending order of length, and the upper quartile length is determined by finding the length at which 75% of the fibers are shorter than this length and 25% of the fibers are longer.
The upper quartile length is an important parameter in the textile industry as it provides information about the quality and performance of the fibers in a given material. Longer fibers are generally considered to be of higher quality as they have a greater strength and are less likely to break during processing and use. The upper quartile length is therefore used as an indicator of the quality of the fibers and can be used to compare different types of textile materials.
The upper quartile length is typically measured using a fiber length analyzer, which is a specialized instrument that is designed to measure the length of individual fibers in a sample of textile material. The results of the analysis can be used to inform decisions about the processing and use of the material, such as whether it is suitable for spinning into high-quality yarns or whether it should be used for lower-grade applications.
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# RoyaltyStat Blog
### Posts by Topic
#### Zero Intercept Linear Profit Function
The typical OECD TNMM (CPM in the U.S.) prescribes a linear statistical function to test the arm's length character ofย โnetโ profits (Y) in terms of the net sales (X):
(1)ย ย ย Yi = ฮฑ Xiย consideringย i = 1, 2, โฆ, N comparables
where ฮฑ is the estimated โnetโ profit margin. For simplification, we set aside a random error term that is added to equation (1). The controlled taxpayer ("tested party") is the case N + 1.
#### Non-Linear Profit Function
Instead of equation (1), "net" profits and sales may be represented by a power function:
(2) ย ย Yi = ฮฑ Xiฮฒ
Power functions are pervasive in economic estimates. Equation (2) states that Yi is proportional to Xiฮฒ . In this case, the profit margin is the slope coefficient of equation (2), which below we show is different from ฮฑ. A power function is appropriate e.g. when the selection of comparables to the tested party includes small and large companies or when the residual variance is not constant.
#### Slope Coefficient
The slope coefficient of equation (2) is the derivative of Y with respect to X:
##### (3) ย ย Yiโ = ฮฑ ฮฒ Xiฮฒ โ 1
where Yiโ (variable followed by an apostrophe) is the first derivative.ย We use two rules of exponents (1/X = X1) and (Xm/Xn = Xmn) to obtain a clearer expression for the slope coefficient of the power function:
##### (4) ย ย Yiโ = (ฮฑ ฮฒ Xiฮฒ / Xi) = ฮฒ (Yi / Xi)
In equation (4), the regression coefficient (ฮฒ) is multiplied by a selected comparable profit margin (Yi / Xi). Itโs sensible to use the comparable median or the average (free of outliers) profit margin of the selected comparables as a multiplicative constant of ฮฒ, and thus determine the armโs length profit margin of the โtested partyโ. In this case, a scatter plot of Yi versusย Xiย canย reveal if the power function is a better (more reliable) fit of the company comparable data than the linear prescription of the OECD.
Itโs easy to compute the slope = ฮฒ (Yi / Xi) using Excel. Starting with the Y and X columns, we can create two additional columns using the Excel natural logarithms function LN(Y) and LN(X). Next, we use the Excel function SLOPE (LN(Y), LN(X)) = ฮฒ. As a final step, we multiply the computed slope by the median or the average (Y/X) in order to obtain ฮฒ (Yi / Xi). To compute the regression coefficients (intercept and slope) together with their standard errors, we can use instead the Excel function LINEST ( ). We think that itโs better to learnย a dedicated statistical package such as Minitab, Systat, or SPSS (instead of Excel) because we can diagnose more effectively the regression residuals and ascertainย the most reliable estimate of the arm's length profit margin. In RoyaltyStat, we have built-in regression functions to estimate the profit margin using the linear, power, exponential, and other statisticalย curves.
Published on May 24, 2016 5:18:06 AM
Ednaldo Silva (Ph.D.) is founder and managing director of RoyaltyStat. He helped draft the US transfer pricing regulations and developed the comparable profits method called TNNM by the OECD. He can be contacted at: esilva@royaltystat.com
RoyaltyStat provides premier online databases of royalty rates extracted from unredacted license agreements
and normalized company financials (income statement, balance sheet, cash flow). We provide high-quality data, built-in analytical tools, customer training and attentive technical support.
Topics: Net Profit Indicator
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Practice on Toph
Participate in exhilarating programming contests, solve unique algorithm and data structure challenges and be a part of an awesome community.
Walk Less
By kazi_nayeem ยท Limits 1s, 512 MB
You live in a city which consists of N points. The points are numbered from 1 to n. The distance between any two points u andย v is |u-v|. Currently, you are on point 1 and you have to go to point n. You can always walk from any point to any other point.
There are M trains that go from a point to another point. You can use these trains any number of times. But if you enter in a train, you can't leave it until it reaches its destination.
Now you have to find, what is the minimum amount of distance you need to walk to go from point 1 to point n?
Input
The first line contains an integer $T$ ($0 < T \le 50$), number of test cases.
The firstย line of each test case contains two integers $N$ ($1 \le N \le 5000$) and $M$ ($0 \le M \le 100000$) where $N$ is the number of points and $M$ is the number of transports. Next $M$ line contains two integers $u$ and $v$ย ($1 \le u, v \le N$)where there is a transport that goes from point $u$ to point $v$.
Output
For every test case, print minimum amount of distance you need to walk to go from point 1 to $n$.
Sample
InputOutput
3
7 2
1 4
3 7
10 3
2 4
3 6
5 10
5 1
1 5
1
2
0
Statistics
54% Solution Ratio
TasdidEarliest, Jan '18
IamHotFastest, 33431.7s
IamHotLightest, 918 kB
Baka_RaffiShortest, 1132B
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# recurrentpseudo: An R package for analysing recurrent events in the presence of terminal events using pseudo-observations
This package computes pseudo-observations for recurrent event data in the presence of terminal events. Three versions exist: One-dimensional, two-dimensional or three-dimensional pseudo-observations.
Following the computation of pseudo-observations, the marginal mean function, survival probability and/or cumulative incidences can be modelled using generalised estimating equations.
See Furberg et al.ย (Bivariate pseudo-observations for recurrent event analysis with terminal events (2021)) for technical details on the procedure.
## Notation
Let $$D^*$$ denote the survival time and let $$N^*(t)$$ denote the number of recurrent events by time $$t$$. Let $$C$$ denote the time of censoring. Due to right-censoring, the data consists of $$X=\lbrace N(\cdot), D, \delta, Z \rbrace$$ where $$N(t) = N^*(t \wedge C)$$, $$D=D^* \wedge C$$, $$\delta = I \left( D^* \leq C \right)$$ and $$Z$$ denotes $$p$$ baseline covariates.
We observe $$X_i=\lbrace N_i(\cdot), D_i, \delta_i, Z_i \rbrace$$ for each individual $$i= 1,\ldots, n$$.
We consider the marginal mean function, $$\mu (t)$$, given by $\mu(t) = E(N^*(t)) = \int_0^t S(u^-) \, d R(u), \quad d R(t) = E(dN^*(t) \mid D^* \geq t)$ and the survival probability, $$S(t)$$, given by $S(t) = P(D^*> t).$
Moreover, we consider the cumulative incidences for death causes 1, $$C_1(t)$$, and 2, $$C_2(t)$$ $C_1(t) = E(I(D^* \leq t, \Delta = 1)), \quad C_2(t) = E(I(D^* \leq t, \Delta = 2))$ where $$\Delta = \lbrace 1, 2 \rbrace$$ represents a cause-of-death indicator.
## Introduction to pseudo-observations
The following section serves as a fast introduction to pseudo-observations, which the methods of this package is based on.
For more detailed information, please see
โข Andersen and Perme (Pseudo-observations in survival analysis (2010)) or
โข Andersen, Klein and Rosthรธj (Generalised linear models for correlated pseudo-observations, with applications to multi-state models (2003))
We wish to formulate a model for $\theta = E(f(X))$ where $$X=X_1, \ldots, X_n$$ denotes a vector of survival times (or other survival data) for $$n$$ individuals and $$f$$ denotes some function. An example would be $$\theta = E(I(D^*>t)) = P(D^*>t)$$.
Assume that a sufficiently nice estimator $$\hat{\theta}$$ of $$\theta$$ exists. For a fixed time, $$t \in [0, \tau]$$, the pseudo-observation for the iโth individual at $$t$$ is given by $\hat{\theta}_i (t)= n \cdot \hat{\theta}(t) - (n-1) \cdot \hat{\theta}^{-i}(t)$ where $$\hat{\theta}(t)$$ denotes the estimate based on the total data set, and $$\hat{\theta}^{-i}(t)$$ denotes the estimate based on the same data set but omitting observations from individual i.
Since the survival times are subject to right-censoring, standard inference on survival data is adjusted to accommodate this, e.g.ย in likelihood estimation.
However, since all subjects has a valid pseudo-observation, $$\hat{\theta}_i (t)$$, at one or more times, these can be used as an outcome variable in a generalised linear model. Note, that this is regardless of the whether a subject is alive, censored or died at time t.
Assume that $$g$$ denotes a link function, then we wish to fit
$g(E(f(X) \mid Z)) = \xi^T Z.$ Following, $$f(X)$$ is replaced by $$\hat{\theta}_i (\cdot)$$ in the model fit.
The model parameters, $$\xi$$, are estimated using generalised estimating equations (GEE), see Liang and Zeger (Longitudinal data analysis using generalized linear models (1986)).
The GEE procedure accommodates the fact that each individual can have several (pseudo-)observations.
## One-dimensional pseudo-observations
The one-dimensional pseudo-observations model is based on the parameter $$\theta = \mu(t)$$, which is estimated by $\hat{\theta} = \hat{\mu}(t) = \int_0^t \hat{S}(u^-) \, d \hat{R}(u),$ where $$\hat{S}(t)$$ denotes the Kaplan-Meier estimator of $$S(t)$$ and $$\hat{R}(t)$$ denotes the Nelson-Aalen estimator of $$R(t)$$.
We assume that $\log \left( \mu(t \mid Z) \right) = \log(\mu_0(t)) + \beta^T Z.$
## Two-dimensional pseudo-observations
The two-dimensional pseudo-observations model is based on the parameter $$\theta = (\mu(t), S(t))$$, which is estimated by $\hat{\theta} = \left( \begin{matrix} \hat{\mu}(t) \\ \hat{S}(t) \end{matrix} \right).$
We assume that $\left( \begin{matrix} \log \left(\mu (t \mid Z) \right) \\ \text{cloglog} \left( S( t \mid Z) \right) \end{matrix} \right) = \left( \begin{matrix} \log \left( \mu_0(t) \right) + {\beta}^T {Z} \\ \log \left(\Lambda_0(t)\right) + {\gamma}^T {Z} \end{matrix} \right).$
## Three-dimensional pseudo-observations
The three-dimensional pseudo-observations model is based on the parameter $$\theta = (\mu(t), C_1(t), C_2(t))$$, which is estimated by $\hat{\theta} = \left( \begin{matrix} \hat{\mu}(t) \\ \hat{C}_1(t) \\ \hat{C}_2(t) \end{matrix} \right)$ where $$\hat{C}_1(t)$$ and $$\hat{C}_2(t)$$ are the Aalen-Johansen estimates of the cumulative incidences for causes 1, $$C_1(t)$$, and 2, $$C_2(t)$$, respectively.
We assume that $\left( \begin{matrix} \log \left(\mu (t \mid Z) \right) \\ \text{cloglog} \left(1- C_1( t \mid Z) \right) \\ \text{cloglog} \left(1- C_2( t \mid Z) \right) \end{matrix} \right) = \left( \begin{matrix} \log \left( \mu_0(t) \right) + {\beta}^T {Z} \\ \log \left(\Lambda_{10}(t)\right) + {\gamma_1}^T {Z} \\ \log \left(\Lambda_{20}(t)\right) + {\gamma_2}^T {Z} \end{matrix} \right)$
# Install package from GitHub
require(devtools)
# devtools::install_github("JulieKFurberg/recurrentpseudo", force = TRUE)
require(recurrentpseudo)
# Main functions
#?pseudo.onedim
#?pseudo.twodim
#?pseudo.threedim
#?pseudo.geefit
# Example - Bladder cancer data from survival package
The functions in recurrentpseudo will be exemplified using the well-known bladder cancer data from the survival package. This data set considers data from a clinical cancer trial conducted by the Veterans Administration Cooperative Urological Research Group (Byar: The veterans administration study of chemoprophylaxis for recurrent stage I bladder tumours: comparisons of placebo, pyridoxine and topical thiotepa (1980)) Here, 118 patients with stage I bladder cancer were randomised to receive placebo, pyridoxine or thiotepa. After randomisation, information on occurrences of superficial bladder tumours and any deaths were collected.
We focus on the comparison between placebo and thiotepa ($$n=86$$ in total). We model recurrent bladder tumours, and adjust for death (cause 1: bladder cancer disease death, cause 2: other causes).
One-, two- and three-dimensional pseudo-observations are computed based on a single time point, $$t=30$$ months.
For the comparison between placebo and thiotepa on recurrent bladder tumours, the effect measure of interest is the mean ratio $$\exp(\beta)$$.
# Example: Bladder cancer data from survival package
require(survival)
#> Indlรฆser krรฆvet pakke: survival
# Make a three level status variable
bladder1$status3 <- ifelse(bladder1$status %in% c(2, 3), 2, bladder1$status) # Add one extra day for the two patients with start=stop=0 # subset(bladder1, stop <= start) bladder1[bladder1$id == 1, "stop"] <- 1
bladder1[bladder1$id == 49, "stop"] <- 1 # Restrict the data to placebo and thiotepa bladdersub <- subset(bladder1, treatment %in% c("placebo", "thiotepa")) # Make treatment variable two-level factor bladdersub$Z <- as.factor(ifelse(bladdersub$treatment == "placebo", 0, 1)) levels(bladdersub$Z) <- c("placebo", "thiotepa")
#> id treatment number size recur start stop status rtumor rsize enum status3
#> 1 1 placebo 1 1 0 0 1 3 . . 1 2
#> 2 2 placebo 1 3 0 0 1 3 . . 1 2
#> 3 3 placebo 2 1 0 0 4 0 . . 1 0
#> 4 4 placebo 1 1 0 0 7 0 . . 1 0
#> 5 5 placebo 5 1 0 0 10 3 . . 1 2
#> 6 6 placebo 4 1 1 0 6 1 1 1 1 1
#> Z
#> 1 placebo
#> 2 placebo
#> 3 placebo
#> 4 placebo
#> 5 placebo
#> 6 placebo
We fit the univariate pseudo-observation model using the binary treatment indicator as covariate, i.e.ย we model $\log \left( \mu(t \mid Z) \right) = \log(\mu_0(t)) + \beta Z$
One-dimensional pseudo-observations and GEE fit can be computed using the following code,
# Pseudo observations at t = 30
pseudo_bladder_1d <- pseudo.onedim(tstart = bladdersub$start, tstop = bladdersub$stop,
status = bladdersub$status3, id = bladdersub$id,
covar_names = "Z",
tk = c(30),
head(pseudo_bladder_1d$outdata) #> mu k ts id Z #> 1 -0.0004269178 1 30 1 placebo #> 2 -0.0004269178 1 30 2 placebo #> 3 1.2359654463 1 30 3 placebo #> 4 1.0739859010 1 30 4 placebo #> 5 -0.0958639918 1 30 5 placebo #> 6 1.0122441163 1 30 6 placebo # GEE fit fit_bladder_1d <- pseudo.geefit(pseudodata = pseudo_bladder_1d, covar_names = c("Z")) fit_bladder_1d #>$xi
#>
#> (Intercept) 0.5590869
#> Zthiotepa -0.4359054
#>
#> $sigma #> (Intercept) Zthiotepa #> (Intercept) 0.02662095 -0.02662095 #> Zthiotepa -0.02662095 0.07934314 #> #> attr(,"class") #> [1] "pseudo.geefit" # Treatment differences xi_diff_1d <- as.matrix(c(fit_bladder_1d$xi[2]), ncol = 1)
mslabels <- c("treat, mu")
rownames(xi_diff_1d) <- mslabels
colnames(xi_diff_1d) <- ""
xi_diff_1d
#>
#> treat, mu -0.4359054
# Variance matrix for differences
sigma_diff_1d <- matrix(c(fit_bladder_1d$sigma[2,2]), ncol = 1, nrow = 1, byrow = T) rownames(sigma_diff_1d) <- colnames(sigma_diff_1d) <- mslabels sigma_diff_1d #> treat, mu #> treat, mu 0.07934314 Thus, the estimated mean ratio is $$\exp(\hat{\beta})=$$ 0.6466789 (standard error and confidence intervals can be found using the Delta method). Alternatively, the bivariate pseudo-observation model using the binary treatment indicator as covariate can be fitted, i.e. $\left( \begin{matrix} \log \left(\mu (t \mid Z) \right) \\ \text{cloglog} \left( S( t \mid Z) \right) \end{matrix} \right) = \left( \begin{matrix} \log \left( \mu_0(t) \right) + {\beta} {Z} \\ \log \left(\Lambda_0(t)\right) + {\gamma} {Z} \end{matrix} \right)$ Two-dimensional pseudo-observations and GEE fit can be computed using the following code # Pseudo observations at t = 30 pseudo_bladder_2d <- pseudo.twodim(tstart = bladdersub$start,
tstop = bladdersub$stop, status = bladdersub$status3,
id = bladdersub$id, covar_names = "Z", tk = c(30), data = bladdersub) head(pseudo_bladder_2d$outdata)
#> mu surv k ts id Z
#> 1 -0.0004269178 1.421085e-14 1 30 1 placebo
#> 2 -0.0004269178 1.421085e-14 1 30 2 placebo
#> 3 1.2359654463 8.170875e-01 1 30 3 placebo
#> 4 1.0739859010 8.170875e-01 1 30 4 placebo
#> 5 -0.0958639918 -5.305763e-02 1 30 5 placebo
#> 6 1.0122441163 -5.305763e-02 1 30 6 placebo
# GEE fit
covar_names = c("Z"))
#> $xi #> #> esttypemu 0.55908687 #> esttypemu:Zthiotepa -0.43590539 #> esttypesurv -1.41652478 #> esttypesurv:Zthiotepa -0.04800778 #> #>$sigma
#> esttypemu esttypemu:Zthiotepa esttypesurv
#> esttypemu 0.026620952 -0.026620952 -0.003481085
#> esttypemu:Zthiotepa -0.026620952 0.079343139 0.003481085
#> esttypesurv -0.003481085 0.003481085 0.123251791
#> esttypesurv:Zthiotepa 0.003481085 0.002758847 -0.123251791
#> esttypesurv:Zthiotepa
#> esttypemu 0.003481085
#> esttypemu:Zthiotepa 0.002758847
#> esttypesurv -0.123251791
#> esttypesurv:Zthiotepa 0.260915569
#>
#> attr(,"class")
#> [1] "pseudo.geefit"
# Treatment differences
xi_diff_2d <- as.matrix(c(fit_bladder_2d$xi[2], fit_bladder_2d$xi[4]), ncol = 1)
mslabels <- c("treat, mu", "treat, surv")
rownames(xi_diff_2d) <- mslabels
colnames(xi_diff_2d) <- ""
xi_diff_2d
#>
#> treat, mu -0.43590539
#> treat, surv -0.04800778
# Variance matrix for differences
sigma_diff_2d <- matrix(c(fit_bladder_2d$sigma[2,2], fit_bladder_2d$sigma[2,4],
fit_bladder_2d$sigma[2,4], fit_bladder_2d$sigma[4,4]),
ncol = 2, nrow = 2,
byrow = T)
rownames(sigma_diff_2d) <- colnames(sigma_diff_2d) <- mslabels
sigma_diff_2d
#> treat, mu treat, surv
#> treat, mu 0.079343139 0.002758847
#> treat, surv 0.002758847 0.260915569
Finally, one could fit the three-dimensional pseudo-observation model to the bladder cancer data.
Three-dimensional pseudo-observations and GEE fit can be computed using the following code
# Add deathtype variable to bladder data
# Deathtype = 1 (bladder disease death), deathtype = 2 (other death reason)
bladdersub$deathtype <- with(bladdersub, ifelse(status == 2, 1, ifelse(status == 3, 2, 0))) table(bladdersub$deathtype, bladdersub$status) #> #> 0 1 2 3 #> 0 55 132 0 0 #> 1 0 0 2 0 #> 2 0 0 0 20 # Pseudo-observations pseudo_bladder_3d <- pseudo.threedim(tstart = bladdersub$start,
tstop = bladdersub$stop, status = bladdersub$status3,
id = bladdersub$id, deathtype = bladdersub$deathtype,
covar_names = "Z",
tk = c(30),
head(pseudo_bladder_3d$outdata_long) #> k ts id esttype y Z #> 1 1 30 1 mu -4.269178e-04 placebo #> 2 1 30 1 surv 1.421085e-14 placebo #> 3 1 30 1 cif1 0.000000e+00 placebo #> 4 1 30 1 cif2 1.000000e+00 placebo #> 5 1 30 2 mu -4.269178e-04 placebo #> 6 1 30 2 surv 1.421085e-14 placebo # GEE fit fit_bladder_3d <- pseudo.geefit(pseudodata = pseudo_bladder_3d, covar_names = c("Z")) fit_bladder_3d #>$xi
#>
#> esttypemu 0.5590869
#> esttypemu:Zthiotepa -0.4359054
#> esttypecif1 -3.7618319
#> esttypecif1:Zthiotepa 0.2930357
#> esttypecif2 -1.5431978
#> esttypecif2:Zthiotepa -0.1005109
#>
#> $sigma #> esttypemu esttypemu:Zthiotepa esttypecif1 #> esttypemu 0.026620952 -0.026620952 0.01663610 #> esttypemu:Zthiotepa -0.026620952 0.079343139 -0.01663610 #> esttypecif1 0.016636098 -0.016636098 1.07839851 #> esttypecif1:Zthiotepa -0.016636098 0.013359996 -1.07839851 #> esttypecif2 -0.006027688 0.006027688 -0.02642283 #> esttypecif2:Zthiotepa 0.006027688 0.001779996 0.02642283 #> esttypecif1:Zthiotepa esttypecif2 esttypecif2:Zthiotepa #> esttypemu -0.01663610 -0.006027688 0.006027688 #> esttypemu:Zthiotepa 0.01336000 0.006027688 0.001779996 #> esttypecif1 -1.07839851 -0.026422825 0.026422825 #> esttypecif1:Zthiotepa 2.01305239 0.026422825 -0.057715255 #> esttypecif2 0.02642283 0.138167379 -0.138167379 #> esttypecif2:Zthiotepa -0.05771525 -0.138167379 0.299045959 #> #> attr(,"class") #> [1] "pseudo.geefit" # Treatment differences xi_diff_3d <- as.matrix(c(fit_bladder_3d$xi[2],
fit_bladder_3d$xi[4], fit_bladder_3d$xi[6]), ncol = 1)
mslabels <- c("treat, mu", "treat, cif1", "treat, cif2")
rownames(xi_diff_3d) <- mslabels
colnames(xi_diff_3d) <- ""
xi_diff_3d
#>
#> treat, mu -0.4359054
#> treat, cif1 0.2930357
#> treat, cif2 -0.1005109
# Variance matrix for differences
sigma_diff_3d <- matrix(c(fit_bladder_3d$sigma[2,2], fit_bladder_3d$sigma[2,4],
fit_bladder_3d$sigma[2,6], fit_bladder_3d$sigma[2,4],
fit_bladder_3d$sigma[4,4], fit_bladder_3d$sigma[4,6],
fit_bladder_3d$sigma[2,6], fit_bladder_3d$sigma[4,6],
fit_bladder_3d$sigma[6,6] ), ncol = 3, nrow = 3, byrow = T) rownames(sigma_diff_3d) <- colnames(sigma_diff_3d) <- mslabels sigma_diff_3d #> treat, mu treat, cif1 treat, cif2 #> treat, mu 0.079343139 0.01336000 0.001779996 #> treat, cif1 0.013359996 2.01305239 -0.057715255 #> treat, cif2 0.001779996 -0.05771525 0.299045959 We can compare the three model fits. Note, that the $$\mu$$ components match each other. # Compare - should match for mu elements xi_diff_1d #> #> treat, mu -0.4359054 xi_diff_2d #> #> treat, mu -0.43590539 #> treat, surv -0.04800778 xi_diff_3d #> #> treat, mu -0.4359054 #> treat, cif1 0.2930357 #> treat, cif2 -0.1005109 sigma_diff_1d #> treat, mu #> treat, mu 0.07934314 sigma_diff_2d #> treat, mu treat, surv #> treat, mu 0.079343139 0.002758847 #> treat, surv 0.002758847 0.260915569 sigma_diff_3d #> treat, mu treat, cif1 treat, cif2 #> treat, mu 0.079343139 0.01336000 0.001779996 #> treat, cif1 0.013359996 2.01305239 -0.057715255 #> treat, cif2 0.001779996 -0.05771525 0.299045959 ### More covariates Assume that we wish to add extra baseline covariates to the model fit. For the sake of illustration, we have simulated a continuous covariate, $$Z_2$$, and a categorical covariate, $$Z_3$$. The covariate $$Z_1$$ corresponds to the binary treatment covariate ($$Z=1$$ is thiotepa and $$Z=0$$ is placebo). In order to make estimation for these models possible, the pseudo-observations are calculated at three time points, namely $$t=20, 30, 40$$ months. For the one-dimensional model for $$\mu$$ it holds that, $\log \left( \mu(t \mid Z) \right) = \log(\mu_0(t)) + \beta_1 Z_1 + \beta_2 Z_2 + \beta_3 Z_3.$ This can be fitted using the below code, ## One-dim # A binary variable, Z1_ # A continuous variable, Z2_ # A categorical variable, Z3_ set.seed(0308) require(magrittr) require(dplyr) bladdersub <- as.data.frame( bladdersub %>% group_by(id) %>% mutate(Z1_ = Z, Z2_ = rnorm(1, mean = 3, sd = 1), Z3_ = sample(x = c("A", "B", "C"), size = 1, replace = TRUE, prob = c(1/4, 1/2, 1/4)) )) # head(bladdersub, 20) # Make pseudo obs at more timepoints (more data) # Pseudo observations at t = 20, 30, 40 pseudo_bladder_1d_3t <- pseudo.onedim(tstart = bladdersub$start,
tstop = bladdersub$stop, status = bladdersub$status3,
id = bladdersub$id, covar_names = c("Z1_", "Z2_", "Z3_"), tk = c(20, 30, 40), data = bladdersub) fit1 <- pseudo.geefit(pseudodata = pseudo_bladder_1d_3t, covar_names = c("Z1_", "Z2_", "Z3_")) fit1$xi
#>
#> (Intercept) 0.39412273
#> Ztime30 0.42336922
#> Ztime40 0.59966454
#> Z1_thiotepa -0.29479824
#> Z2_ -0.08253287
#> Z3_B 0.01965615
#> Z3_C -0.38332247
fit1$sigma #> (Intercept) Ztime30 Ztime40 Z1_thiotepa Z2_ #> (Intercept) 0.136673486 -0.0061702979 -0.0109943095 -0.036100986 -0.0247560186 #> Ztime30 -0.006170298 0.0046658382 0.0061013009 0.001167246 0.0002746085 #> Ztime40 -0.010994310 0.0061013009 0.0100442253 0.005504053 -0.0004015364 #> Z1_thiotepa -0.036100986 0.0011672458 0.0055040527 0.090063224 -0.0078557067 #> Z2_ -0.024756019 0.0002746085 -0.0004015364 -0.007855707 0.0109918233 #> Z3_B -0.055441417 0.0005892506 0.0051883496 0.037829933 -0.0035769385 #> Z3_C -0.041748863 0.0075014896 0.0141048652 0.023020550 -0.0082026411 #> Z3_B Z3_C #> (Intercept) -0.0554414168 -0.041748863 #> Ztime30 0.0005892506 0.007501490 #> Ztime40 0.0051883496 0.014104865 #> Z1_thiotepa 0.0378299330 0.023020550 #> Z2_ -0.0035769385 -0.008202641 #> Z3_B 0.0869004239 0.050773663 #> Z3_C 0.0507736625 0.200303239 fit1$xi[4]
#> [1] -0.2947982
Or for two-dimensional pseudo-observations, it holds that
$\left( \begin{matrix} \log \left(\mu (t \mid Z) \right) \\ \text{cloglog} \left( S( t \mid Z) \right) \end{matrix} \right) = \left( \begin{matrix} \log \left( \mu_0(t) \right) + \beta_1 {Z_1} + \beta_2 {Z_2} + \beta_3 {Z_3} \\ \log \left(\Lambda_0(t)\right) + {\gamma_1} Z_1 + {\gamma_2} Z_2 + {\gamma_3} Z_3 \end{matrix} \right).$ Or for three-dimensional pseudo-observations, it holds that $\left( \begin{matrix} \log \left(\mu (t \mid Z) \right) \\ \text{cloglog} \left( 1-C_1( t \mid Z) \right) \\ \text{cloglog} \left( 1-C_2( t \mid Z) \right) \end{matrix} \right) = \left( \begin{matrix} \log \left( \mu_0(t) \right) + {\beta_1} {Z_1} + {\beta_2} {Z_2} + {\beta_3} Z_3\\ \log \left(\Lambda_{10}(t)\right) + \gamma_{11} {Z_1} + \gamma_{12} {Z_2} + \gamma_{13} {Z_3} \\ \log \left(\Lambda_{20}(t)\right) + \gamma_{21} {Z_1} + \gamma_{22} {Z_2} + \gamma_{23} {Z_3} \end{matrix} \right).$
These two models are fitted using the below code,
## Two-dim
# Pseudo observations at t = 20, 30, 40
pseudo_bladder_2d_3t <- pseudo.twodim(tstart = bladdersub$start, tstop = bladdersub$stop,
status = bladdersub$status3, id = bladdersub$id,
covar_names = c("Z1_", "Z2_", "Z3_"),
tk = c(20, 30, 40),
covar_names = c("Z1_", "Z2_", "Z3_"))
# fit2$xi # fit2$sigma
## Three-dim
pseudo_bladder_3d_3t <- pseudo.threedim(tstart = bladdersub$start, tstop = bladdersub$stop,
status = bladdersub$status3, id = bladdersub$id,
covar_names = c("Z1_", "Z2_", "Z3_"),
deathtype = bladdersub$deathtype, tk = c(20, 30, 40), data = bladdersub) fit3 <- pseudo.geefit(pseudodata = pseudo_bladder_3d_3t, covar_names = c("Z1_", "Z2_", "Z3_")) # fit3$xi
# fit3$sigma ## Compare for mu fit1$xi[4]
#> [1] -0.2947982
fit2$xi[4] #> [1] -0.2947982 fit3$xi[4]
#> [1] -0.2948043
# Citation
To cite the recurrentpseudo package please use the following references,
Julie K. Furberg, Per K. Andersen, Sofie Korn, Morten Overgaard, Henrik Ravn: Bivariate pseudo-observations for recurrent event analysis with terminal events (Lifetime Data Analysis, 2021)
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https://justaaa.com/statistics-and-probability/512947-a-researcher-is-looking-into-whether-who-wear
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Question
# A researcher is looking into whether who wear Crocs sandals are less cool than people who...
A researcher is looking into whether who wear Crocs sandals are less cool than people who wear other sandals. He knows that the mean โcoolnessโ score for people who wear other sandals is 21, with a population standard deviation of 6. He takes a sample of 9 people who wear Crocs and finds a sample mean coolness score of 17.4. What is the probability of finding this result or lower, if the null hypothesis is true?
a. .6844
b. .9912
c. .0792
d. .0359
Solution :
= 21
=17.4
=6
n = 9
This is the left tailed test .
The null and alternative hypothesis is ,
H0 :ย ย ย ย = 21
Ha : < 21
Test statistic = z
= ( - ) / / n
= (17.4-21) / 6 / 9
= -1.8
Test statistic = z = -1.8
P(z < -1.8 ) = 0.0359
P-value = 0.0359
Option d ) is correct.
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http://t.zoukankan.com/ytytzzz-p-6836564.html
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zoukankan ย ย ย ย ย htmlย ย cssย ย jsย ย c++ย ย java
โข Codeforces Round #400 (Div. 1 + Div. 2, combined)โโABCDE
้ข็ฎๆณ่ฟ้
A.A Serial Killer
้ข็ฎๆ่ฟฐไผผไนๅพๆถๅฟ๏ผ็ปๅๆ ทไพๅๆ ทไพ่งฃ้็ๆต็้ขๆ
ไฝฟ็จC++11็autoๅฏไปฅๆฅไธๆ้ชๆไฝ
```#include <bits/stdc++.h>
using namespace std;
int n;
string s[4];
map <string, int> p;
int main() {
cin >> s[0] >> s[1];
p[s[0]] = p[s[1]] = 1;
cin >> n;
cout << s[0] << " " <<s[1] << endl;
while(n --) {
cin >> s[2] >> s[3];
p[s[2]] ++, p[s[3]] ++;
for(auto iter : p)
if(iter.second == 1) cout << iter.first << " ";
puts("");
}
return 0;
}```
View Code
ๅ
ถๅฎ็ญไปทไบ่ฟๆ ทๅ
```#include <bits/stdc++.h>
using namespace std;
int n;
string s[4];
map <string, int> p;
int main() {
cin >> s[0] >> s[1];
p[s[0]] = p[s[1]] = 1;
cin >> n;
cout << s[0] << " " <<s[1] << endl;
while(n --) {
cin >> s[2] >> s[3];
p[s[2]] ++, p[s[3]] ++;
for(map <string, int>::iterator iter = p.begin();iter != p.end();iter ++)
if(iter -> second == 1) cout << iter -> first << " ";
puts("");
}
return 0;
}```
View Code
B.Sherlock and his girlfriend
ๅพ่ ข็ไธ้ข๏ผ่ดจๆฐๆ 1๏ผๅๆฐๆ 2ๅฐฑๅฅฝไบ
```#include <bits/stdc++.h>
#define rep(i, j, k) for(int i = j;i <= k;i ++)
#define rev(i, j, k) for(int i = j;i >= k;i --)
using namespace std;
typedef long long ll;
const int maxn = 100010;
int n, a[maxn];
int main() {
ios::sync_with_stdio(false);
cin >> n;
if(n < 3) puts("1");
else puts("2");
for(int i = 2;i <= n + 1;i ++)
if(a[i] != 2) {
a[i] = 1;
for(int j = i << 1;j <= n + 1;j += i)
a[j] = 2;
}
for(int i = 2;i <= n + 1;i ++)
printf("%d ", a[i]);
return 0;
}```
View Code
C.Molly's Chemicals
ๆ้ฃไนไธ็นๆๆ็้ข็ฎ
ๆฑๆๅคๅฐๆฎต่ฟ็ปญๅญๆฎตๅไธบk็้่ดpower
ๆพ็ถkไธบ2็่ฏ๏ผๅคงๆฆ่ฝ2^0 - 2^50ๅทฆๅณๅง
ๆไปฅ็ดๆฅๆไธพ k^p ๅณๅฏ
ๅทๆๅฅไธชmap๏ผๅคๆๅบฆO(n(logn)^2)
ๆณจๆ:
1.้่ดpower๏ผๅ
ๆฌ1
2. |k| = 1 ็นๅค๏ผๅฆๅๆญปๅพช็ฏ
```#include <bits/stdc++.h>
#define rep(i, j, k) for(int i = j;i <= k;i ++)
#define rev(i, j, k) for(int i = j;i >= k;i --)
using namespace std;
typedef long long ll;
int n, t;
ll k, s[100010];
map <ll, int> p;
int main() {
ios::sync_with_stdio(false);
int x;
cin >> n >> t;
rep(i, 1, n) cin >> x, s[i] = s[i - 1] + x;
for(ll j = 1;abs(j) <= 100000000000000ll;j *= t) {
p.clear(), p[0] = 1;
rep(i, 1, n) {
k += p[s[i] - j];
p[s[i]] ++;
}
if(t == 1 || (t == -1 && j == -1)) break;
}
cout << k;
return 0;
}```
View Code
D.The Door Problem
ๅบ่ฏฅๆณจๆๅฐeach door is controlled byย exactly twoย switches
ๆไปฅๆพ็ถๅฏนไบไธๅผๅง้ไธ็้จ๏ผๅช่ฝ้ๆฉไธไธชๅผๅ
ณ
ไธๅผๅงๆๅผ็้จ๏ผๅฏไปฅ้ๆฉ้ฝไธ้ๆ่
้ฝ้
ไบๆฏๆไปฌๅฏไปฅๆณๅฐ2-satๆฅ่งฃๅณ
ๅฎ้
ไธ2-satไน็็กฎๅฏไปฅ่งฃๅณ
ไฝๆฏๆไปฌๆณจๆๅฐ่ฟไธช2-sat็็นๆฎๆง
ๆฏ็ปไธญ็ไธคไธช้ๆฉๅจๆ็ง็จๅบฆไธๆฏ็ญไปท็
่ๆไปฌๅนณๆถๅ็ Ai ไธ Aiโ ๆฏไธ็ญไปท็
ไธคไธช้ๆฉ็ญไปทๆๅณ็่ฟ็่พนๅทฒ็ปๆฏๆ ๅ่พน
ๅณ่ฅๆAi -> Aj๏ผๅๅฟ
ๆAj -> Ai
่ฟๆ ทๅฐฑไธ้่ฆๅtarjan
็ดๆฅๅนถๆฅ้ๅฐฑๅฏไปฅ่งฃๅณไบ
```#include <cstdio>
const int maxn = 100010;
int n, m, f[maxn << 1], a[2][maxn];
bool op[maxn];
int find_(int x) {
if(f[x] != x) return f[x] = find_(f[x]);
return x;
}
void union_(int x, int y) {
x = find_(x), y = find_(y);
if(x != y) f[x] = y;
}
int main() {
scanf("%d %d", &n, &m);
for(int i = 1;i <= n;i ++) scanf("%d", &op[i]);
for(int k, j, i = 1;i <= m;i ++) {
scanf("%d", &j);
while(j --) {
scanf("%d", &k);
if(a[0][k]) a[1][k] = i;
else a[0][k] = i;
}
}
for(int i = m << 1;i;i --) f[i] = i;
for(int i = 1;i <= n;i ++)
if(op[i]) union_(a[0][i], a[1][i]), union_(a[0][i] + m, a[1][i] + m);
else union_(a[0][i], a[1][i] + m), union_(a[0][i] + m, a[1][i]);
for(int i = 1;i <= m;i ++)
if(find_(i) == find_(i + m)) {
puts("NO");
return 0;
}
puts("YES");
return 0;
}```
View Code
E.The Holmes Children
ๆๅจ่ฎก็ฎๅ็ฐ f ๅฝๆฐไธบๆฌงๆๅฝๆฐ
gcd(x, y) = 1
x + y = n
=> gcd(x, x + y) = 1 ๅณ gcd(x, n) = 1
g(n) = n ๏ผๅฉไธ้จๅๅพๅฅฝ่งฃๅณ
```#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
const int mod_ = 1e9 + 7;
ll f(ll x) {
ll ret = x;
for(ll i = 2;i * i <= x;i ++)
if(x % i == 0) {
ret /= i, ret *= (i - 1);
while(x % i == 0) x /= i;
}
if(x != 1) ret /= x, ret *= (x - 1);
return ret;
}
int main(){
ll n, k;
cin >> n >> k;
k = (k + 1) >> 1;
for(int i = 1;i <= k;i ++) {
n = f(n);
if(n == 1) break;
}
cout << n % mod_;
return 0;
}```
View Code
โข ็ธๅ
ณ้
่ฏป:
as3 return่ฏญๅฅไธญ็่ฟ็ฎ็ฌฆ
AIR custom ApplicationUpdaterUI
Flash Builder ๆ ๆณ่ฟๆฅๅฐๅบ็จ็จๅบไปฅ่ฎฟๅญๆฆ่ฆๅๆๆฐๆฎ
Android็ๆๆๆ้่ฏดๆ
Tomcatๆฐๆฎๆบ้
็ฝฎ
hibernate ๅปถ่ฟๅ ่ฝฝ(ๆๅ ่ฝฝ)
Android SQLiteๆฐๆฎๅบๆไฝ
Androidไธ่ฝฝๆๆฌๆไปถๅmp3ๆไปถ
JPAๆณจ่งฃ
Android Intentไผ ๅผไธๅฎ็ฐ็ชไฝ่ทณ่ฝฌ
โข ๅๆๅฐๅ๏ผhttps://www.cnblogs.com/ytytzzz/p/6836564.html
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| 2.546875
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CC-MAIN-2024-10
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latest
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en
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|
http://book.caltech.edu/bookforum/printthread.php?s=43eb9f2bf03e41be288eef749a9237f2&t=967
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LFD Book Forum (http://book.caltech.edu/bookforum/index.php)
- ย General comments on the course (http://book.caltech.edu/bookforum/forumdisplay.php?f=140)
- ย - ย Pre-requisite help with Math etc... (http://book.caltech.edu/bookforum/showthread.php?t=967)
itooam 08-08-2012 06:57 AM
Pre-requisite help with Math etc...
I wish we were taught Calculus when I was at school (in UK)...
For all the other's who have muddled through the difficult Math, here are some pretty good videos that I have started watching to get to grips with the more complicated aspects of this course:
Highlights of Calculus - Gilbert Strang
So far I seem to be doing okay on the questions where the Math can be modelled, but things like "set theory" etc, I simply don't have enough Math knowledge to tackle.
Anyone else know of any other good resources etc which you have/think may be useful/beneficial for use with this "Learning From Data" course?
munchkin 08-08-2012 04:30 PM
Re: Pre-requisite help with Math etc...
Online textbook covers a huge chunk of the math used in computer science and courses such as this one. Good for review or finding out something new:
http://www.cs.princeton.edu/courses/...433/mathcs.pdf
A primer on conditional probability.
http://www.math.uh.edu/~weizhang/M33...eNotes_2_2.pdf
Keith 08-10-2012 06:01 AM
Re: Pre-requisite help with Math etc...
Here is a free online tool for help with symbolic calculus http://mathomatic.orgserve.de/CGI/math.php This script finds the partial derivative but in a "simplified" form:
(u*e^v+2*v*e^-u)^2
differentiate u
itooam 08-11-2012 01:41 AM
Re: Pre-requisite help with Math etc...
Keith, thanks for the link, was looking for something like that a few months back. It is brilliant, I wish I'd seen that before doing q4... confirms my working though. A great proofing tool. Do you know if I can use that tool to explicitly give me values of u and v (so I can proof translations)? The best I found was to use the command "calculate" which gives a more user friendly formula (easier to plug into code).
itooam 08-11-2012 08:50 AM
Re: Pre-requisite help with Math etc...
Keith, your formula above should have been:
(u*e^v-2*v*e^-u)^2
Caused me a bit of initial confusion lol. That'll serve me right for not proofing myself.
Keith 08-11-2012 09:37 AM
Re: Pre-requisite help with Math etc...
Whoops, sorry about that! Too bad that copy and paste doen't work well between many programs resulting in typos that waste time and cause confusion.
Mayson Lancaster 08-12-2012 08:53 PM
Re: Pre-requisite help with Math etc...
Quote:
Originally Posted by itooam (Post 3898) Anyone else know of any other good resources etc which you have/think may be useful/beneficial for use with this "Learning From Data" course?
Here's a useful site for calculus, and for plotting:
http://calc.matthen.com
depinski 08-13-2012 06:50 PM
Re: Pre-requisite help with Math etc...
If your calculus is very rusty, like mine, I suggest using wolframalpha to check your answer for the derivative
ex:
http://www.wolframalpha.com/input/?i...h+respect+to+u
All times are GMT -7. The time now is 10:21 AM.
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http://oeis.org/A181370
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crawl-data/CC-MAIN-2020-34/segments/1596439738888.13/warc/CC-MAIN-20200812083025-20200812113025-00075.warc.gz
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The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.
Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
A181370 Square array read by antidiagonals: T(m,n) is the number of L-convex polyominoes with m rows and n columns. 0
1, 1, 1, 1, 5, 1, 1, 11, 11, 1, 1, 19, 42, 19, 1, 1, 29, 110, 110, 29, 1, 1, 41, 235, 402, 235, 41, 1, 1, 55, 441, 1135, 1135, 441, 55, 1, 1, 71, 756, 2709, 4070, 2709, 756, 71, 1, 1, 89, 1212, 5740, 11982, 11982, 5740, 1212, 89, 1, 1, 109, 1845, 11124, 30618, 42510 (list; table; graph; refs; listen; history; text; internal format)
OFFSET 1,5 COMMENTS An L-convex polyomino is a convex polyomino where any two cells can be connected by a path internal to the polyomino and which has at most 1 change of direction (i.e., one of the four orientation of the letter L). LINKS G. Castiglione, A. Frosini, A. Restivo and S. Rinaldi, Enumeration of L-convex polyominoes by rows and columns, Theor. Comp. Sci., 347, 2005, 336-352. G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28, 2007, 1724-1741 (see Section 3.1.2). G. Castiglione and A. Restivo, Reconstruction of L-convex polyominoes, Electronic Notes in Discrete Mathematics, Vol. 12, Elsevier Science, 2003. FORMULA T(m,n) = Sum(2^k*(k^2+(m+n-2)*k+mn-1)*binomial(m+n-2,2k)*binomial(m+n-2-2k,m-1-k)/[(m+n-2)(2k+1)], k=0..min(m-1,n-1)) ((m,n)!=(1,1)); T(1,1)=1. T(n,n) = A126765(n-1). G.f.: G(x,y) = Sum_{m>=1, n>=1} T(m,n)*x^m*y^n = xy(1-x)(1-y)/(1-2x-2y+x^2+y^2) (see 2nd Maple program). EXAMPLE T(2,2)=5 because we have the 2 X 2 square and the four polyominoes obtained by removing 1 square from the four squares of the 2 X 2 square. Square array starts: ย ย 1,ย ย 1,ย ย ย 1,ย ย ย ย 1,ย ย ย ย 1, ...; ย ย 1,ย ย 5,ย ย 11,ย ย ย 19,ย ย ย 29, ...; ย ย 1, 11,ย ย 42,ย ย 110,ย ย 235, ...; ย ย 1, 19, 110,ย ย 402, 1135, ...; ย ย 1, 29, 235, 1135, 4070, ...; MAPLE T := proc (m, n) if m = 1 and n = 1 then 1 else sum(2^k*(k^2+(m+n-2)*k+m*n-1)*binomial(m+n-2, 2*k)*binomial(m+n-2-2*k, m-1-k)/((m+n-2)*(2*k+1)), k = 0 .. min(m-1, n-1)) end if end proc: matrix(9, 9, T); # yields first 9 rows and 9 columns of the square array G := x*y*(1-x)*(1-y)/(1-2*x-2*y+x^2+y^2): a := proc (m, n) options operator, arrow; coeff(series(coeff(simplify(series(G, x = 0, 20)), x, m), y = 0, 20), y, n) end proc: matrix(9, 9, a); # yields first 9 rows and 9 columns of the square array CROSSREFS Cf. A126765. Sequence in context: A173043 A082046 A132787 * A119307 A296039 A296974 Adjacent sequences:ย ย A181367 A181368 A181369 * A181371 A181372 A181373 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Oct 18 2010 STATUS approved
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Last modified August 12 05:33 EDT 2020. Contains 336438 sequences. (Running on oeis4.)
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https://coinbae.org/what-is-arithmetic-progression/
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# What is arithmetic progression?
## What is arithmetic progression?
In todayโs article, I am going to provide you with a equipo of solved arithmetic progression exercises, with the aim of learning what an arithmetic progression is and preparing you to take the next step with geometric progressions.
Remember that Knowing what a progression is cรกnido help you better understand what an annuity is. Sure, annuities are mostly based on geometric progressions, but I think sometimes itโs better to go from fรกcil to more complicated.
Therefore, first of all, it is better that you understand what a arithmetic progression and then you perro move on to geometric progressions.
I hope you find it useful.
## What is a progression?
A progression cรกnido be seen as an uninterrupted series or sequence of numbers.
Some aspects that you have to consider are the following:
1. Each number belonging to the sequence is called a term.
2. In every progression there must be a first term.
3. Each progression has a criterion, which helps us to determine each of the following terms of the sequence.
4. The terms that make up the progression cรกnido be obtained by difference (arithmetic progression) or by quotient (geometric progression).
## Examples of progressions
Some examples of progressions are:
โข 1, 2, 3, 4, 5, 6, 7, 8, 9,โฆ
โข 1, 3, 5, 7, 9, 11, 13, 15,โฆ
โข 2, 4, 6, 8, 10, 12, 14, 16,โฆ
โข 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,โฆ
## Types of progressions
We perro find arithmetic progressions and geometric progressions.
## What is an arithmetic progression?
Arithmetic progressions are progressions whose terms integrate by difference. I know that it is possible that it is better that I explain it to you with an example.
Do you remember that in elementary school they made us find the next number in a sequence?
What number or term follows in the following sequence? 2, 4, 6, 8, __
Sure, it is a very fรกcil sequence, but how would you go about finding the answer?
One way to do it is by finding the constant difference that exists between the terms of the sequence, that is, by subtraction.
Following the previous example, the following subtractions cรกnido be done:
โข 4 โ 2 = 2
โข 6 โ 4 = 2
โข 8 โ 6 = 2
Therefore, by subtraction we have been able to find the constant difference of the arithmetic progression. In this case, the constant difference is equal to 2.
So, to find the next term in the sequence, all you have to do is add 2 to the last term of the sequence you have, which is 8. Therefore, the next term of the sequence is 10.
It should be noted that it has been a very fรกcil sequence and it has not been necessary to have a elabora, but What would happen if I ask you to give me the term 100 of that sequence. In the example above I have asked you for the fifth term, but what is the 100th term or the 1000th term?
Well, for that we cรกnido use a elabora that I am going to espectรกculo you later. For now, I want you to stick with the fact that in arithmetic progressions you are going to use difference (subtraction) to get the different terms of the progression.
Note: remember that we are talking about a constant difference.
For example, in the previous case, it does not matter which terms you choose from the sequence, since the difference will always be 2.
## literals
Before continuing, it is important that we agree on the literals that are going to be used and that you know their meaning.
By the way, Iโm going to continue using the example: 2, 4, 6, 8, __
โข a = first term of the sequence.
Remember that I said at the beginning that you should always have a first term.
Continuing with the previous example, the first term is the value 2.
โข d = constant difference. Simply put, it is the value found by subtracting its terms.
In the previous example it was the value 2 (4 โ 2).
โข n = number of terms that the progression contains. Continuing with the example, the sequence has 4 terms and we are asked for the fifth term.
โข I = x. It is the term of the sequence that we are looking for or the last term of the sequence.
Note: Remember that the progression cรกnido be finite or infinite.
## Asรญ expression of the arithmetic progression
So that you cรกnido see the way in which the different terms of an arithmetic progression are found, then, I am going to give you the following table.
Continuing with the same example (2, 4, 6, 8, __), then, it is as follows:
## Arithmetic progression elabora
As you cรกnido see, in order to find the nth term of a sequence, all you need is the following elabora:
โข a = first term of the sequence.
โข d = constant difference.
โข n = number of terms that the progression contains.
โข I = term that we are looking for or the last term of the sequence.
Therefore, if you want to find the 100th term of the sequence: 2, 4, 6, 8, __
2 + (100 โ 1)2 = 200
## Clearances of the arithmetic progression elabora
Just as they perro ask us to provide the last number of the sequence (I), they perro also ask us to provide the first term or the constant difference.
Therefore, next I am going to provide you with the different clearances of the arithmetic progression.
### How to find the first term?
If you want to find the first term of a progression, then the clearance is as follows:
### How to find the constant difference?
The clearance of the constant difference is as follows:
### How to find the number of terms of an arithmetic progression?
The elabora (clearance) that will help you find the number of terms that an arithmetic progression has is the following:
## What if the constant difference is positive?
If the constant difference of the sequence is positive, then it is an increasing progression.
### example of increasing progression
Suppose you have the sequence: 4, 8, 12, 16, 20,
If we obtain the constant difference, we will obtain as a result that the constant difference is equal to 4 TRUE? In fact, just by looking at the sequence we cรกnido see that the terms are increasing by 4 by four.
Of course, here they have given us the succession directly, but there are times when they will only give us the data.
For example:
In this way, we already know that the progression has the number 4 as its first term; it is a growing sequence; increases from 4 to four and has 5 terms.
## What if the constant difference is negative?
In this case, the opposite happens to the previous case.
So that if the constant difference is negative, then the progression is decreasing.
### Example of decreasing progression
Suppose you are given the following data:
Therefore, the arithmetic progression has as its first term the number 40; has a constant difference of โ5 (so it is decreasing) and has 5 terms.
Thus, the sequence is as follows:
40, 35, 30, 25, 20
By the way, to know any value of the sequence, you cรกnido use the expression that I gave you before.
Letโs suppose that you are asked to find the next term of the previous sequence (sixth term).
Note: remember that we are going to use I = a + (n โ 1) d
40 + ( 6 โ 1) -5 = 15
As you perro see, the next term of the sequence is 15.
In fact, since Iโm using fรกcil sequences, itโs very easy to predict the next term.
That way you perro verify that it works.
## What is the sum of an arithmetic progression?
As its name already indicates, it is the sum of all the terms that a certain progression (sequence) has. In fact, there is a very interesting theorem that tells us the following:
โIn every limited arithmetic progression, the sum of the means equidistant from the extremes is equal to the sum of the extremesโ
To exemplify the theorem, I will take the following sequence as an example: 5, 10, 15, 20, 25, 30
Now, letโs start with the simplest part of the previous theoremโฆ Who are the ends of a progression? Well, they are the first and last terms of the sequence, right? After all, we are talking about extremes.
As we have already seen in the elabora, the foreground of a sequence is equal to the letter a. He Last term is equal to the letter I. You already know them because they are in the elabora.
Note: remember that I used those letters, which I have seen in books, but in theory you cรกnido take whatever letters you want.
As long as you know what each letter means.
Howeverโฆ What are the means equidistant from the extremes? Well, in this case, we are talking about two terms that are the same distance from the extremes.
Following the example sequence, the second term (10) of the sequence and the fifth term (25) of the sequence comply with the above.
After all, the second term is next to the first term, and the fifth term is next to the last term.
Likewise, the third and fourth terms are also means equidistant from the extremes.
### Does the theorem hold?
Now we are going to verify that the theorem is true.
โข The sum of the extremes is equal to 35. a = 5; I = 30.
Therefore, 5 + 30 equals 35.
โข Now, I am going to use the second term and the fifth term of the sequence as means equidistant from the extremes.
The second term is equal to 10; the fifth term is equal to 25.
Therefore, 10 + 25 = 35. As you perro see, it is also true.
You cรกnido try to make the sum of the other equidistant means so that you cรกnido see that it is fulfilled.
## Arithmetic Progression Sum Elabora
Since we have seen that the previous theorem is true, now I am going to provide you with two formulas that will help you find the sum of an arithmetic progression.
### If you know the last term of the sequence
If you know the last term of the progression, then you perro use the following elabora to find the sum of the progression:
As an example I will use the following sequence: 5, 10, 15, 20, 25, 30
As you perro see, we already have all the data we need, that is:
Therefore: S = (5 + 30) 6 / 2
The answer is equal to 105. In fact, if you think itโs not true, then you cรกnido do the sum: 5 + 10 + 15 + 20 + 25 + 30 = 105.
That way, if they put you to add all the terms of a sequence that has 100 or 1000 terms, it wonโt take you all day to do the sum.
### If you do not know the last term of the sequence
If you do not know the last term of the sequence and do not want to obtain it, then you perro use the following elabora:
I am going to use the same sequence as in the previous example, but I am going to imagine that I do not know the last term (30).
Therefore, the sequence looks like this: 5, 10, 15, 20, 25, __
So the data we have is:
Therefore: S = (6/2)(2*5 + (6 -1) 5) = 105.
As you perro see, it gives us the same result.
Although, you cรกnido choose the elabora you want to use.
After all, you cรกnido get the data you need.
## Solved exercises of arithmetic progression
Well, at this point you already know enough to be able to start solving some exercises.
Therefore, next, I am going to provide you with some solved exercises of arithmetic progression.
I recommend that you try to solve them first and then check your result.
### Exercise 1
The data for the first year are as follows:
โข a = 3.
โข d = 3.
โข n=10.
โข I = ? What is the last term of the sequence?
To find the last term of the progression, what you have to do is the following:
I = a + (n โ 1) d
I = 3 + (10 โ 1) 3
Therefore, the last term of the progression is equal to 30. In fact, it is the table of 3, but so that you perro see that it is true, I am going to put the ten terms of the sequence: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
### Exercise 2
For exercise two, the data is as follows:
โข to = ?
โข d = 7.
โข n=5.
โข I = 42.
To find the first term of an arithmetic progression, then, you have to do the following:
a = I โ (n โ 1) d
a = 42 โ (5 โ 1) 7
Therefore, the first term is equal to 14.
In such a way that the sequence is as follows: 14, 21, 28, 35, 42
### Exercise 3
For exercise 3, I will use the following data:
โข a = 50.
โข I = 10.
โข n=6.
โข d = ?
So that you cรกnido find the constant difference in a progression, what you have to do is the following:
d = (I โ a) / (N โ 1)
D = (10-50) / (6-1)
Therefore, the answer of the third exercise of arithmetic progression is equal to โ 8.
### Exercise 4
The last exercise has the following data:
โข a = 79.
โข d = -9.
โข I = 25.
โข n = ?
In order for you to find the number of terms in the sequence, you have to do the following:
n = ( (Ia) /(d) ) + 1
n = ((25 โ 79) / -9) + 1
Therefore, the number of terms in the sequence is equal to 7.
We hope you liked our article What is arithmetic progression?
and everything related to earning money, getting a job, and the economy of our house.
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Mathbox for Jonathan Ben-Naim < Previousย ย Next > Nearby theorems Mirrorsย > ย Homeย > ย MPE Homeย > ย Th. Listย > ย Mathboxesย > ย bnj1198 Structured versionย ย Visualization versionย ย GIF version
Theorem bnj1198ย 32141
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1198.1 (๐ โ โ๐ฅ๐)
bnj1198.2 (๐โฒ๐)
Assertion
Ref Expression
bnj1198 (๐ โ โ๐ฅ๐โฒ)
Proof of Theorem bnj1198
StepHypRef Expression
1ย bnj1198.1 . 2 (๐ โ โ๐ฅ๐)
2ย bnj1198.2 . . 3 (๐โฒ๐)
32exbiiย 1849 . 2 (โ๐ฅ๐โฒ โ โ๐ฅ๐)
41, 3sylibrย 237 1 (๐ โ โ๐ฅ๐โฒ)
Colors of variables: wff setvar class Syntax hints: ย โ wiย 4 ย โ wbย 209 ย โwexย 1781 This theorem was proved from axioms: ย ax-mpย 5 ย ax-1ย 6 ย ax-2ย 7 ย ax-3ย 8 ย ax-genย 1797 ย ax-4ย 1811 This theorem depends on definitions: ย df-biย 210 ย df-exย 1782 This theorem is referenced by: ย bnj1209 ย 32142 ย bnj1275 ย 32159 ย bnj1340 ย 32169 ย bnj1345 ย 32170 ย bnj605 ย 32253 ย bnj607 ย 32262 ย bnj906 ย 32276 ย bnj908 ย 32277 ย bnj1189 ย 32355 ย bnj1450 ย 32396 ย bnj1312 ย 32404
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# An algebra which is a direct sum of simple sub-bimodules over a subalgebra
Let $$A$$ be an infinite-dimensional noncommutative algebra over a field, let $$B$$ be an infinite-dimensional subalgebra of $$A$$, and let $$A$$ be a direct sum of projective simple $$B$$-sub-bimodules. Then can one conclude that $$A$$, or indeed $$B$$, is a semisimple ring?
EDIT: I should highlight that I am interested only in the case where $$A$$ and $$B$$ are infinite-dimensional algebras.
โข It is B that must be semisimple and I guess separable โย Benjamin Steinberg Jun 13 at 16:33
โข Ah ok, so to get this straight: From the assumptions above on $A$ and $B$, we can conclude that $B$ is a semisimple ring? โย Boris Henriques Jun 13 at 16:35
โข I think so. B will be a semisimple bimodule and the simple summands are a subset of those appearing in A. Thus B is a projective B-B bimodule and hence separable. Any separable algebra over a field is semisimple. I'm assuming that in these bimodules the left and right actions of k agree which I guess is fine since this is true for the bimodule structure on A. โย Benjamin Steinberg Jun 13 at 16:42
โข Great! Please put this as an answer and I will accept it! โย Boris Henriques Jun 13 at 16:51
โข I probably won't have a chance to write a detailed answer until later. โย Benjamin Steinberg Jun 13 at 17:06
@BugsBunny answered the original version of the question. I'll answer the new version. The algebra $$B$$ must be finite dimensional and semisimple under these hypothesis, and even stronger, it must be separable meaning that it remains semisimple even under base extension.
Let $$B^{e}=B\otimes_k B^{op}$$ be the enveloping algebra. Note that (left) $$B^e$$-modules equal $$B$$-$$B$$-bimodules in which the left and right actions of $$k$$ coincide. In particular $$A$$ and $$B$$ are $$B^e$$-modules.
Recall that $$B$$ is separable over $$k$$ if $$B$$ is a projective left $$B^e$$-module. This is well known to be equivalent to $$B$$ being finite dimensional over $$k$$ and for each field extension $$L/K$$, the algebra $$L\otimes_k B$$ is semisimple. All these things can be found in Pierce's book.
Now under your assumption, $$A$$ is a direct sum of simple $$B^e$$-modules that are projective. Thus $$A$$ is a semisimple $$B^e$$-module and hence the same is true for its $$B^e$$-submodule $$B$$. Moreover, if $$S$$ is a simple $$B^e$$-submodule of $$B$$, then it must be nontrivial under the one of the projections of $$A$$ onto a simple $$B^e$$-summand and so $$S$$ is isomorphic to one of the simple $$B^e$$-summands in $$A$$ by Schur's lemma. Therefore, $$B$$ is a direct sum of projective $$B^e$$-modules and hence is projective. Thus $$B$$ is separable over $$K$$ and hence finite dimensional and semisimple (even after base change).
So your desired situation cannot occur if $$B$$ is infinite dimensional over $$k$$.
If you drop the projective hypothesis you could take $$B$$ a finite direct product of simple $$k$$-algebras at least one of which is infinite dimensional and take $$A=B$$ and $$A$$ will be a finite direct sum of simple $$B$$-bimodules. You can even make $$B$$ finitely presented as a $$k$$-algebra.
โข Ok, so now I see the issue. I meant projective as a left module, and projective as a right module, not projective as a bimodule. But as I wrote it indicates projective as a bimodule. But it's too late to change to change it now, so thanks a lot for the answer! โย Boris Henriques Jun 13 at 21:50
โข Maybe you should ask a question about the specific algebras you are interested in. You want A a direct sum of simple B-bimodules and B projective as a left and right B-module? โย Benjamin Steinberg Jun 13 at 23:12
โข I meant A projective as a left and right B-module. โย Benjamin Steinberg Jun 14 at 0:35
โข @BugsBunny I believe the issue with your counterexample is writing A as a direct sum of projective simple B-bimodules. I dont think your B is a projective B-bimodules. There is a big difference between projective as a one sided module and as a bimodule. If you take $B\otimes_{\mathbb Q} B$ it contains $\mathbb C\otimes_{\mathbb Q} \mathbb C$ as a summand which is some fairly huge and complicated commutative algebra which is uncountably dimension over $\mathbb C$. Is it semisimple? โย Benjamin Steinberg Jun 15 at 14:56
โข The OP wasn't clear what kind of bimodules are allowed. Since he is taking about k-algebras I interpreted bimodules as $B^e$-modules. I am not convinced of you take $k=\mathbb Q$ and $B=\mathbb C=A$ that $B$ is a projective $\mathbb C\otimes_Q \mathbb C=B^e$-module. But since projectivity depends on the category of bimodules the OP should have been clearer. โย Benjamin Steinberg Jun 15 at 15:00
No. Take any $$A$$ and take the ground field $$k=k1_A$$ as $$B$$.
โข For latecomers, please note that this answered the original version of the question which has since changed. โย Benjamin Steinberg Jun 13 at 20:43
โข It did not change that much :-)) โย Bugs Bunny Jun 15 at 14:17
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MATH1231 - Tangent Planes
1. Take partial derivative of x (โz/โx)
โข This will be in terms of x and y, so sub in the points you have for them
โข Use the point gradient formula with z and x, leaving y to its original value:
(1)
\begin{align} z โ z_1 = \frac {โz}{โx}(x-x_1), y = y_1 \end{align}
โข Set $ฮป = (x โ x_1)$, and then put the equation in parametric vector form:
(2)
\begin{align} (x, y, z) = (x_1, y_1, z_1) + ฮป(1, 0, \frac {โz}{โx}) \end{align}
2. Take partial derivative of y (โz/โy)
โข This will be in terms of x and y, so sub in the points you have for them.
โข Use the point gradient formula with z and y, leaving x in its original form:
(3)
\begin{align} z โ z_1 = \frac {โz}{โy}(y-y_1), x = x_1 \end{align}
โข Set $ฮผ = (y โ y_1)$, and then put the equation in parametric vector form:
(4)
\begin{align} (x, y, z) = (x_1, y_1, z_1) + ฮผ(0, 1, \frac {โz}{โy}) \end{align}
3. Finding the Tangent Plane
โข Both of these lines lie in the tangent plane, and since the two vectors are non-parallel, the tangent plane to the surface at $(x_1, y_1, z_1)$ is equal to
(5)
\begin{align} (x, y, z) = (x_1, y_1, z_1) + ฮป(1, 0, \frac {โz}{โx}) + ฮผ(0, 1, \frac {โz}{โy}) \end{align}
4. Finding the Normal
โข We now need to find the normal to the surface at the point $(x_1, y_1, z_1)$
โข This equals the cross product of $(0, 1, \frac {โz}{โy}) and (1, 0, \frac {โz}{โx})$
โข So equals $(\frac {โz}{โx}, \frac {โz}{โy}, -1)$
โข Hence tangent plane: $n \cdot (x โ x_1, y โ y_1, z โ z_1) = 0$
page revision: 9, last edited: 12 Aug 2011 01:52
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# TECHNIQUES OF INTEGRATION
## Presentation on theme: "TECHNIQUES OF INTEGRATION"โ Presentation transcript:
TECHNIQUES OF INTEGRATION
8 TECHNIQUES OF INTEGRATION
As we have seen, integration is more challenging than differentiation.
TECHNIQUES OF INTEGRATION As we have seen, integration is more challenging than differentiation. In finding the derivative of a function, it is obvious which differentiation formula we should apply. However, it may not be obvious which technique we should use to integrate a given function.
Until now, individual techniques have been applied in each section.
TECHNIQUES OF INTEGRATION Until now, individual techniques have been applied in each section. For instance, we usually used: Substitution in Exercises 5.5 Integration by parts in Exercises 7.1 Partial fractions in Exercises 7.4
Strategy for Integration
TECHNIQUES OF INTEGRATION 7.5 Strategy for Integration In this section, we will learn about: The techniques to evaluate miscellaneous integrals.
STRATEGY FOR INTEGRATION
In this section, we present a collection of miscellaneous integrals in random order. The main challenge is to recognize which technique or formula to use.
STRATEGY FOR INTEGRATION
No hard and fast rules can be given as to which method applies in a given situation. However, we give some advice on strategy that you may find useful.
STRATEGY FOR INTEGRATION
A prerequisite for strategy selection is a knowledge of the basic integration formulas.
STRATEGY FOR INTEGRATION
In the upcoming table, we have collected: The integrals from our previous list Several additional formulas we have learned in this chapter
STRATEGY FOR INTEGRATION
Most should be memorized. It is useful to know them all. However, the ones marked with an asterisk need not be memorizedโthey are easily derived.
TABLE OF INTEGRATION FORMULAS
TABLE OF INTEGRATION FORMULAS
TABLE OF INTEGRATION FORMULAS
TABLE OF INTEGRATION FORMULAS
Formula 19 can be avoided by using partial fractions. Trigonometric substitutions can be used instead of Formula 20.
STRATEGY FOR INTEGRATION
Once armed with these basic integration formulas, you might try this strategy: Simplify the integrand if possible. Look for an obvious substitution. Classify the integrand according to its form. Try again.
1. SIMPLIFY THE INTEGRAND
Sometimes, the use of algebraic manipulation or trigonometric identities will simplify the integrand and make the method of integration obvious.
1. SIMPLIFY THE INTEGRAND
Here are some examples:
2. LOOK FOR OBVIOUS SUBSTITUTION
Try to find some function u = g(x) in the integrand whose differential du = gโ(x) dx also occurs, apart from a constant factor. For instance, in the integral , notice that, if u = x2 โ 1, then du = 2x dx. So, we use the substitution u = x2 โ 1 instead of the method of partial fractions.
3. CLASSIFY THE FORM If Steps 1 and 2 have not led to the solution, we take a look at the form of the integrand f(x).
3 a. TRIGONOMETRIC FUNCTIONS
We use the substitutions recommended in Section 7.2 if f(x) is a product of: sin x and cos x tan x and sec x cot x and csc x
3 b. RATIONAL FUNCTIONS If f is a rational function, we use the procedure of Section 7.4 involving partial fractions.
3 c. INTEGRATION BY PARTS If f(x) is a product of a power of x (or a polynomial) and a transcendental function (a trigonometric, exponential, or logarithmic function), we try integration by parts. We choose u and dv as per the advice in Section 7.1 If you look at the functions in Exercises 7.1, you will see most of them are the type just described.
3 d. RADICALS Particular kinds of substitutions are recommended when certain radicals appear. If occurs, we use a trigonometric substitution according to the table in section 7.3 If occurs, we use the rationalizing substitution More generally, this sometimes works for
4. TRY AGAIN If the first three steps have not produced the answer, remember there are basically only two methods of integration: Substitution Parts
4 a. TRY SUBSTITUTION Even if no substitution is obvious (Step 2), some inspiration or ingenuity (or even desperation) may suggest an appropriate substitution.
4 b. TRY PARTS Though integration by parts is used most of the time on products of the form described in Step 3 c, it is sometimes effective on single functions. Looking at Section 7.1, we see that it works on tan-1x, sin-1x, and ln x, and these are all inverse functions.
4 c. MANIPULATE THE INTEGRAND
Algebraic manipulations (rationalizing the denominator, using trigonometric identities) may be useful in transforming the integral into an easier form. These manipulations may be more substantial than in Step 1 and may involve some ingenuity.
4 c. MANIPULATE THE INTEGRAND
Here is an example:
4 d. RELATE TO PREVIOUS PROBLEMS
When you have built up some experience in integration, you may be able to use a method on a given integral that is similar to a method you have already used on a previous integral. You may even be able to express the given integral in terms of a previous one.
4 d. RELATE TO PREVIOUS PROBLEMS
For instance, โซ tan2x sec x dx is a challenging integral. If we make use of the identity tan2x = sec2x โ 1, we can write: Then, if โซ sec3x dx has previously been evaluated, that calculation can be used in the present problem.
Sometimes, two or three methods are required to evaluate an integral.
4 e. USE SEVERAL METHODS Sometimes, two or three methods are required to evaluate an integral. The evaluation could involve several successive substitutions of different types. It might even combine integration by parts with one or more substitutions.
STRATEGY FOR INTEGRATION
In the following examples, we indicate a method of attack, but do not fully work out the integral.
SIMPLIFY INTEGRAND Example 1 In Step 1, we rewrite the integral: It is now of the form โซtanmx secnx dx with m odd. So, we can use the advice in Section 7.2
Suppose, in Step 1, we had written:
TRY SUBSTITUTION Example 1 Suppose, in Step 1, we had written: Then, we could have continued as follows.
TRY SUBSTITUTION Example 1 Substitute u = cos x:
TRY SUBSTITUTION Example 2 According to (ii) in Step 3 d, we substitute u = โx. Then, x = u2, so dx = 2u du and The integrand is now a product of u and the transcendental function eu. So, it can be integrated by parts.
RATIONAL FUNCTIONS Example 3 No algebraic simplification or substitution is obvious. So, Steps 1 and 2 donโt apply here. The integrand is a rational function. So, we apply the procedure of Section 7.4, remembering that the first step is to divide.
Here, Step 2 is all that is needed.
TRY SUBSTITUTION Example 4 Here, Step 2 is all that is needed. We substitute u = ln x, because its differential is du = dx/x, which occurs in the integral.
MANIPULATE INTEGRAND Example 5 Although the rationalizing substitution works here [(ii) in Step 3 d], it leads to a very complicated rational function. An easier method is to do some algebraic manipulation (either as Step 1 or as Step 4 c).
Multiplying numerator and denominator by , we have:
MANIPULATE INTEGRAND Example 5 Multiplying numerator and denominator by , we have:
CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS?
The question arises: Will our strategy for integration enable us to find the integral of every continuous function? For example, can we use it to evaluate โซ ex2dx?
CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS?
The answer is โNo.โ At least, we cannot do it in terms of the functions we are familiar with.
ELEMENTARY FUNCTIONS The functions we have been dealing with in this book are called elementary functions. For instance, the function is an elementary function.
If f is an elementary function, then fโ is an elementary function.
ELEMENTARY FUNCTIONS If f is an elementary function, then fโ is an elementary function. However, โซ f(x) dx need not be an elementary function.
Consider f(x) = ex2. ELEMENTARY FUNCTIONS
Since f is continuous, its integral exists. If we define the function F by then we know from Part 1 of the Fundamental Theorem of Calculus (FTC1) that Thus, f(x) = ex2 has an antiderivative F.
However, it has been proved that F is not an elementary function.
ELEMENTARY FUNCTIONS However, it has been proved that F is not an elementary function. This means that, however hard we try, we will never succeed in evaluating โซ ex2dx in terms of the functions we know. In Chapter 11, however, we will see how to express โซ ex2dx as an infinite series.
The same can be said of the following integrals:
ELEMENTARY FUNCTIONS The same can be said of the following integrals:
ELEMENTARY FUNCTIONS In fact, the majority of elementary functions donโt have elementary antiderivatives. You may be assured, though, that the integrals in the exercises are all elementary functions.
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The OEIS is supported by the many generous donors to the OEIS Foundation.
Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
A192271 Anti-weird numbers. 0
11, 12, 13, 14, 15, 18, 20, 21, 25, 27, 28, 30, 37, 40, 42, 43, 46, 47, 48, 50, 55, 57, 58, 62, 65, 66, 75, 78, 80, 84, 86, 87, 90, 91, 92, 93, 97, 99, 100, 107, 111, 113, 118, 119, 120, 121, 124, 125, 126, 128, 132, 133, 135, 136, 140, 142, 145, 152, 153, 155, 160, 161, 163, 168, 170, 173, 177, 180, 181, 183, 184, 186, 188, 190, 192, 196, 197, 198, 204, 205, 208, 210, 212, 213, 218, 222, 223 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,1 COMMENTS Like A006037 but using anti-divisors: Anti-weird numbers are anti-abundant (A192268) but not pseudo anti-perfect (A192270). LINKS EXAMPLE 25 is an anti-weird number because it is anti-abundant (its anti-divisors are 2, 3, 7, 10, 17 and their sum is 39 > 25) and no subsets of its anti-divisors add up to 25. MAPLE # see A066272 isA192270 := proc(n) local a, S ; a := antidivisors(n) ;ย ย S := combinat[subsets](a) ; while not S[finished] do if convert(S[nextvalue](), `+`) = n then return true; end if; end do; false ; end proc: isA192268 := proc(n) A066417(n) > n ; end proc: isA192271 := proc(n) isA192268(n) and not isA192270(n) ; end proc: for n from 1 to 40 do if isA192271(n) then printf("%d, ", n) ; end if; end do: # R. J. Mathar, Jul 04 2011 CROSSREFS Cf. A006037, A066272, A192268, A192270. Sequence in context: A296710 A297143 A138595 * A214423 A185300 A097932 Adjacent sequences:ย ย A192268 A192269 A192270 * A192272 A192273 A192274 KEYWORD nonn AUTHOR Paolo P. Lava, Jun 28 2011 STATUS approved
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Last modified May 25 19:14 EDT 2022. Contains 354071 sequences. (Running on oeis4.)
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# Calculus for the Social Sciences
Effective Date:
Course
Discontinued
No
Course Code
MATH 1125
Descriptive
Calculus for the Social Sciences
Department
Mathematics
Faculty
Science & Technology
Credits
3.00
Start Date
End Term
Not Specified
PLAR
No
Semester Length
15 weeks
Max Class Size
35
Contact Hours
4 hours lecture + 1 hour tutorial /week
Method Of Instruction
Lecture
Tutorial
Methods Of Instruction
Lectures, tutorials,ย problem sessions and assignments
Course Description
This course is an introduction to differential calculus for students in business, social sciences and biological sciences. Topics include limits, differentiation techniques for algebraic, logarithmic, exponential and trigonometric functions, mathematical modeling, applications to graphing and optimization, implicit differentiation and differentials.
Course Content
1. Limits and Limit Laws
2. Continuity
3. Tangent Lines and the Derivative
4. Differentiation Rules and Implicit Differentiation
5. Related Rates
6. Marginal Analysis and Differentials
7. Applications to Graphing Functions
8. Determining the Extrema of Functions
Learning Outcomes
Upon completion of MATH 1125 the student should be able to:
โข evaluate elementary limits involving algebraic, exponential, logarithmic and trigonometric functions
โข describe the concept of continuity and determine intervals upon which a function is continuous
โข apply the intermediate value theorem
โข find average and instantaneous rates of change
โข find derivatives and relate them to tangent lines and instantaneous rates of change
โข use differentiation rules to compute the derivatives of algebraic, exponential, logarithmic, trigonometric and implicit functions
โข formulate and solve problems involving marginal analysis, elasticity, points of diminishing returns, and other forms of economic modeling
โข apply the concepts of differentials and linear approximations
โข sketch graphs of functions by applying first and second derivative techniques as well as analysis of vertical, horizontal and slant asymptotes
โข use differentiation to determine the local and absolute extrema of functions
โข use calculus methods to solve problems of time value of money: interest, annuities, loans, investments and the value of a continuous money flow
Additional topics that may be included in the course:
โข compute the definite and indefinite integral of a function
โข use integration techniques (substitution, integration by parts and others) to compute integrals
โข apply the integral to problems in Business and the Social Sciences
โข use Newtonโs method to determine points of intersection
โข solving problems involving Markov Chains, Linear Programming and Game Theory
Means of Assessment
Evaluation will be carried out in accordance with Douglas College policy.ย The instructor will present a written course outline with specific evaluation criteria at the beginning of the semester.ย Evaluation will be based on some of the following:
Weekly tests 0-40% Term tests 20-70% Assignments 0-20% Attendance/participation 0-5% Tutorials 0-10% Final examination 30-40%
Textbook Materials
Textbook varies by semester, please see College Bookstore for current version.
Typical text:
Hoffmann and Bradley, Applied Calculus, current edition, McGraw Hill
Prerequisites
MATH 1105; or MATH 1110; or Principles of Math 12 with a B or better or an approved equivalent; or Precalculus 12 with a B or better.
Which Prerequisite
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# Radon-Nikodym derivative of product measure
For $j=1,2$, let $\nu_{j},\mu_{j}$ be $\sigma$-finite measures on $(X_{j},\mathcal{M}_{j})$ such that $\nu_{j}\ll\mu_{j}$. I want to show that $\nu_{1}\times\nu_{2}\ll\mu_{1}\times\mu_{2}$ and that $\frac{d(\nu_{1}\times\nu_{2})}{d(\mu_{1}\times\mu_{2})}(x_{1},x_{2})=\frac{d\nu_{1}}{d\mu_{1}}(x_{1})\frac{d\nu_{2}}{d\mu_{2}}(x_{2})$.
I tried to show that if $E\in\mathcal{M}_{1}\otimes\mathcal{M}_{2}$ and $\mu_{1}\times\mu_{2}(E)=0$, then $\nu_{1}\times\nu_{2}(E)=0$. This criterion holds for measurable rectangles so then I tried considering $\{E\in\mathcal{M}_{1}\otimes\mathcal{M}_{2}|\mu_{1}\times\mu_{2}(E)>0\;\text{or}\;\nu_{1}\times\nu_{2}(E)=0\}$ and show that it is a $\sigma$-algebra containing all measurable rectangles but I guess it didn't quite work.
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Maybe you could use Tonelli? โย copper.hat Nov 4 '12 at 16:45
Denote $X=X_1\times X_2$, $\mu=\mu_1\times\mu_2$, $\nu=\nu_1\times\nu_2$, $\mathcal{M}=\mathcal{M}_1\times\mathcal{M}_2$ and $$f(x_1,x_2)=\dfrac{d\nu_1}{d\mu_1}(x_1)\dfrac{d\nu_2}{d\mu_2}(x_2).$$ Evidently $f$ is non-negative, $\mathcal{M}$ measurable and $\mu$ locally integrable. Then as @copper.hat commented, by Fubini's theorem(or Tonelli's theorem instead), we have:
$$\nu(E_1\times E_2)=\nu_1(E_1)\cdot\nu_2(E_2)=\int_{E_1}\dfrac{d\nu_1}{d\mu_1}d\mu_1\times\int_{E_2}\dfrac{d\nu_2}{d\mu_2}d\mu_2=\int_{E_1\times E_2}f~d\mu.$$
Consider the collection
$$\mathcal{C}:=\{E\in \mathcal{M}:\nu(E)=\int_E f~d\mu\}$$ and we want to show $\mathcal{C}=\mathcal{M}$. We have shown $\Pi:=\{E\in\mathcal{M}:E=E_1\times E_2\}\subset \mathcal{C}$, it suffices to show $\mathcal{C}$ is a $\sigma$-algebra. First let us assume that $\mu(X)<\infty$. Since $\Pi$ is a $\pi$-system, we only need to show $\mathcal{C}$ is a $\lambda$-system: (i) $X\in\mathcal{C}$; (ii) if $D,E\in\mathcal{C}$ with $D\subset E$, then $E\setminus D\in\mathcal{C}$; and if $\{E_n:n\ge 1\}\subset \mathcal{C}$ is an increasing sequence, then $\cup_n E_n\in\mathcal{C}$. The first two items are obvious and the last one follows from monotone convergence theorem. Now we have proved $\mathcal{C}=\mathcal{M}$ with the additional assumption $\mu(X)<\infty$. For general $X$, let $\{F_n=F_{1,n}\times F_{2,n}:n\ge 1\}\subset\Pi\subset\mathcal{C}$ be such that $\mu(F_n)<\infty$ for every $n$ and $\cup_n F_n=X$. Then the argument before implies that for every $n$, $\{E\in \mathcal{C}:E\subset F_n\}=\{E\in \mathcal{M}:E\subset F_n\}$. Then given $E\in\mathcal{M}$, $E\subset F_n\in\mathcal{C}$ for every $n$, so again by monotone convergence theorem, $E\in\mathcal{C}$, which completes the proof.
Therefore, when $E\in\mathcal{M}$, $\nu(E)=\int_E f~d\mu$. The conclusion follows.
-
When you use the monotone convergence theorem to show that $\bigcup_{n}E_{n}\in\mathcal{C}$, are you considering the sequence $(\bigcup_{n=1}^{k}E_{n})_{k=1}^{\infty}$? Do we need to first show that $\mathcal{C}$ is closed under finite unions? โย user44532 Nov 4 '12 at 22:23
@CYC: Yes, you are correct, and it seems not so easy to prove that directly. Thank you for pointing it out. Maybe to save words, I should use $\pi-\lambda$ theorem to fix this problem. โย 23rd Nov 5 '12 at 0:10
What is this $\pi-\lambda$ theorem that you mentioned? โย user44532 Nov 5 '12 at 0:44
@CYC: See the updated answer for details. โย 23rd Nov 5 '12 at 0:46
I tried showing that $\mathcal{C}$ is a monotone class containing all finite disjoint unions of rectangles, then apply the monotone class lemma. It seems to work as well. โย user44532 Nov 5 '12 at 4:45
We use the following lemma:
Let $(S,\mathcal A)$ a measurable space, and $\mu$, $\nu$ two finite measures. We have $\nu\ll\mu$ if and only if for all $\varepsilon>0$, we can find $\delta>0$ such that if $A\in\mathcal A$ and $\mu(A)\leq\delta$ then $\nu(A)\leq \varepsilon$.
Let $\{E_i\}\subset\mathcal M_1$, $\{F_j\}\subset \mathcal M_2$, where $\mu_1(E_i)+\nu_1(E_i)$ and $\mu_2(F_j)+\nu_2(F_j)$ are finite for all $i,j$ and $\bigcup_{i=1}^{+\infty}E_i=X_1$, $\bigcup_{j=1}^{+\infty}F_j=X_2$. Define the finite measures $\mu^{(i,j)}(S):=\mu_1\otimes \mu_2(E_i\times F_j\cap S)$, and $\nu^{(i,j)}(S):=\nu_1\otimes \nu_2(E_i\times F_j\cap S)$. We just have to show that $\nu^{(i,j)}\ll\mu^{(i,j)}$.
Let $\mu_1^{i}(S):=\mu_1(E_i\cap S)$ and similarly for the other measures.
Let $\varepsilon>0$, $i,j$ fixed integers and $\sqrt\delta$ working for $\varepsilon$.
As $\nu^{(i,j)}+\mu^{(i,j)}$ is finite, if $\mu^{(i,j)}(S)\leq \delta/2$, we can find, by this result, $S':=\bigsqcup_{k=1}^NA_k\times B_k$ where $$\sum_{k=1}^N\nu_1(E_i\cap A_k)\nu_2(F_j\cap B_k)+\sum_{k=1}^N\mu_1(E_i\cap A_k)\mu_2(F_j\cap B_k)\leq \delta.$$ Indeed, we take $S'$ of the previous form such that $(\nu^{(i,j)}+\mu^{(i,j)})(S\Delta S')\lt\delta/2$. This implies that $$(\nu^{(i,j)}+\mu^{(i,j)})(S')\leqslant (\nu^{(i,j)}+\mu^{(i,j)})(S)+\delta/2\leqslant \delta.$$
Let $I:=\{k\in [N],\mu_1(E_i\cap A_k)\leq \sqrt \delta\}$, and $I':=\{k\in [N],\mu_2(F_j\cap B_k)\leq \sqrt \delta\}$. Then $I\cup I'=[N]$, which gives \begin{align} \nu^{(i,j)}(S)&\leq \delta+\nu^{(i,j)}(S')\\ &=\delta+\varepsilon\sum_{k\in I}\nu_2(F_j\cap B_k)+\varepsilon\sum_{k\in I'}\nu_1(E_i\cap A_k)\\ &\leq \varepsilon(1+\mu_1(E_i)+\mu_2(F_j)), \end{align} as we can assume WLOG $\delta\leq\varepsilon$.
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I do not understand how you arrive at the first main equation, ending with ''$\leq \delta$''. Can you explain more? โย Doug Nov 7 '14 at 1:15
@DanDouglas I've edited. Is it clearer? โย Davide Giraudo Nov 7 '14 at 10:09
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Skip to main content
Subsection1.2.4The vector $p$-norms
A vector norm is a measure of the magnitude of a vector. The Euclidean norm (length) is merely the best known such measure. There are others. A simple alternative is the 1-norm.
Definition1.2.4.1. Vector 1-norm.
The vector 1-norm, $\| \cdot \|_1 : \C^m \rightarrow \mathbb R \text{,}$ is defined for $x \in \C^m$ by
\begin{equation*} \| x \|_1 = \vert \chi_0 \vert + \vert \chi_1 \vert + \cdots + \vert \chi_{m-1} \vert = \sum_{i=0}^{m-1} \vert \chi_i \vert. \end{equation*}
Homework1.2.4.1.
Prove that the vector 1-norm is a norm.
Solution
We show that the three conditions are met:
Let $x, y \in \C^m$ and $\alpha \in \mathbb C$ be arbitrarily chosen. Then
โข $x \neq 0 \Rightarrow \| x \|_1 > 0$ ($\| \cdot \|_1$ is positive definite):
Notice that $x \neq 0$ means that at least one of its components is nonzero. Let's assume that $\chi_j \neq 0 \text{.}$ Then
\begin{equation*} \| x \|_1 = \vert \chi_0 \vert + \cdots + \vert \chi_{m-1} \vert \geq \vert \chi_j \vert > 0 . \end{equation*}
โข $\| \alpha x \|_1 = \vert \alpha \vert \| x \|_1$ ($\| \cdot \|_1$ is homogeneous):
\begin{equation*} \begin{array}{l} \| \alpha x \|_1 ~~~=~~~~ \lt \mbox{ scaling~a~vector-scales-its-components; definition} \gt \\ \vert \alpha \chi_0 \vert + \cdots + \vert \alpha \chi_{m-1} \vert \\ ~~~=~~~~ \lt \mbox{ algebra} \gt \\ \vert \alpha \vert \vert \chi_0 \vert + \cdots + \vert \alpha \vert \vert \chi_{m-1} \vert \\ ~~~=~~~~ \lt \mbox{ algebra} \gt \\ \vert \alpha \vert ( \vert \chi_0 \vert + \cdots + \vert \chi_{m-1} \vert ) \\ ~~~=~~~~ \lt \mbox{ definition} \gt \\ \vert \alpha \vert \| x \|_1. \end{array} \end{equation*}
โข $\| x + y \|_1 \leq \| x \|_1 + \| y \|_1$ ($\| \cdot \|_1$ obeys the triangle inequality):
\begin{equation*} \begin{array}{l} \| x + y \|_1 \\ ~~~=~~~~ \lt \mbox{ vector~addition;~definition~of~1-norm} \gt \\ \vert \chi_0 + \psi_0 \vert + \vert \chi_1 + \psi_1 \vert + \cdots + \vert \chi_{m-1} + \psi_{m-1} \vert \\ ~~~\leq~~~~ \lt \mbox{ algebra} \gt \\ \vert \chi_0 \vert + \vert \psi_0 \vert + \vert \chi_1 \vert + \vert \psi_1 \vert + \cdots + \vert \chi_{m-1} \vert + \vert \psi_{m-1} \vert \\ ~~~=~~~~ \lt \mbox{ commutivity} \gt \\ \vert \chi_0 \vert + \vert \chi_1 \vert + \cdots + \vert \chi_{m-1} \vert + \vert \psi_0 \vert + \vert \psi_1 \vert + \cdots + \vert \psi_{m-1} \vert \\ ~~~= ~~~~ \lt \mbox{ associativity;~definition} \gt \\ \| x \|_1 + \| y \|_1. \end{array} \end{equation*}
The vector 1-norm is sometimes referred to as the "taxi-cab norm". It is the distance that a taxi travels, from one point on a street to another such point, along the streets of a city that has square city blocks.
Another alternative is the infinity norm.
Definition1.2.4.2. Vector $\infty$-norm.
The vector $\infty$-norm, $\| \cdot \|_\infty : \C^m \rightarrow \mathbb R \text{,}$ is defined for $x \in \C^m$ by
\begin{equation*} \| x \|_\infty = \max( \vert \chi_0 \vert, \ldots , \vert \chi_{m-1} ) = \max_{i=0}^{m-1} \vert \chi_i \vert. \end{equation*}
The infinity norm simply measures how large the vector is by the magnitude of its largest entry.
Homework1.2.4.2.
Prove that the vector $\infty$-norm is a norm.
Solution
We show that the three conditions are met:
Let $x, y \in \C^m$ and $\alpha \in \mathbb C$ be arbitrarily chosen. Then
โข $x \neq 0 \Rightarrow \| x \|_\infty > 0$ ($\| \cdot \|_\infty$ is positive definite):
Notice that $x \neq 0$ means that at least one of its components is nonzero. Let's assume that $\chi_j \neq 0 \text{.}$ Then
\begin{equation*} \| x \|_\infty = \max_{i=0}^{m-1} \vert \chi_i \vert \ge \vert \chi_j \vert > 0. \end{equation*}
โข $\| \alpha x \|_\infty = \vert \alpha \vert \| x \|_\infty$ ($\| \cdot \|_\infty$ is homogeneous):
\begin{equation*} \begin{array}{lcl} \| \alpha x \|_\infty \amp =\amp \max_{i=0}^{m-1} \vert \alpha \chi_i \vert \\ \amp =\amp \max_{i=0}^{m-1} \vert \alpha \vert \vert \chi_i \vert \\ \amp =\amp \vert \alpha \vert \max_{i=0}^{m-1} \vert \chi_i \vert \\ \amp =\amp \vert \alpha \vert \| x \|_\infty. \end{array} \end{equation*}
โข $\| x + y \|_\infty \leq \| x \|_\infty + \| y \|_\infty$ ($\| \cdot \|_\infty$ obeys the triangle inequality):
\begin{equation*} \begin{array}{lcl} \| x + y \|_\infty \amp =\amp \max_{i=0}^{m-1} \vert \chi_i + \psi_i \vert \\ \amp \leq\amp \max_{i=0}^{m-1} ( \vert \chi_i \vert + \vert \psi_i \vert ) \\ \amp \leq\amp \max_{i=0}^{m-1} \vert \chi_i \vert + \max_{i=0}^{m-1} \vert \psi_i \vert \\ \amp = \amp \| x \|_\infty + \| y \|_\infty. \end{array} \end{equation*}
In this course, we will primarily use the vector 1-norm, 2-norm, and $\infty$-norms. For completeness, we briefly discuss their generalization: the vector $p$-norm.
Definition1.2.4.3. Vector $p$-norm.
Given $p \geq 1 \text{,}$ the vector $p$-norm $\| \cdot \|_p : \C^m \rightarrow \mathbb R$ is defined for $x \in \C^m$ by
\begin{equation*} \| x \|_p = \sqrt[p]{\vert \chi_0 \vert^p + \cdots + \vert \chi_{m-1} \vert^p} = \left( \sum_{i=0}^{m-1} \vert \chi_i \vert^p \right)^{1/p}. \end{equation*}
The proof of this result is very similar to the proof of the fact that the 2-norm is a norm. It depends on Hรถlder's inequality, which is a generalization of the Cauchy-Schwartz inequality:
We skip the proof of Hรถlder's inequality and Theoremย 1.2.4.4. You can easily find proofs for these results, should you be interested.
Remark1.2.4.6.
The vector 1-norm and 2-norm are obviously special cases of the vector $p$-norm. It can be easily shown that the vector $\infty$-norm is also related:
\begin{equation*} \lim_{p \rightarrow \infty} \| x \|_p = \| x \|_{\infty}. \end{equation*}
Ponder This1.2.4.3.
Consider Homeworkย 1.2.3.3. Try to elegantly formulate this question in the most general way you can think of. How do you prove the result?
Ponder This1.2.4.4.
Consider the vector norm $\| \cdot \|: \Cm \rightarrow \mathbb R \text{,}$ the matrix $A \in \mathbb C^{m \times n}$ and the function $f: \Cn \rightarrow \mathbb R$ defined by $f( x ) = \| A x \| \text{.}$ For what matrices $A$ is the function $f$ a norm?
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NVIDIA Modulus Sym (Latest Release)
Sym (Latest Release)
# Turbulent physics: Zero Equation Turbulence Model
## Introduction
This tutorial walks you through the process of adding a algebraic (zero equation) turbulence model to the Modulus Sym simulations. In this tutorial you will learn the following:
1. How to use the Zero equation turbulence model in Modulus Sym.
2. How to create nodes in the graph for arbitrary variables.
Note
This tutorial assumes that you have completed the Introductory Example tutorial on Lid Driven Cavity Flow and have familiarized yourself with the basics of the Modulus Sym APIs.
## Problem Description
In this tutorial you will add the zero equation turbulence for a Lid Driven Cavity flow. The problem setup is very similar to the one found in the Introductory Example. The Reynolds number is increased to 1000. The velocity profile is kept the same as before. To increase the Reynolds Number, the viscosity is reduced to 1ย รย 10โ4 $$m^2/s$$.
## Case Setup
The case set up in this tutorial is very similar to the example in Introductory Example. It only describes the additions that are made to the previous code.
Note
The python script for this problem can be found at examples/ldc/ldc_2d_zeroEq.py
### Importing the required packages
Import Modulus Symโ ZeroEquation to help setup the problem. Other import are very similar to previous LDC.
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# limitations under the License.
import os
import warnings
from sympy import Symbol, Eq, Abs
import torch
import modulus.sym
from modulus.sym.hydra import to_absolute_path, instantiate_arch, ModulusConfig
from modulus.sym.utils.io import csv_to_dict
from modulus.sym.solver import Solver
from modulus.sym.domain import Domain
from modulus.sym.geometry.primitives_2d import Rectangle
from modulus.sym.domain.constraint import (
PointwiseBoundaryConstraint,
PointwiseInteriorConstraint,
)
from modulus.sym.domain.monitor import PointwiseMonitor
from modulus.sym.domain.validator import PointwiseValidator
from modulus.sym.domain.inferencer import PointwiseInferencer
from modulus.sym.eq.pdes.navier_stokes import NavierStokes
from modulus.sym.eq.pdes.turbulence_zero_eq import ZeroEquation
### Defining the Equations, Networks and Nodes
In addition to the Navier-Stokes equation, the Zero Equation turbulence model is included by instantiating the ZeroEquation equation class. The kinematic viscosity $$\nu$$ in the Navier-Stokes equation is a now a sympy expression given by the ZeroEquation. The ZeroEquation turbulence model provides the effective viscosity $$(\nu+\nu_t)$$ to the Navier-Stokes equations. The kinematic viscosity of the fluid calculated based on the Reynolds number is given as an input to the ZeroEquation class.
The Zero Equation turbulence model is defined in the equations below. Note, $$\mu_t = \rho \nu_t$$.
(144)$\mu_t=\rho l_m^2 \sqrt{G}$
(145)$G=2(u_x)^2 + 2(v_y)^2 + 2(w_z)^2 + (u_y + v_x)^2 + (u_z + w_x)^2 + (v_z + w_y)^2$
(146)$l_m=\min (0.419d, 0.09d_{max})$
Where, $$l_m$$ is the mixing length, $$d$$ is the normal distance from wall, $$d_{max}$$ is maximum normal distance and $$\sqrt{G}$$ is the modulus of mean rate of strain tensor.
The zero equation turbulence model requires normal distance from no slip walls to compute the turbulent viscosity. For most examples, signed distance field (SDF) can act as a normal distance. When the geometry is generated using either the Modulus Symโ geometry module/tesselation module you have access to the sdf variable similar to the other coordinate variables when used in interior sampling. Since zero equation also computes the derivatives of the viscosity, when using the PointwiseInteriorConstraint, you can pass an argument that says compute_sdf_derivatives=True. This will compute the required derivatives of the SDF like sdf__x, sdf__y, etc.
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def run(cfg: ModulusConfig) -> None:
# make geometry
height = 0.1
width = 0.1
x, y = Symbol("x"), Symbol("y")
rec = Rectangle((-width / 2, -height / 2), (width / 2, height / 2))
# make list of nodes to unroll graph on
ze = ZeroEquation(nu=1e-4, dim=2, time=False, max_distance=height / 2)
ns = NavierStokes(nu=ze.equations["nu"], rho=1.0, dim=2, time=False)
flow_net = instantiate_arch(
input_keys=[Key("x"), Key("y")],
output_keys=[Key("u"), Key("v"), Key("p")],
cfg=cfg.arch.fully_connected,
)
nodes = (
### Setting up domain, adding constraints and running the solver
This section of the code is very similar to LDC tutorial, so the code and final results is presented here.
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)
# make ldc domain
ldc_domain = Domain()
# top wall
top_wall = PointwiseBoundaryConstraint(
nodes=nodes,
geometry=rec,
outvar={"u": 1.5, "v": 0},
batch_size=cfg.batch_size.TopWall,
lambda_weighting={"u": 1.0 - 20 * Abs(x), "v": 1.0}, # weight edges to be zero
criteria=Eq(y, height / 2),
)
# no slip
no_slip = PointwiseBoundaryConstraint(
nodes=nodes,
geometry=rec,
outvar={"u": 0, "v": 0},
batch_size=cfg.batch_size.NoSlip,
criteria=y < height / 2,
)
# interior
interior = PointwiseInteriorConstraint(
nodes=nodes,
geometry=rec,
outvar={"continuity": 0, "momentum_x": 0, "momentum_y": 0},
batch_size=cfg.batch_size.Interior,
compute_sdf_derivatives=True,
lambda_weighting={
"continuity": Symbol("sdf"),
"momentum_x": Symbol("sdf"),
"momentum_y": Symbol("sdf"),
},
)
file_path = "openfoam/cavity_uniformVel_zeroEqn_refined.csv"
if os.path.exists(to_absolute_path(file_path)):
mapping = {
"Points:0": "x",
"Points:1": "y",
"U:0": "u",
"U:1": "v",
"p": "p",
"d": "sdf",
"nuT": "nu",
}
openfoam_var = csv_to_dict(to_absolute_path(file_path), mapping)
openfoam_var["x"] += -width / 2 # center OpenFoam data
openfoam_var["y"] += -height / 2 # center OpenFoam data
openfoam_var["nu"] += 1e-4 # effective viscosity
openfoam_invar_numpy = {
key: value
for key, value in openfoam_var.items()
if key in ["x", "y", "sdf"]
}
openfoam_outvar_numpy = {
key: value for key, value in openfoam_var.items() if key in ["u", "v", "nu"]
}
openfoam_validator = PointwiseValidator(
nodes=nodes,
invar=openfoam_invar_numpy,
true_outvar=openfoam_outvar_numpy,
batch_size=1024,
plotter=ValidatorPlotter(),
)
grid_inference = PointwiseInferencer(
nodes=nodes,
invar=openfoam_invar_numpy,
output_names=["u", "v", "p", "nu"],
batch_size=1024,
plotter=InferencerPlotter(),
)
else:
warnings.warn(
)
global_monitor = PointwiseMonitor(
rec.sample_interior(4000),
output_names=["continuity", "momentum_x", "momentum_y"],
metrics={
"mass_imbalance": lambda var: torch.sum(
var["area"] * torch.abs(var["continuity"])
),
"momentum_imbalance": lambda var: torch.sum(
var["area"]
* (torch.abs(var["momentum_x"]) + torch.abs(var["momentum_y"]))
),
},
nodes=nodes,
)
# make solver
slv = Solver(cfg, ldc_domain)
# start solver
slv.solve()
if __name__ == "__main__":
run()
The results of the turbulent lid driven cavity flow are shown below.
Fig. 65 Visualizing variables from Inference domain
Fig. 66 Comparison with OpenFOAM data. Left: Modulus Sym Prediction. Center: OpenFOAM, Right: Difference
Previous Coupled Spring Mass ODE System
Next Scalar Transport: 2D Advection Diffusion
ยฉ Copyright 2023, NVIDIA Modulus Team. Last updated on Jul 25, 2024.
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# Examen 2 Parcial De Tic Excel
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Examen Tic, 1 Semestre de los Grupos A,B,C y D de Cecytes SPS. Tipo 1.
โข 1.
### Es la intersecciรณn de una fila por una columna
โข A.
Casilla
โข B.
Celda
โข C.
Renglรณn
โข D.
B. Celda
Explanation
The correct answer is "Celda" because it refers to the intersection of a row and a column in a table or grid. It is a term commonly used in spreadsheet programs or databases to represent a single unit of data within a larger structure.
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โข 2.
### Es la selecciรณn de varias celdas
โข A.
Columnas
โข B.
Rango
โข C.
Dialogos
โข D.
B. Rango
Explanation
The correct answer is "Rango". This is because "Rango" refers to a range, which is the selection of multiple cells in a spreadsheet. The other options mentioned, such as "Columnas" (columns), "Dialogos" (dialogs), and "Lista numerada" (numbered list), do not accurately describe the selection of multiple cells. Therefore, "Rango" is the most appropriate answer.
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โข 3.
### Las Columnas se ordenan por nรบmeros y las Filas por letras
โข A.
โข B.
Falso
B. Falso
Explanation
The statement is false because columns are not ordered by numbers and rows are not ordered by letters. In a typical table or grid, columns are usually labeled with letters and rows are labeled with numbers.
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โข 4.
### Nos muestra el contenido de la celda activa, es decir, la casilla donde estamos situados
โข A.
โข B.
Barra de Formulas
โข C.
Rango
โข D.
B. Barra de Formulas
Explanation
The correct answer is "Barra de Formulas". The barra de fรณrmulas, also known as the formula bar, is a toolbar located at the top of the Excel window. It displays the contents of the active cell, allowing users to view and edit the formulas or data in that cell. This feature is helpful for users to easily see and modify the formulas or data in a cell without directly editing the cell itself.
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โข 5.
### El nombre de un archivo en Excel cuando no ha sido guardado se muestra en la barra de titulo como:
โข A.
Documento
โข B.
Hoja
โข C.
Libro
โข D.
Presentacion
C. Libro
Explanation
Cuando se abre un archivo en Excel y aรบn no se ha guardado, el nombre del archivo se muestra en la barra de tรญtulo como "Libro". Esto se debe a que en Excel, un archivo se conoce como un "libro" que contiene varias hojas de cรกlculo. Por lo tanto, hasta que se le asigne un nombre y se guarde, se muestra como "Libro" en la barra de tรญtulo.
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โข 6.
### La barra de Etiquetas nos muestra:
โข A.
Celdas
โข B.
Rangos
โข C.
Libros
โข D.
Hojas
D. Hojas
Explanation
The correct answer is "Hojas" because the question is asking what the Tags bar shows, and among the options given, "Hojas" is the only one that fits. The Tags bar typically displays the different sheets or tabs within a workbook or spreadsheet, allowing users to navigate and organize their data.
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โข 7.
### Elemento de la ventana de Excel que indica cuรกl esย la celda Activa
โข A.
โข B.
โข C.
Barra de formulas
โข D.
Barra activa
Explanation
The cuadro de nombres, or name box, is an element of the Excel window that indicates which cell is currently active. It displays the cell reference or name of the active cell, making it easier for users to identify and work with specific cells in their worksheets. The name box is located next to the formula bar at the top of the Excel window.
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โข 8.
### El cuadro de diรกlogo de formato de celdas se divide en las siguientes pestaรฑas:
โข A.
Nรบmero, Celdas, Fuente, Bordes, Rangosy Proteger.
โข B.
Nรบmero, Letras, Fuente, Bordes, Relleno y Calcular.
โข C.
Nรบmero, Alineaciรณn, Fuente, Bordes, Relleno y Proteger.
โข D.
Nรบmero, Eliminar, Fuente, Insertar, Relleno y Proteger.
C. Nรบmero, Alineaciรณn, Fuente, Bordes, Relleno y Proteger.
Explanation
The correct answer is "Nรบmero, Alineaciรณn, Fuente, Bordes, Relleno y Proteger." This is because the "Format Cells" dialog box in Excel is divided into different tabs or sections, and these tabs include options for formatting cells. The "Nรบmero" tab allows you to format the number style and display, the "Alineaciรณn" tab allows you to adjust the alignment of the cell contents, the "Fuente" tab allows you to change the font style and size, the "Bordes" tab allows you to add or remove borders around cells, the "Relleno" tab allows you to fill cells with colors or patterns, and the "Proteger" tab allows you to protect cells from editing or make them read-only.
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โข 9.
### La ficha tรฉcnica de la caja de diรกlogo Formato de Celdas que permite bloquear u ocultar celdas de la hoja de cรกlculo es
โข A.
Formato
โข B.
Numero
โข C.
Proteger
โข D.
Ocultar
C. Proteger
Explanation
The correct answer is "Proteger" because the question is asking for the option in the "Formato de Celdas" dialog box that allows the user to lock or hide cells in the spreadsheet. The option "Proteger" is the most appropriate choice as it refers to the action of protecting cells, which includes both locking and hiding them.
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โข 10.
### Combinaciรณn de teclas utilizada para acceder al cuadro de dialogo Formato de celdas
โข A.
Ctrl + F1
โข B.
Alt + 1
โข C.
Ctrl + B1
โข D.
Ctrl + 1
D. Ctrl + 1
Explanation
The correct answer is Ctrl + 1. This keyboard combination is used to access the Format Cells dialog box in various software applications, such as Microsoft Excel. By pressing Ctrl + 1, users can quickly open the dialog box and make changes to the formatting of selected cells, including number format, alignment, font, border, and fill options. This shortcut is commonly used by professionals to efficiently customize the appearance of their data.
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โข 11.
### Al Comenzar una Funciรณn o Formula comenzamos a editar con el caracter
โข A.
*
โข B.
/
โข C.
=
โข D.
?
C. =
Explanation
When starting a function or formula, we begin editing with the character "=" because it is the symbol used to indicate that we are entering a formula or function in a spreadsheet or programming language. The "=" sign tells the software to interpret the following characters as a command or calculation rather than plain text.
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โข 12.
### Ordenar una base de datos Ascendente con la observaciรณn que se usara la 2 Columna de la base de datos para ordenar todo se tiene que usar la herramienta......
โข A.
Orden A a Z
โข B.
Orden Z a A
โข C.
โข D.
Filtro
Explanation
To sort a database in ascending order using the second column, the "Orden personalizado" (Custom Order) tool needs to be used. This tool allows the user to define their own specific order for sorting the data. It provides flexibility in arranging the data according to specific requirements or preferences, unlike the predefined options of sorting in alphabetical or reverse alphabetical order.
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โข 13.
### Las categorรญas de formato ย celda: ย General, Moneda, Contabilidad, Fecha, pertenecen a
โข A.
Bordes
โข B.
Alineacion
โข C.
Fuente
โข D.
Numero
D. Numero
Explanation
The categories of cell format, such as General, Currency, Accounting, and Date, belong to the "Number" category. This category is used to format cells that contain numerical data, allowing users to customize how numbers are displayed in the cell.
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โข 14.
### Funciรณn que muestra el mayorย dato numรฉrico de una serie de celdas:
โข A.
Suma
โข B.
Producto
โข C.
Max
โข D.
Contar
C. Max
Explanation
The correct answer is "Max" because the function "Max" is used to find the highest numerical value in a range of cells. It returns the maximum value among the given arguments.
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โข 15.
### Fรณrmula que se utiliza para calcular el promedio de un rango de celdas seleccionadas:
โข A.
=promedio / total valor
โข B.
=promedio (rango)
โข C.
=prom (a1:a20)
โข D.
=calcular.promedio (rango)
B. =promedio (rango)
Explanation
The correct answer is "=promedio (rango)". This formula is used to calculate the average of a selected range of cells. The "promedio" function in Excel is used to calculate the average, and "rango" refers to the range of cells that you want to include in the calculation. By using this formula, you can easily find the average value of a specific range of cells in Excel.
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โข 16.
### Se le llama asรญ a las lรญneas que se pueden colocar para dividir las celdas, y aparezcan al momento de imprimir.
โข A.
Lรญneas de selecciรณn
โข B.
โข C.
Margenes
โข D.
Bordes
D. Bordes
Explanation
The correct answer is "Bordes". This is because the question states that it refers to lines that can be placed to divide cells and appear when printing. "Bordes" translates to "borders" in English, which are lines that can be added to the edges of cells to separate them visually.
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โข 17.
### Selecciona la respuesta correcta, en la letra A tomando en cuenta las coordenadas de la celdas
โข A.
=A3+C3
โข B.
=A3*C3
โข C.
=A3*A6
โข D.
=Suma (A3*C3)
B. =A3*C3
Explanation
The correct answer is =A3*C3. This is because the formula multiplies the values in cell A3 and C3. The other options either add or multiply different cells, which are not specified in the question.
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โข 18.
### Selecciona la respuesta correcta, en la letraย B tomando en cuenta las coordenadas de la celdas
โข A.
=SUM(D3..D6)
โข B.
=D3:D6
โข C.
=PROMEDIO (D3:D6)
โข D.
=SUMA(D3:D6)
D. =SUMA(D3:D6)
Explanation
The correct answer is =SUMA(D3:D6). This formula calculates the sum of the values in cells D3 to D6.
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โข 19.
### EN LA LETRA C, LA FORMULA QUE SE NECESITA ES.......
โข A.
=D7*0.16
โข B.
=D7* IVA
โข C.
=D7*D9
โข D.
=D7*CELDAIVA
A. =D7*0.16
Explanation
The correct answer is =D7*0.16. This formula is used to calculate the value of 16% of the value in cell D7. It multiplies the value in cell D7 by 0.16 to determine the result.
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โข 20.
โข A.
=D7+B7
โข B.
=D8+D8
โข C.
=D7+D8
โข D.
=D7+D9
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## MA.912.DP.4.2: Determine if events A and B are independent by calculating the product of their probabilities.
There are 31 resources.
Title Description Thumbnail Image Curriculum Topics
## Math Examples Collection: Probability
Overview ThiThis collection aggregates all the math examples around the topic of Probability. There are a total of 28 Math Examples. Probability
## Definition--Statistics and Probability Concepts--Dependent Event
Definition--Statistics and Probability Concepts--Dependent Event
This is part of a collection of definitions on the topic of probability and statistics. Each definition includes an example of the term.
Probability
## Definition--Statistics and Probability Concepts--Independent Event
Definition--Statistics and Probability Concepts--Independent Event
This is part of a collection of definitions on the topic of probability and statistics. Each definition includes an example of the term.
Probability
## Math Example--Probability Concepts--Probability: Example 1
Math Example--Probability Concepts--Probability: Example 1
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 10
Math Example--Probability Concepts--Probability: Example 10
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 11
Math Example--Probability Concepts--Probability: Example 11
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 12
Math Example--Probability Concepts--Probability: Example 12
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 13
Math Example--Probability Concepts--Probability: Example 13
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 14
Math Example--Probability Concepts--Probability: Example 14
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 15
Math Example--Probability Concepts--Probability: Example 15
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 16
Math Example--Probability Concepts--Probability: Example 16
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 17
Math Example--Probability Concepts--Probability: Example 17
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 18
Math Example--Probability Concepts--Probability: Example 18
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 19
Math Example--Probability Concepts--Probability: Example 19
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 2
Math Example--Probability Concepts--Probability: Example 2
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 20
Math Example--Probability Concepts--Probability: Example 20
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 21
Math Example--Probability Concepts--Probability: Example 21
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 22
Math Example--Probability Concepts--Probability: Example 22
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 23
Math Example--Probability Concepts--Probability: Example 23
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 24
Math Example--Probability Concepts--Probability: Example 24
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 25
Math Example--Probability Concepts--Probability: Example 25
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 26
Math Example--Probability Concepts--Probability: Example 26
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 27
Math Example--Probability Concepts--Probability: Example 27
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 28
Math Example--Probability Concepts--Probability: Example 28
This is part of a collection of math examples that explore different aspects of probability.
Probability
## Math Example--Probability Concepts--Probability: Example 3
Math Example--Probability Concepts--Probability: Example 3
This is part of a collection of math examples that explore different aspects of probability.
Probability
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Home / Expert Answers / Economics / suppose-the-price-of-good-x-is-2-and-the-price-of-good-y-is-3-also-suppose-mux-y-and-muy-x-w-pa488
# (Solved): Suppose the price of good x is \$2 and the price of good y is \$3. Also suppose Mux = y and MUy = x. W ...
Suppose the price of good x is \$2 and the price of good y is \$3. Also suppose Mux = y and MUy = x. Which of the following baskets could be utility maximizing? X = 2, y=4 X=7.5, y=5 X=6, y=8 X=10, y=15
We have an Answer from Expert
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HomeLesson Worksheet โ 0 7+ Fresh Fractions Of A Group Worksheets
# 7+ Fresh Fractions Of A Group Worksheets
Worksheet for Third Grade Math. Worksheet 1 Worksheet 2.
Learn Fractions Worksheet Fractions Worksheets Fractions Fraction Worksheets Fractions of a group worksheets
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Fractions Of A Group โ Displaying top 8 worksheets found for this concept. These fractions worksheets will use 12s 14s 18s. Fraction Of Groups โ Displaying top 8 worksheets found for this concept.
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6 Bob got 8. Finding Fraction Of A Group Displaying top 8 worksheets found for โ Finding Fraction Of A Group. Some of the worksheets for this concept are Fractions of groups Fractions of groups Fractions of a group or set work Fractions as part of a group of things Mega fun fractions Lesson plan fractions Coloring fractions a Fractional part of a set work.
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# Definition:Inner Automorphism
## Definition
Let $G$ be a group.
Let $x \in G$.
Let the mapping $\kappa_x: G \to G$ be defined such that:
$\forall g \in G: \map {\kappa_x} g = x g x^{-1}$
$\kappa_x$ is called the inner automorphism of $G$ (given) by $x$.
The set of inner automorphisms of $G$ is denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ as $\Inn G$.
## Also denoted as
While $\kappa$ is the symbol generally used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote an inner automorphism, this is not universal in the literature.
Different sources use different symbols, for example $\alpha$ as used by Allan Clark: Elements of Abstract Algebra.
The set of inner automorphisms of $G$ can be found denoted in a number of ways, for example:
$\map {\mathscr I} G$
$\map I G$
## Also see
โข Results about inner automorphisms can be found here.
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# Civil Engineering - Theory of Structures - Discussion
Discussion Forum : Theory of Structures - Section 1 (Q.No. 17)
17.
The force in BF of the truss shown in given figure, is
4t tension
4t compression
4.5t tension
4.5t compression
zero.
Explanation:
No answer description is available. Let's discuss.
Discussion:
10 comments Page 1 of 1.
Patel said: ย 1 decade ago
Reactions are va+ab = 5t.
Moment at A= 4t*x+5t*x-vb*2x = 0.
So 9t = 2vb.
vb = 4.5t.
So force in compression member BF = 4.5t compression.
(1)
P.K. Nayak said: ย 8 years ago
Taking moment about A.
Vb * 2 = 5 * 1 + 4 * 1.
=> Vb = 4.5T Upward.
Considering joint B
Fbc = 0 & Fbf = 4.5T c.
(2)
Krunal said: ย 7 years ago
How to determine it as compression or tension?
Manoj paridwal said: ย 7 years ago
Is sign is - then compression?
If sign positive then tension.
Abhi said: ย 6 years ago
How to consider the length of this? Please explain me in detail.
Nath said: ย 6 years ago
Here, Assume x as the length of each segment.
Gajanan todeti said: ย 5 years ago
Answer is A) 4T tension.
Michael said: ย 3 years ago
Why zero can't be? Explain, please.
Sk said: ย 3 years ago
Thanks everyone for explaining.
Mahesh said: ย 2 years ago
By equilibrium conditions.
All vertical forces=0,
Va + Vb = 5t, ----> eqn(1).
Taking moments about A,
Vb*2=5*1+4*1,
Vb = 4.5t
Va = 4.5t.
(3)
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# Null hypothesis.
Theย null hypothesisย is the center of the hypothesis testing, which seeks to be able to reject it when statistical significance is reached.
The null hypothesis, you familiarly call it H0, has a misleading name. Despite what one might think, that improper name doesnโt prevent it to be the core of all hypothesis testing.
And, what is hypothesis testing?. Let us see an example.
Let us suppose we want to know if residents (as they believe) are smarter than attending physicians. We pick out a random sample composed by 30 assistants and 30 residents from our hospital and we measure their IQ. We come up with an average value of 110 for assistants and 98 for residents (sorry, Iโm an assistant and, as it happens, Iโm writing this example).
In view of these results we ask ourselves: what is the probability that the group of assistants selected are smarter than the residents of our example?. The answer is simple: 100% (of course, provided that everyone have passed an intelligence test and not a satisfaction survey). But the problem is that we are interested in knowing if assistant physicians (in overall) are smarter than residents (in overall). We have only measured the IQ of 60 people and, of course, we want to know what happens in the general population.
## Null hypothesis
At this point we consider two hypotheses:
1. The two groups are equally intelligent (this example is pure fiction) and the differences that we have found are due to chance (random). This, ladies and gentlemen, is the null hypothesis or H0. We state it in this way:
H0: CIA = CIR
1. Actually, the two groups are not equally intelligent. This will be the alternative hypothesis:
H1: CIA โ ย CIR
We could have stated this hypothesis in a different way, considering that IQ from one people being greater o smaller than other peopleโs, but letโs leave it this way for now.
At first, we always assume that H0 is true (and they call it null), so when we run our statistical software and compare the two means we come up with a statistical parameter (which one depend on the test we use) with the probability that differences observed are due to chance (the famous p).
If we get a p lower than 0.05 (this is the value usually chosen by convention) we can say that the probability that H0 is true is lower than 5%, so we reject the null hypothesis. Letโs suppose that we do the test and come up with a p = 0.02. Weโll draw the conclusion that it is not true that both groups are equally clever and that the observed difference is not due to chance (in this case the result was evident from the beginning, but in other scenarios it wouldnโt be so clear).
## Weโre leavingโฆ
And what happens if p is greater than 0.05?. Does it mean that the null hypothesis is true?. Well, maybe yes, maybe no. All that we can say is that the study is no powerful enough to reject the null hypothesis. But if we accept it as true without further considerations we will run the risk of blunder committing a type II error. But thatโs another storyโฆ
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Prove that the square of an even or odd function is always even
Arun Kumar IIT Delhi
10 years ago
Hello Student,
f(-x)=f(x)
let g(x)= (f(x))^2
=>Replacing x by -x we get
g(-x)=(f(-x))^2
=> g(-x)=( f(x) )^2
=> g(-x)=g(x)
let f(x) is an odd function
then by def f(-x)= -f(x)
let g(x)=(f(x))^2
Replacing x by -x we get
g(-x)=(f(-x))^2
=>g(-x)=(-f(x))^2=(f(x))^2
=>g(-x)= (g(x))^2
prooved.
Thanks & Regards
Arun Kumar
Btech, IIT Delhi
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โข 1 Vote(s) - 2 Average
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interesting pattern in hyper-operations Base-Acid Tetration Fellow Posts: 94 Threads: 15 Joined: Apr 2009 04/30/2009, 04:15 AM (This post was last modified: 04/30/2009, 04:34 AM by Base-Acid Tetration.) I just wanted to know if pentation would have any asymptotes like tetration does and I just discovered this interesting property. So, tet(1) for base b = b, tet(0) = 1 and tet(-1) = 0, so tetra-logarithm (superlogarithm) of b is 1, tetlog(1) is 0, and tetlog(0)=-1. pent(1) = b for any b, pent(0) = tetlog(pent(1)) = tetlog(b) = 1, pent(-1) = tetlog(1) = 0, pent(-2) = -1. pentlog(b) = 1 pentlog(1) = 0 pentlog(0) = -1 pentlog(-1) = -2 Continuing this for higher operations (verification is left for the reader), for base greater than 1, for hexation, hex(1) = b, hex(0)=1, hex(-1)=0, hex(-2)=(-1), hex(-3)=-2; for heptation, hept(-1) = 0, hept(-2)=-1, hept(-3)=-2, hept(-4)=-3; oct(-5)=-4, non(-6)=-5, dec(-7)=-6, ... So here is my conjecture (theorem?)... for n>=3, b[n]-n+3 = -n+4; for n>=4, b[n]-n+2=-n+3, etc. If you graph these hyper operation functions for integers, you will notice a linear-ish part in a larger domain for increasing n. (specifically around the domain [-n+3,0]), so my conjecture can be stated as: for k>=3, we have b[k]n=n+1 for any natural n which is in the interval [-k+3,0]. What is the implication of the growing quasi-linear part for the real or complex analytic extensions of those higher hyper-operations pentation, hexation, etc? Is it a good thing or a bad thing? Also would pentation have any asymptotes? ยซ Next Oldest | Next Newest ยป
Messages In This Thread interesting pattern in hyper-operations - by Base-Acid Tetration - 04/30/2009, 04:15 AM RE: interesting pattern in hyper-operations - by andydude - 04/30/2009, 05:35 AM RE: interesting pattern in hyper-operations - by andydude - 04/30/2009, 05:53 AM RE: interesting pattern in hyper-operations - by BenStandeven - 04/30/2009, 10:41 PM RE: interesting pattern in hyper-operations - by Base-Acid Tetration - 05/01/2009, 10:22 AM RE: interesting pattern in hyper-operations - by bo198214 - 05/01/2009, 01:34 PM RE: interesting pattern in hyper-operations - by BenStandeven - 05/02/2009, 08:11 PM RE: interesting pattern in hyper-operations - by bo198214 - 05/03/2009, 08:20 PM RE: interesting pattern in hyper-operations - by BenStandeven - 05/04/2009, 09:15 PM
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# algebra 2
complete the square
2x^2-4x-1=0
1. ๐ 0
2. ๐ 0
3. ๐ 102
1. solve by completing the square ??
first divide by 2 and move the constant to the right ...
x^2 - 2x = 1
x^2 - 2x + 1 = 1 + 1
(x-1)^2 = 2
x-1 = ยฑ โ2
x = 1 ยฑ โ2
1. ๐ 0
2. ๐ 0
posted by Reiny
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6. ### Algebra
My homework is to solve by completing the square. But how can you complete the square on this problem? -3x^2 + 7x - 8 The -3 doesn't factor out.
asked by kaylee on October 25, 2007
7. ### Algebra
Complete the square for the binomial. Then factor the resulting perfect square trinomial. x^2 -2/11X
asked by Anonymous on March 30, 2010
8. ### PreCalc
I am supposed to graph this y= -x^2+2x he told us to coplete the square to get us the vertex, but how do i complete the square with a - in front of the x^2??
asked by Lauren on January 2, 2009
9. ### math
i need to solve this equation by completing the square: x^2-14x+1=0 i don't' know how to do this, nor do i know what complete the square means. please help.
asked by Nic on November 4, 2008
10. ### Algebra
Complete the square for the binomial. Then factor the resulting perfect square trinomial. x^2 -2/11X
asked by Anonymous on April 1, 2010
11. ### Pre-cac
Find a such that f(x)= ax^2-3x+5 has a maximum value of 15. I'm stuck because how am I suppose to complete the square? 3/2= 1.5? then square 1.5? So lost.
asked by Sam on March 23, 2017
More Similar Questions
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Examples Examples
Find the slope of the equation: 3x+3y = 39
First you need it in slope-intercept form.
3x + 3y = 39
(3x + 3y)/3 = 39/3
x + y = 39/3
y = 39/3 - x
The slope would be the coefficent of x, which is -1
More Examples
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# Nov 26 30 MATH by h5UEWFg4
VIEWS: 2 PAGES: 2
โข pg 1
``` Lesson Plan for Second Grade
Week of: Week 7 (Nov 26-30) Teacher: ALL
Subject: Math
Day of the Week: All
Standard(s): 2.G.2 Partition a rectangle into rows and columns of same-size
squares and count to find the total number of them.
EQ-What strategies can I use to count the total number of squares in a rectangle?
Lesson โI canโ:
I can separate a rectangle into equal parts.
I can count the total number of rows and columns in a rectangle.
Assessment (formative, benchmark, summative):
๏ท Math Investigations assessment
๏ท Teacher anecdotal notes
๏ท Zoo Trouble assessment
๏ท โGrandmaโs Quiltsโ quilts recording sheet
๏ท Making Rectangles recording sheet
Essential Vocabulary Materials
Rows Each Orange has 8 Slices
Columns Remainder of One
Rectangle Georgia Math Site
Separate Math journals
Recording sheets
Colored square tiles
Grid paper
Whole Group Teaching Strategies (including writing):
Small Group
Partners Monday: Georgia Math Site- Grandmaโs Quilt, part one and two
Independent Tuesday: Georgia Math Site- Making Rectangles
Stations Wednesday: Georgia Math Site- Zoo Trouble
Thursday: Review Math skills covered during first and second
quarters through Math Jeopardy on the Smartboard
Friday: FIELD TRIP
Math Stations:
Guided Math
Math Writing: (Students will be given a prompt and asked to
write a response).
Computers: Study Island
Problem Solving/Games: Review sheets, addition fluency
practice with flashcards/ word problems
Differentiation
Guided Math groups
Partner help in stations
Colored tiles available for Zoo Trouble assessment
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# Error in paragraph
Can you spot what's wrong with the following sentence?
PETER WAS ANGRY AT
HIMSELF BECAUSE HE
ACCIDENTALLY LEFT THE
THE KEYS TO HIS CAR
AT HIS FRIEND'S HOUSE.
# Similar Riddles
##### Trap in Room Riddle
A man is trapped in a room. The room has only two possible exits doors. Through the first door there is a room constructed from magnifying glass. The blazing hot sun instantly fries anything or anyone that enters. Through the second door there is a fire-breathing dragon. How does the man escape?
Asked by Neha on 09 Jan 2021
##### Make the Line Sorter
Birbal was jester, counsellor, and fool to the great Moghul emperor, Akbar.
The villagers loved to talk of Birbal's wisdom and cleverness,
and the emperor loved to try to outsmart him.
One day Akbar (emperor) drew a line across the floor.
"Birbal," he ordered, "you must make this line shorter, but you cannot erase any bit of it."
Everyone present thought the emperor had finally outsmarted Birbal.
It was clearly an impossible task.
Yet within moments the emperor and everyone else present had to agree that Birbal had made the line shorter without erasing any of it.
How could this be?
Asked by Neha on 17 May 2021
##### Servants with multiple legs
The king of Octopuses has servants who have six, seven or eight legs. The distinguishing characteristics of the servants is that the one with seven legs always lie but the one with either six or eight legs speak the truth always.
One day, four servants meet and converse:
The black one says, 'We have 28 legs altogether.'
The green one says, 'We have 27 legs altogether.'
The yellow one says, 'We have 26 legs altogether.'
The red one says, 'We have 25 legs altogether.'
Can you identify the colour of the servant who is speaking the truth?
Asked by Neha on 09 Apr 2021
##### I Like Mornings
ฯยยย I am a 7 letter word.
ฯยยย I like mornings
ฯยยย If you remove my 1st letter you can drink me
ฯยยย If you remove my 1st & 2nd letters ฯยยย you may not like me
ฯยยย If you remove my last letter, you will see me on television
Let us see who solves this....
Asked by Neha on 14 May 2021
##### Make the Meaningful Words
_ _ _ IE _ _
_ _ _ IE _
_ _ IE _ _
_ _ IE _
_ _ _ _ IE
Like you see, some letters have gone missing from these words that contain the IE pair at some or the other place. The letters that will be used to fill the blanks are given below. Use them and form meaningful words. Can you do that?
A, C, D, F, H, K, L, M, N, N, O, R, R, S, S, S, T, T, Y and Y.
Asked by Neha on 22 Jul 2021
##### Solve the Rebus
What does below Rebus say?
XLR8
Asked by Neha on 10 May 2021
##### Once Akbar summoned Birbal
Akbar summoned Birbal out of anger.
He told him that he will have to face death.
He asked him to make a statement and if the statement is true he will be buried alive and if the statement is false, he will be thrown at lions.
After hearing Birbalฮฒยยs statement, Akbar could do nothing but smile.
He gave him 5 gold bars and let him go.
What did Birbal say?
Asked by Neha on 17 May 2021
##### Monkey Gear riddle
When monkey rotate the gear, which mark will be hit 1 or 2 ?
Asked by Neha on 30 Apr 2021
##### Find the way to Village
Mr. Buttons was all set to go to the village of Buttonland to meet his friend. So, he packed his bags and left for the village at 5 in the morning. Upon travelling on a road for miles, he came across a point where the road diverged into two. He was confused on which road to take. He gazed around and he saw two owls sitting on a branch. He thought he could ask for directions for the village from the two owls. So he went to the tree. There he saw a sign which read, "One owl always lies, and one is always truthful. They both fly away if you ask them more than 1 question."
Mr. Buttons was caught in the dilemma of what to ask? And from which owl to ask, since he only had one question. What should Mr. Buttons ask?
Asked by Neha on 18 Aug 2021
##### Count the F Word
Count the number of times the letter "F" appears in the following paragraph:
FAY FRIED FIFTY POUNDS OF
SALTED FISH AND THREE POUNDS
OF DRY FENNEL FOR DINNER FOR
FORTY MEMBERS OF HER FATHER'S FAMILY.
Asked by Neha on 05 May 2022
### Amazing Facts
###### Challenging
There is a cryptic organization called Cicada 3301 that posts challenging puzzles online, possibly to recruit codebreakers and linguists.
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Using superscript โปยน as a postfix operator for inv?
Nice. I see theyโre using the latest unicode.
Does anyone know if it will be possible (an supported by Julia) to express negative one โ-1โ as a superscript w/ unicode?
X^{-1} is a lot leaner than inv(X) or X^-1
1 Like
Slightly off topic, but the real answer is to never use inv. It is almost never the right answer.
8 Likes
Off-topic, but you can already do that with X\^-\^1 but the โปยน is part of the variable name so you can write Xโปยน = [...], same as the more common ฯยฒ = .... It would be a breaking change to make these into operators, and the idea was previously rejected: Unicode superscript numbers for exponentiation ยท Issue #9806 ยท JuliaLang/julia ยท GitHub .
6 Likes
See an article claiming that inv(A)'s bad reputation is not deserved. At the end of section 2, the author writes:
It appears that this easy-to-derive but loose bound gave the matrix inverse its bad reputation.
2 Likes
More serious is the expense, IMO.
Solving a linear system by Gaussian elimination is a constant factor 4 faster than finding the inverse [Trefethen, Numerical Linear Algebra].
3 Likes
The original author did not suggest to use inv() for solving linear systems. There are perfectly valid reasons for inverting matrices other than solving linear systems. E.g. if you need to compute its trace, or you need the inverse of a Hessian as a covariance matrix in a likelihood optimization, or a Fisher information matrix, or to compute the projection matrix on a column space, or โฆ whatever.
Also, inv() is not a matrix-thing, it can equally well be used for things like finite fields, where its implementation can be via power operators or a generalized gcd.
Personally I do not like filling programs with utf8. It looks nice, but the next programmer may not be able to find all the funny symbols on the keyboard (like guessing that รฅ is written \aa, which every Dane or Norwegian born before 1870 will easily guess). Or it may even be ambiguous because the glyphs look similar for two different utf8-symbols. Like m and \ttm looks different with my current font setup, but looks exactly the same with another font setup, and will of course be two different variables. And things like \epsilon in the ess julia mode in emacs is \varepsilon in the REPL.
5 Likes
I think Unicode in text files is great (when used within reason, as all things). Mostly, because it can significantly improve readability, and that is more important than writeability.
No guessing required!
help?> รฅ
"รฅ" can be typed by \aa<tab>
Well, thatโs if you can copy-paste from the source of course. But if youโre looking at code in a picture or on paper, it doesnโt matter anyway, you can choose the symbols when writing your own code, as usual.
Totally agree. In this case Unicode actually hurts readability. Thatโs where โwithin reasonโ comes in. Note that this has always been a problem, not specific to Unicode symbols: maybe I and l, or O and 0 are easily distinguishable in your font but not in your collaboratorโs bad font. But Unicode can definitely take this problem to a new level.
Thatโs a rather minor issueโฆ Surely Emacs has a mode for writing Julia code using the same symbol names as in the REPL? In any case it is unfortunate that Emacs binds \epsilon to ฯต in the so-called โTeX input modeโ.
1 Like
I am aware that both countries have outstanding life expectancy, but I would guess that the set of such people is empty in 2020.
7 Likes
If โปยน is an operator, you canโt use it as part of a variable name:
Xโปยน = inv(X)
use(Xโปยน )
reuse(Xโปยน )
and_again(Xโปยน )
vs
X_inv = Xโปยน
use(X_inv)
reuse(X_inv)
and_again(X_inv)
In general I think it is more valuable to be able to use unicode super/subscripts as part of variable names, then for operator names.
3 Likes
As a side note, I learned typewriting quite some years ago, on a real typewriter. It didnโt have 0 and 1, to save mechanical costs. It was quite common at the time. We were instructed to write O and l instead. It created some unexpected problems in the computer science course the following year, it turned out that the ROM based Basic-interpreter differentiated between 0 and O, and 1 and l.
4 Likes
You can actually do this with no language changes if youโre willing to write (A)โปยน rather than Aโปยน. Hereโs an example at the REPL:
julia> begin
struct Inverter end
const โปยน = Inverter()
Base.:(*)(A, ::Inverter) = inv(A)
end
julia> A = rand(3,3)
3ร3 Array{Float64,2}:
0.51549 0.0616877 0.530502
0.783366 0.0598045 0.697045
0.381722 0.833891 0.632847
julia> (A)โปยน
3ร3 Array{Float64,2}:
-13.5368 10.0475 0.280811
-5.72129 3.08199 1.4014
15.704 -10.1216 -0.435822
julia> ((A)โปยน)*A โ I(3)
true
This works because (a)b is parsed as a * b:
julia> quote (a)b end
quote
#= REPL[7]:1 =#
a * b
end
so we need only define an object โปยน which when multiplied by A on the left produces inv(A), trivially easy with multiple dispatch.
One could do the same with ยฒ to make an squaring operator if they were unable to stomach reading x^2 or x*x
16 Likes
The solution is great but how can I get
to work? I get
syntax: invalid character "ยฒ" near column 10
Stacktrace:
[1] top-level scope
@ In[31]:3
while the inverter works perfectly.
2 Likes
The character ยฒ can not be used as a variable name by itself (or as a first character of a variable name). The same thing happens to to subindex โ to mean x[1]. The parser would have to be changed. I donโt know what are the ramifications of such a change.
julia> ยฒ = 10
ERROR: syntax: invalid character "ยฒ" near column 1
Stacktrace:
[1] top-level scope
@ none:1
julia> โX = 10
ERROR: syntax: invalid character "โ" near column 1
Stacktrace:
[1] top-level scope
@ none:1
julia> Xโ = 10
10
Paulo
1 Like
1 Like
[EDIT: I first meant to respond in the thread, then decided on a private message to Mason, but posted here anyway by accident. I could delete it.]
FYI (A)โปยน still works in 1.10.0-beta1 (and using LinearAlgebra; ((A)โปยน)*A โ I(3) which you implied and may should have written), while on my modified master:
julia> (A)โปยน
ERROR: MethodError: no method matching inv(::Matrix{Float64})
I was going to post this to the thread and maybe file a regression issue, until I remembered my master is my experiment of (LinearAlgebra) excision, and this works if I first do using LinearAlgebra along with A*A.
I knew about A*A not working in my โJulia 2.0โ, and it seemed a worthy tradeoff given my 8% if I recall faster startup. I still believe all using LinearAlgebra recovers all compatibility. In practice isnโt that what you would do in all real-world code anyway? Unless you use none of it, then of course not an issue. You can still define (multi-dimensional) arrays fine.
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# Cross-correlation questions
33 views (last 30 days)
Cside on 20 Feb 2020
Commented: Adam Danz on 20 Feb 2020
Hi, I have 2 spike trains and would like to use cross-correlation to find out their time lag. Are there any resources out there that may guide me on cross correlation and explain how to further interpret the results into smt useful? Also, I have a few questions.
1) why is it that there is still a peak r value with 2 spike trains that are not correlated?
2) Do I have to normalize the spike train?
3) xcorr gives me the r and lag value. But how may I utilize this information to make it useful? ie is it significant?
4) why is it that a xcorr value (i input what mathworks suggested xcorr(x,y), returns me a correlation sequence) can be >1/-1? shouldnt a r value be bounded within that range?
Thank you and i appreciate any help for a beginner here :)
Adam Danz on 20 Feb 2020
Edited: Adam Danz on 20 Feb 2020
Perhaps working through one of the examples in the xcorr documentation would be helpful.
1) why is it that there is still a peak r value with 2 spike trains that are not correlated?
You can cross correlate any two vectors and there will be a maximum correlation at some time point (unless both vectors are flat).
Produce this plot, for example.
theta = linspace(0,2*pi,100);
x1 = sin(theta);
x2 = sin(1.2*theta);
[r, lags] = xcorr(x1,x2);
clf()
tiledlayout(2,1)
nexttile
hold on
plot(theta, x1, 'DisplayName', 'x1')
plot(theta, x2, 'DisplayName', 'x2')
legend()
nexttile
plot(lags,r, 'ks')
grid on
xlabel('lag')
ylabel('cross-corr')
2) Do I have to normalize the spike train?
You don't have to but it might be a good idea to normalize the amplitudes, especially if you're comparing responses from two different sources (ie, two different neurons). I also advise you to look at the literature from your field to see the common approach taken.
3) xcorr gives me the r and lag value. But how may I utilize this information to make it useful? ie is it significant?
After you get the lag, use circshift to circularly shift the 2nd vector so that it aligns to the first vector at the maximum correlation. Plot out the shifted data so you can visually confirm that the shift was done correctly. Then use [rho,pval] = corr(X,Y) to compute the correlation and the corresponding p-value for the shifted vectors.
Adam Danz on 20 Feb 2020
I also want to add that circularly shifting the data is useful if you're dealing with circular data. You might just need a linear shift of the data which would likely result in left and right components of the data that aren't paired due to the shift.
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# How Your Finance Charge is Calculated
Have you ever looked at your credit card statementโs finance charge and wondered, โJust how, exactly, did they come up with that?โ Or perhaps you are card shopping and all those APRโs have made you go cross-eyed. There actually are different ways a credit card company can calculate your finance charge and it is important to understand how they make those calculations as it directly affects your balance and minimum payments.
When you are comparing credit cards or making a decision as to consolidation the APR is the most important detail you should look at. Credit card companies may employ different ways to calculate and use that APR. The methods of calculation include โaverage daily balanceโ, โadjusted balanceโ, โprevious balanceโ and the lesser known โtwo-cycle or double-cycle balanceโ. How these calculations affect you is equally important.
Calculation Methods
The Two-Cycle or Double-Cycle method is the least favored method, especially by consumers who carry a balance on their cards. They normally offer lower APRS that look favorable; but are actually only favorable to those who pay off their balances each month. This method uses the activity from the last two monthsโ billing cycles which negates the interest free period that most cards offer (grace period). Therefore, if you carry a balance or decide to make a minimum payment instead of paying off the balance you are actually penalized. The wording on this specific computation may be misleading or hard to understand. For example, you may see โrolling consecutive twelve billing cycle period.โ If you notice this method being used or are having trouble understanding the verbiage, then your best bet would be to call the card issuer directly and have a customer service agent explain it to you before you commit to it.
The โAdjusted Balanceโ method is the most beneficial method for cardholders. The balance is calculated by subtracting payments or credits made during the current billing cycle from the balance at the end of the previous cycle. With this method, cardholders can avoid interest charges on current charges by making a payment before the end of the cycle. The โPrevious Balanceโ method is similar in that it uses balances from the previous billing cycle. The difference is that it does not include payments, credits or purchases made during the current billing cycle.
The โAverage Daily Balanceโ
is the most commonly used calculation and is based on the beginning balance each day of the billing cycle. The card company takes the total balance each day minus any credits and adds them all together divided by the number of days in the billing cycle and then applies the APR based on that result.
The Difference is in the Results
The differences in the above methods may seem subtle, but really are not. When considering the methods it can mean great variation in your finance charge amount. The Federal Trade Commission uses the explanation below:
โSuppose your monthly interest rate is 1.5 percent, your APR is 18 percent, and your previous balance is \$400. On the 15th day of your billing cycle, the card issuer receives and posts your payment of \$300. On the 18th day, you make a \$50 purchase.
โข Average Daily Balance method (including new purchases), your finance charge would be \$4.05.
โข Average Daily Balance method (excluding new purchases), your finance charge would be \$3.75.
โข Average Daily Balance Double Cycle method (including new purchase and the previous monthโs balance), your finance charge would be \$6.53.
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Purchase Solution
# Lower Hemicontinuity
Not what you're looking for?
Please determine whether or not the 2 correspndences is lower hemicontinuous and please justify why (using definition/proof of lower hemicontinuity):
1)F:R^2->R^2, F(u)={x: x o u =0}
2)F:R^n{0}->R^n, F(x)=B(x;||x||), the closed ball centred at x with radius ||x||.
Thanks
Note: o is the dot product
is the complement
||x|| is the distance
My definiton of lower hemicontinuity is:
F:A->B is a correspondence, A is contained in R^n, B is contained in R^k, p belongs to A. I say that F is lowerhemicontinuous at p if: for all sequences {x^m} in A with x^m->p and for all q belonging to F(p), there exists {y^m} in B with y^m belonging to F(x^m) for all m and y^m->q.
##### Solution Summary
Lower Hemicontinuity is investigated. The solution is detailed and well presented. The expert examines dot products for complements.
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1. ,
Proof: We consider a sequence and . Then we consider an arbitrary point , then we have . This implies and for some real number . Now we ...
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src/HOL/Arith.ML
author paulson Wed Sep 23 10:12:01 1998 +0200 (1998-09-23) changeset 5537 c2bd39a2c0ee parent 5497 497215d66441 child 5598 6b8dee1a6ebb permissions -rw-r--r--
deleted needless parentheses
``` 1 (* Title: HOL/Arith.ML
```
``` 2 ID: \$Id\$
```
``` 3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory
```
``` 4 Copyright 1998 University of Cambridge
```
``` 5
```
``` 6 Proofs about elementary arithmetic: addition, multiplication, etc.
```
``` 7 Some from the Hoare example from Norbert Galm
```
``` 8 *)
```
``` 9
```
``` 10 (*** Basic rewrite rules for the arithmetic operators ***)
```
``` 11
```
``` 12
```
``` 13 (** Difference **)
```
``` 14
```
``` 15 qed_goal "diff_0_eq_0" thy
```
``` 16 "0 - n = 0"
```
``` 17 (fn _ => [induct_tac "n" 1, ALLGOALS Asm_simp_tac]);
```
``` 18
```
``` 19 (*Must simplify BEFORE the induction! (Else we get a critical pair)
```
``` 20 Suc(m) - Suc(n) rewrites to pred(Suc(m) - n) *)
```
``` 21 qed_goal "diff_Suc_Suc" thy
```
``` 22 "Suc(m) - Suc(n) = m - n"
```
``` 23 (fn _ =>
```
``` 24 [Simp_tac 1, induct_tac "n" 1, ALLGOALS Asm_simp_tac]);
```
``` 25
```
``` 26 Addsimps [diff_0_eq_0, diff_Suc_Suc];
```
``` 27
```
``` 28 (* Could be (and is, below) generalized in various ways;
```
``` 29 However, none of the generalizations are currently in the simpset,
```
``` 30 and I dread to think what happens if I put them in *)
```
``` 31 Goal "0 < n ==> Suc(n-1) = n";
```
``` 32 by (asm_simp_tac (simpset() addsplits [nat.split]) 1);
```
``` 33 qed "Suc_pred";
```
``` 34 Addsimps [Suc_pred];
```
``` 35
```
``` 36 Delsimps [diff_Suc];
```
``` 37
```
``` 38
```
``` 39 (**** Inductive properties of the operators ****)
```
``` 40
```
``` 41 (*** Addition ***)
```
``` 42
```
``` 43 qed_goal "add_0_right" thy "m + 0 = m"
```
``` 44 (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
``` 45
```
``` 46 qed_goal "add_Suc_right" thy "m + Suc(n) = Suc(m+n)"
```
``` 47 (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
``` 48
```
``` 49 Addsimps [add_0_right,add_Suc_right];
```
``` 50
```
``` 51 (*Associative law for addition*)
```
``` 52 qed_goal "add_assoc" thy "(m + n) + k = m + ((n + k)::nat)"
```
``` 53 (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
``` 54
```
``` 55 (*Commutative law for addition*)
```
``` 56 qed_goal "add_commute" thy "m + n = n + (m::nat)"
```
``` 57 (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
``` 58
```
``` 59 qed_goal "add_left_commute" thy "x+(y+z)=y+((x+z)::nat)"
```
``` 60 (fn _ => [rtac (add_commute RS trans) 1, rtac (add_assoc RS trans) 1,
```
``` 61 rtac (add_commute RS arg_cong) 1]);
```
``` 62
```
``` 63 (*Addition is an AC-operator*)
```
``` 64 val add_ac = [add_assoc, add_commute, add_left_commute];
```
``` 65
```
``` 66 Goal "(k + m = k + n) = (m=(n::nat))";
```
``` 67 by (induct_tac "k" 1);
```
``` 68 by (Simp_tac 1);
```
``` 69 by (Asm_simp_tac 1);
```
``` 70 qed "add_left_cancel";
```
``` 71
```
``` 72 Goal "(m + k = n + k) = (m=(n::nat))";
```
``` 73 by (induct_tac "k" 1);
```
``` 74 by (Simp_tac 1);
```
``` 75 by (Asm_simp_tac 1);
```
``` 76 qed "add_right_cancel";
```
``` 77
```
``` 78 Goal "(k + m <= k + n) = (m<=(n::nat))";
```
``` 79 by (induct_tac "k" 1);
```
``` 80 by (Simp_tac 1);
```
``` 81 by (Asm_simp_tac 1);
```
``` 82 qed "add_left_cancel_le";
```
``` 83
```
``` 84 Goal "(k + m < k + n) = (m<(n::nat))";
```
``` 85 by (induct_tac "k" 1);
```
``` 86 by (Simp_tac 1);
```
``` 87 by (Asm_simp_tac 1);
```
``` 88 qed "add_left_cancel_less";
```
``` 89
```
``` 90 Addsimps [add_left_cancel, add_right_cancel,
```
``` 91 add_left_cancel_le, add_left_cancel_less];
```
``` 92
```
``` 93 (** Reasoning about m+0=0, etc. **)
```
``` 94
```
``` 95 Goal "(m+n = 0) = (m=0 & n=0)";
```
``` 96 by (induct_tac "m" 1);
```
``` 97 by (ALLGOALS Asm_simp_tac);
```
``` 98 qed "add_is_0";
```
``` 99 AddIffs [add_is_0];
```
``` 100
```
``` 101 Goal "(0<m+n) = (0<m | 0<n)";
```
``` 102 by (simp_tac (simpset() delsimps [neq0_conv] addsimps [neq0_conv RS sym]) 1);
```
``` 103 qed "add_gr_0";
```
``` 104 AddIffs [add_gr_0];
```
``` 105
```
``` 106 (* FIXME: really needed?? *)
```
``` 107 Goal "((m+n)-1 = 0) = (m=0 & n-1 = 0 | m-1 = 0 & n=0)";
```
``` 108 by (exhaust_tac "m" 1);
```
``` 109 by (ALLGOALS (fast_tac (claset() addss (simpset()))));
```
``` 110 qed "pred_add_is_0";
```
``` 111 Addsimps [pred_add_is_0];
```
``` 112
```
``` 113 (* Could be generalized, eg to "k<n ==> m+(n-(Suc k)) = (m+n)-(Suc k)" *)
```
``` 114 Goal "0<n ==> m + (n-1) = (m+n)-1";
```
``` 115 by (exhaust_tac "m" 1);
```
``` 116 by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_Suc]
```
``` 117 addsplits [nat.split])));
```
``` 118 qed "add_pred";
```
``` 119 Addsimps [add_pred];
```
``` 120
```
``` 121 Goal "m + n = m ==> n = 0";
```
``` 122 by (dtac (add_0_right RS ssubst) 1);
```
``` 123 by (asm_full_simp_tac (simpset() addsimps [add_assoc]
```
``` 124 delsimps [add_0_right]) 1);
```
``` 125 qed "add_eq_self_zero";
```
``` 126
```
``` 127
```
``` 128 (**** Additional theorems about "less than" ****)
```
``` 129
```
``` 130 (*Deleted less_natE; instead use less_eq_Suc_add RS exE*)
```
``` 131 Goal "m<n --> (? k. n=Suc(m+k))";
```
``` 132 by (induct_tac "n" 1);
```
``` 133 by (ALLGOALS (simp_tac (simpset() addsimps [le_eq_less_or_eq])));
```
``` 134 by (blast_tac (claset() addSEs [less_SucE]
```
``` 135 addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1);
```
``` 136 qed_spec_mp "less_eq_Suc_add";
```
``` 137
```
``` 138 Goal "n <= ((m + n)::nat)";
```
``` 139 by (induct_tac "m" 1);
```
``` 140 by (ALLGOALS Simp_tac);
```
``` 141 by (etac le_trans 1);
```
``` 142 by (rtac (lessI RS less_imp_le) 1);
```
``` 143 qed "le_add2";
```
``` 144
```
``` 145 Goal "n <= ((n + m)::nat)";
```
``` 146 by (simp_tac (simpset() addsimps add_ac) 1);
```
``` 147 by (rtac le_add2 1);
```
``` 148 qed "le_add1";
```
``` 149
```
``` 150 bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans)));
```
``` 151 bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans)));
```
``` 152
```
``` 153 Goal "(m<n) = (? k. n=Suc(m+k))";
```
``` 154 by (blast_tac (claset() addSIs [less_add_Suc1, less_eq_Suc_add]) 1);
```
``` 155 qed "less_iff_Suc_add";
```
``` 156
```
``` 157
```
``` 158 (*"i <= j ==> i <= j+m"*)
```
``` 159 bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans));
```
``` 160
```
``` 161 (*"i <= j ==> i <= m+j"*)
```
``` 162 bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans));
```
``` 163
```
``` 164 (*"i < j ==> i < j+m"*)
```
``` 165 bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans));
```
``` 166
```
``` 167 (*"i < j ==> i < m+j"*)
```
``` 168 bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans));
```
``` 169
```
``` 170 Goal "i+j < (k::nat) ==> i<k";
```
``` 171 by (etac rev_mp 1);
```
``` 172 by (induct_tac "j" 1);
```
``` 173 by (ALLGOALS Asm_simp_tac);
```
``` 174 by (blast_tac (claset() addDs [Suc_lessD]) 1);
```
``` 175 qed "add_lessD1";
```
``` 176
```
``` 177 Goal "~ (i+j < (i::nat))";
```
``` 178 by (rtac notI 1);
```
``` 179 by (etac (add_lessD1 RS less_irrefl) 1);
```
``` 180 qed "not_add_less1";
```
``` 181
```
``` 182 Goal "~ (j+i < (i::nat))";
```
``` 183 by (simp_tac (simpset() addsimps [add_commute, not_add_less1]) 1);
```
``` 184 qed "not_add_less2";
```
``` 185 AddIffs [not_add_less1, not_add_less2];
```
``` 186
```
``` 187 Goal "m+k<=n --> m<=(n::nat)";
```
``` 188 by (induct_tac "k" 1);
```
``` 189 by (ALLGOALS (asm_simp_tac (simpset() addsimps le_simps)));
```
``` 190 qed_spec_mp "add_leD1";
```
``` 191
```
``` 192 Goal "m+k<=n ==> k<=(n::nat)";
```
``` 193 by (full_simp_tac (simpset() addsimps [add_commute]) 1);
```
``` 194 by (etac add_leD1 1);
```
``` 195 qed_spec_mp "add_leD2";
```
``` 196
```
``` 197 Goal "m+k<=n ==> m<=n & k<=(n::nat)";
```
``` 198 by (blast_tac (claset() addDs [add_leD1, add_leD2]) 1);
```
``` 199 bind_thm ("add_leE", result() RS conjE);
```
``` 200
```
``` 201 (*needs !!k for add_ac to work*)
```
``` 202 Goal "!!k:: nat. [| k<l; m+l = k+n |] ==> m<n";
```
``` 203 by (auto_tac (claset(),
```
``` 204 simpset() delsimps [add_Suc_right]
```
``` 205 addsimps [less_iff_Suc_add,
```
``` 206 add_Suc_right RS sym] @ add_ac));
```
``` 207 qed "less_add_eq_less";
```
``` 208
```
``` 209
```
``` 210 (*** Monotonicity of Addition ***)
```
``` 211
```
``` 212 (*strict, in 1st argument*)
```
``` 213 Goal "i < j ==> i + k < j + (k::nat)";
```
``` 214 by (induct_tac "k" 1);
```
``` 215 by (ALLGOALS Asm_simp_tac);
```
``` 216 qed "add_less_mono1";
```
``` 217
```
``` 218 (*strict, in both arguments*)
```
``` 219 Goal "[|i < j; k < l|] ==> i + k < j + (l::nat)";
```
``` 220 by (rtac (add_less_mono1 RS less_trans) 1);
```
``` 221 by (REPEAT (assume_tac 1));
```
``` 222 by (induct_tac "j" 1);
```
``` 223 by (ALLGOALS Asm_simp_tac);
```
``` 224 qed "add_less_mono";
```
``` 225
```
``` 226 (*A [clumsy] way of lifting < monotonicity to <= monotonicity *)
```
``` 227 val [lt_mono,le] = Goal
```
``` 228 "[| !!i j::nat. i<j ==> f(i) < f(j); \
```
``` 229 \ i <= j \
```
``` 230 \ |] ==> f(i) <= (f(j)::nat)";
```
``` 231 by (cut_facts_tac [le] 1);
```
``` 232 by (asm_full_simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
```
``` 233 by (blast_tac (claset() addSIs [lt_mono]) 1);
```
``` 234 qed "less_mono_imp_le_mono";
```
``` 235
```
``` 236 (*non-strict, in 1st argument*)
```
``` 237 Goal "i<=j ==> i + k <= j + (k::nat)";
```
``` 238 by (res_inst_tac [("f", "%j. j+k")] less_mono_imp_le_mono 1);
```
``` 239 by (etac add_less_mono1 1);
```
``` 240 by (assume_tac 1);
```
``` 241 qed "add_le_mono1";
```
``` 242
```
``` 243 (*non-strict, in both arguments*)
```
``` 244 Goal "[|i<=j; k<=l |] ==> i + k <= j + (l::nat)";
```
``` 245 by (etac (add_le_mono1 RS le_trans) 1);
```
``` 246 by (simp_tac (simpset() addsimps [add_commute]) 1);
```
``` 247 qed "add_le_mono";
```
``` 248
```
``` 249
```
``` 250 (*** Multiplication ***)
```
``` 251
```
``` 252 (*right annihilation in product*)
```
``` 253 qed_goal "mult_0_right" thy "m * 0 = 0"
```
``` 254 (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
``` 255
```
``` 256 (*right successor law for multiplication*)
```
``` 257 qed_goal "mult_Suc_right" thy "m * Suc(n) = m + (m * n)"
```
``` 258 (fn _ => [induct_tac "m" 1,
```
``` 259 ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
```
``` 260
```
``` 261 Addsimps [mult_0_right, mult_Suc_right];
```
``` 262
```
``` 263 Goal "1 * n = n";
```
``` 264 by (Asm_simp_tac 1);
```
``` 265 qed "mult_1";
```
``` 266
```
``` 267 Goal "n * 1 = n";
```
``` 268 by (Asm_simp_tac 1);
```
``` 269 qed "mult_1_right";
```
``` 270
```
``` 271 (*Commutative law for multiplication*)
```
``` 272 qed_goal "mult_commute" thy "m * n = n * (m::nat)"
```
``` 273 (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
``` 274
```
``` 275 (*addition distributes over multiplication*)
```
``` 276 qed_goal "add_mult_distrib" thy "(m + n)*k = (m*k) + ((n*k)::nat)"
```
``` 277 (fn _ => [induct_tac "m" 1,
```
``` 278 ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
```
``` 279
```
``` 280 qed_goal "add_mult_distrib2" thy "k*(m + n) = (k*m) + ((k*n)::nat)"
```
``` 281 (fn _ => [induct_tac "m" 1,
```
``` 282 ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
```
``` 283
```
``` 284 (*Associative law for multiplication*)
```
``` 285 qed_goal "mult_assoc" thy "(m * n) * k = m * ((n * k)::nat)"
```
``` 286 (fn _ => [induct_tac "m" 1,
```
``` 287 ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib]))]);
```
``` 288
```
``` 289 qed_goal "mult_left_commute" thy "x*(y*z) = y*((x*z)::nat)"
```
``` 290 (fn _ => [rtac trans 1, rtac mult_commute 1, rtac trans 1,
```
``` 291 rtac mult_assoc 1, rtac (mult_commute RS arg_cong) 1]);
```
``` 292
```
``` 293 val mult_ac = [mult_assoc,mult_commute,mult_left_commute];
```
``` 294
```
``` 295 Goal "(m*n = 0) = (m=0 | n=0)";
```
``` 296 by (induct_tac "m" 1);
```
``` 297 by (induct_tac "n" 2);
```
``` 298 by (ALLGOALS Asm_simp_tac);
```
``` 299 qed "mult_is_0";
```
``` 300 Addsimps [mult_is_0];
```
``` 301
```
``` 302 Goal "m <= m*(m::nat)";
```
``` 303 by (induct_tac "m" 1);
```
``` 304 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_assoc RS sym])));
```
``` 305 by (etac (le_add2 RSN (2,le_trans)) 1);
```
``` 306 qed "le_square";
```
``` 307
```
``` 308
```
``` 309 (*** Difference ***)
```
``` 310
```
``` 311
```
``` 312 qed_goal "diff_self_eq_0" thy "m - m = 0"
```
``` 313 (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
``` 314 Addsimps [diff_self_eq_0];
```
``` 315
```
``` 316 (*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
```
``` 317 Goal "~ m<n --> n+(m-n) = (m::nat)";
```
``` 318 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
``` 319 by (ALLGOALS Asm_simp_tac);
```
``` 320 qed_spec_mp "add_diff_inverse";
```
``` 321
```
``` 322 Goal "n<=m ==> n+(m-n) = (m::nat)";
```
``` 323 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1);
```
``` 324 qed "le_add_diff_inverse";
```
``` 325
```
``` 326 Goal "n<=m ==> (m-n)+n = (m::nat)";
```
``` 327 by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1);
```
``` 328 qed "le_add_diff_inverse2";
```
``` 329
```
``` 330 Addsimps [le_add_diff_inverse, le_add_diff_inverse2];
```
``` 331
```
``` 332
```
``` 333 (*** More results about difference ***)
```
``` 334
```
``` 335 Goal "n <= m ==> Suc(m)-n = Suc(m-n)";
```
``` 336 by (etac rev_mp 1);
```
``` 337 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
``` 338 by (ALLGOALS Asm_simp_tac);
```
``` 339 qed "Suc_diff_le";
```
``` 340
```
``` 341 Goal "n<=(l::nat) --> Suc l - n + m = Suc (l - n + m)";
```
``` 342 by (res_inst_tac [("m","n"),("n","l")] diff_induct 1);
```
``` 343 by (ALLGOALS Asm_simp_tac);
```
``` 344 qed_spec_mp "Suc_diff_add_le";
```
``` 345
```
``` 346 Goal "m - n < Suc(m)";
```
``` 347 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
``` 348 by (etac less_SucE 3);
```
``` 349 by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq])));
```
``` 350 qed "diff_less_Suc";
```
``` 351
```
``` 352 Goal "m - n <= (m::nat)";
```
``` 353 by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
```
``` 354 by (ALLGOALS Asm_simp_tac);
```
``` 355 qed "diff_le_self";
```
``` 356 Addsimps [diff_le_self];
```
``` 357
```
``` 358 (* j<k ==> j-n < k *)
```
``` 359 bind_thm ("less_imp_diff_less", diff_le_self RS le_less_trans);
```
``` 360
```
``` 361 Goal "!!i::nat. i-j-k = i - (j+k)";
```
``` 362 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
```
``` 363 by (ALLGOALS Asm_simp_tac);
```
``` 364 qed "diff_diff_left";
```
``` 365
```
``` 366 Goal "(Suc m - n) - Suc k = m - n - k";
```
``` 367 by (simp_tac (simpset() addsimps [diff_diff_left]) 1);
```
``` 368 qed "Suc_diff_diff";
```
``` 369 Addsimps [Suc_diff_diff];
```
``` 370
```
``` 371 Goal "0<n ==> n - Suc i < n";
```
``` 372 by (exhaust_tac "n" 1);
```
``` 373 by Safe_tac;
```
``` 374 by (asm_simp_tac (simpset() addsimps le_simps) 1);
```
``` 375 qed "diff_Suc_less";
```
``` 376 Addsimps [diff_Suc_less];
```
``` 377
```
``` 378 Goal "i<n ==> n - Suc i < n - i";
```
``` 379 by (exhaust_tac "n" 1);
```
``` 380 by (auto_tac (claset(),
```
``` 381 simpset() addsimps [Suc_diff_le]@le_simps));
```
``` 382 qed "diff_Suc_less_diff";
```
``` 383
```
``` 384 Goal "m - n <= Suc m - n";
```
``` 385 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
``` 386 by (ALLGOALS Asm_simp_tac);
```
``` 387 qed "diff_le_Suc_diff";
```
``` 388
```
``` 389 (*This and the next few suggested by Florian Kammueller*)
```
``` 390 Goal "!!i::nat. i-j-k = i-k-j";
```
``` 391 by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1);
```
``` 392 qed "diff_commute";
```
``` 393
```
``` 394 Goal "k<=j --> j<=i --> i - (j - k) = i - j + (k::nat)";
```
``` 395 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
```
``` 396 by (ALLGOALS Asm_simp_tac);
```
``` 397 by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1);
```
``` 398 qed_spec_mp "diff_diff_right";
```
``` 399
```
``` 400 Goal "k <= (j::nat) --> (i + j) - k = i + (j - k)";
```
``` 401 by (res_inst_tac [("m","j"),("n","k")] diff_induct 1);
```
``` 402 by (ALLGOALS Asm_simp_tac);
```
``` 403 qed_spec_mp "diff_add_assoc";
```
``` 404
```
``` 405 Goal "k <= (j::nat) --> (j + i) - k = i + (j - k)";
```
``` 406 by (asm_simp_tac (simpset() addsimps [add_commute, diff_add_assoc]) 1);
```
``` 407 qed_spec_mp "diff_add_assoc2";
```
``` 408
```
``` 409 Goal "(n+m) - n = (m::nat)";
```
``` 410 by (induct_tac "n" 1);
```
``` 411 by (ALLGOALS Asm_simp_tac);
```
``` 412 qed "diff_add_inverse";
```
``` 413 Addsimps [diff_add_inverse];
```
``` 414
```
``` 415 Goal "(m+n) - n = (m::nat)";
```
``` 416 by (simp_tac (simpset() addsimps [diff_add_assoc]) 1);
```
``` 417 qed "diff_add_inverse2";
```
``` 418 Addsimps [diff_add_inverse2];
```
``` 419
```
``` 420 Goal "i <= (j::nat) ==> (j-i=k) = (j=k+i)";
```
``` 421 by Safe_tac;
```
``` 422 by (ALLGOALS Asm_simp_tac);
```
``` 423 qed "le_imp_diff_is_add";
```
``` 424
```
``` 425 Goal "(m-n = 0) = (m <= n)";
```
``` 426 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
``` 427 by (ALLGOALS Asm_simp_tac);
```
``` 428 qed "diff_is_0_eq";
```
``` 429 Addsimps [diff_is_0_eq RS iffD2];
```
``` 430
```
``` 431 Goal "m-n = 0 --> n-m = 0 --> m=n";
```
``` 432 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
``` 433 by (REPEAT(Simp_tac 1 THEN TRY(atac 1)));
```
``` 434 qed_spec_mp "diffs0_imp_equal";
```
``` 435
```
``` 436 Goal "(0<n-m) = (m<n)";
```
``` 437 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
``` 438 by (ALLGOALS Asm_simp_tac);
```
``` 439 qed "zero_less_diff";
```
``` 440 Addsimps [zero_less_diff];
```
``` 441
```
``` 442 Goal "i < j ==> ? k. 0<k & i+k = j";
```
``` 443 by (res_inst_tac [("x","j - i")] exI 1);
```
``` 444 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, less_not_sym]) 1);
```
``` 445 qed "less_imp_add_positive";
```
``` 446
```
``` 447 Goal "Suc(m)-n = (if m<n then 0 else Suc(m-n))";
```
``` 448 by (simp_tac (simpset() addsimps [leI, Suc_le_eq, Suc_diff_le]) 1);
```
``` 449 qed "if_Suc_diff_le";
```
``` 450
```
``` 451 Goal "Suc(m)-n <= Suc(m-n)";
```
``` 452 by (simp_tac (simpset() addsimps [if_Suc_diff_le]) 1);
```
``` 453 qed "diff_Suc_le_Suc_diff";
```
``` 454
```
``` 455 Goal "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)";
```
``` 456 by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
```
``` 457 by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac));
```
``` 458 qed "zero_induct_lemma";
```
``` 459
```
``` 460 val prems = Goal "[| P(k); !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
```
``` 461 by (rtac (diff_self_eq_0 RS subst) 1);
```
``` 462 by (rtac (zero_induct_lemma RS mp RS mp) 1);
```
``` 463 by (REPEAT (ares_tac ([impI,allI]@prems) 1));
```
``` 464 qed "zero_induct";
```
``` 465
```
``` 466 Goal "(k+m) - (k+n) = m - (n::nat)";
```
``` 467 by (induct_tac "k" 1);
```
``` 468 by (ALLGOALS Asm_simp_tac);
```
``` 469 qed "diff_cancel";
```
``` 470 Addsimps [diff_cancel];
```
``` 471
```
``` 472 Goal "(m+k) - (n+k) = m - (n::nat)";
```
``` 473 val add_commute_k = read_instantiate [("n","k")] add_commute;
```
``` 474 by (asm_simp_tac (simpset() addsimps [add_commute_k]) 1);
```
``` 475 qed "diff_cancel2";
```
``` 476 Addsimps [diff_cancel2];
```
``` 477
```
``` 478 (*From Clemens Ballarin, proof by lcp*)
```
``` 479 Goal "[| k<=n; n<=m |] ==> (m-k) - (n-k) = m-(n::nat)";
```
``` 480 by (REPEAT (etac rev_mp 1));
```
``` 481 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
``` 482 by (ALLGOALS Asm_simp_tac);
```
``` 483 (*a confluence problem*)
```
``` 484 by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1);
```
``` 485 qed "diff_right_cancel";
```
``` 486
```
``` 487 Goal "n - (n+m) = 0";
```
``` 488 by (induct_tac "n" 1);
```
``` 489 by (ALLGOALS Asm_simp_tac);
```
``` 490 qed "diff_add_0";
```
``` 491 Addsimps [diff_add_0];
```
``` 492
```
``` 493
```
``` 494 (** Difference distributes over multiplication **)
```
``` 495
```
``` 496 Goal "!!m::nat. (m - n) * k = (m * k) - (n * k)";
```
``` 497 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
``` 498 by (ALLGOALS Asm_simp_tac);
```
``` 499 qed "diff_mult_distrib" ;
```
``` 500
```
``` 501 Goal "!!m::nat. k * (m - n) = (k * m) - (k * n)";
```
``` 502 val mult_commute_k = read_instantiate [("m","k")] mult_commute;
```
``` 503 by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1);
```
``` 504 qed "diff_mult_distrib2" ;
```
``` 505 (*NOT added as rewrites, since sometimes they are used from right-to-left*)
```
``` 506
```
``` 507
```
``` 508 (*** Monotonicity of Multiplication ***)
```
``` 509
```
``` 510 Goal "i <= (j::nat) ==> i*k<=j*k";
```
``` 511 by (induct_tac "k" 1);
```
``` 512 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono])));
```
``` 513 qed "mult_le_mono1";
```
``` 514
```
``` 515 (*<=monotonicity, BOTH arguments*)
```
``` 516 Goal "[| i <= (j::nat); k <= l |] ==> i*k <= j*l";
```
``` 517 by (etac (mult_le_mono1 RS le_trans) 1);
```
``` 518 by (rtac le_trans 1);
```
``` 519 by (stac mult_commute 2);
```
``` 520 by (etac mult_le_mono1 2);
```
``` 521 by (simp_tac (simpset() addsimps [mult_commute]) 1);
```
``` 522 qed "mult_le_mono";
```
``` 523
```
``` 524 (*strict, in 1st argument; proof is by induction on k>0*)
```
``` 525 Goal "[| i<j; 0<k |] ==> k*i < k*j";
```
``` 526 by (eres_inst_tac [("m1","0")] (less_eq_Suc_add RS exE) 1);
```
``` 527 by (Asm_simp_tac 1);
```
``` 528 by (induct_tac "x" 1);
```
``` 529 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_less_mono])));
```
``` 530 qed "mult_less_mono2";
```
``` 531
```
``` 532 Goal "[| i<j; 0<k |] ==> i*k < j*k";
```
``` 533 by (dtac mult_less_mono2 1);
```
``` 534 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [mult_commute])));
```
``` 535 qed "mult_less_mono1";
```
``` 536
```
``` 537 Goal "(0 < m*n) = (0<m & 0<n)";
```
``` 538 by (induct_tac "m" 1);
```
``` 539 by (induct_tac "n" 2);
```
``` 540 by (ALLGOALS Asm_simp_tac);
```
``` 541 qed "zero_less_mult_iff";
```
``` 542 Addsimps [zero_less_mult_iff];
```
``` 543
```
``` 544 Goal "(m*n = 1) = (m=1 & n=1)";
```
``` 545 by (induct_tac "m" 1);
```
``` 546 by (Simp_tac 1);
```
``` 547 by (induct_tac "n" 1);
```
``` 548 by (Simp_tac 1);
```
``` 549 by (fast_tac (claset() addss simpset()) 1);
```
``` 550 qed "mult_eq_1_iff";
```
``` 551 Addsimps [mult_eq_1_iff];
```
``` 552
```
``` 553 Goal "0<k ==> (m*k < n*k) = (m<n)";
```
``` 554 by (safe_tac (claset() addSIs [mult_less_mono1]));
```
``` 555 by (cut_facts_tac [less_linear] 1);
```
``` 556 by (blast_tac (claset() addIs [mult_less_mono1] addEs [less_asym]) 1);
```
``` 557 qed "mult_less_cancel2";
```
``` 558
```
``` 559 Goal "0<k ==> (k*m < k*n) = (m<n)";
```
``` 560 by (dtac mult_less_cancel2 1);
```
``` 561 by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
```
``` 562 qed "mult_less_cancel1";
```
``` 563 Addsimps [mult_less_cancel1, mult_less_cancel2];
```
``` 564
```
``` 565 Goal "(Suc k * m < Suc k * n) = (m < n)";
```
``` 566 by (rtac mult_less_cancel1 1);
```
``` 567 by (Simp_tac 1);
```
``` 568 qed "Suc_mult_less_cancel1";
```
``` 569
```
``` 570 Goalw [le_def] "(Suc k * m <= Suc k * n) = (m <= n)";
```
``` 571 by (simp_tac (simpset_of HOL.thy) 1);
```
``` 572 by (rtac Suc_mult_less_cancel1 1);
```
``` 573 qed "Suc_mult_le_cancel1";
```
``` 574
```
``` 575 Goal "0<k ==> (m*k = n*k) = (m=n)";
```
``` 576 by (cut_facts_tac [less_linear] 1);
```
``` 577 by Safe_tac;
```
``` 578 by (assume_tac 2);
```
``` 579 by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac));
```
``` 580 by (ALLGOALS Asm_full_simp_tac);
```
``` 581 qed "mult_cancel2";
```
``` 582
```
``` 583 Goal "0<k ==> (k*m = k*n) = (m=n)";
```
``` 584 by (dtac mult_cancel2 1);
```
``` 585 by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
```
``` 586 qed "mult_cancel1";
```
``` 587 Addsimps [mult_cancel1, mult_cancel2];
```
``` 588
```
``` 589 Goal "(Suc k * m = Suc k * n) = (m = n)";
```
``` 590 by (rtac mult_cancel1 1);
```
``` 591 by (Simp_tac 1);
```
``` 592 qed "Suc_mult_cancel1";
```
``` 593
```
``` 594
```
``` 595 (** Lemma for gcd **)
```
``` 596
```
``` 597 Goal "m = m*n ==> n=1 | m=0";
```
``` 598 by (dtac sym 1);
```
``` 599 by (rtac disjCI 1);
```
``` 600 by (rtac nat_less_cases 1 THEN assume_tac 2);
```
``` 601 by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1);
```
``` 602 by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1);
```
``` 603 qed "mult_eq_self_implies_10";
```
``` 604
```
``` 605
```
``` 606 (*** Subtraction laws -- mostly from Clemens Ballarin ***)
```
``` 607
```
``` 608 Goal "[| a < (b::nat); c <= a |] ==> a-c < b-c";
```
``` 609 by (subgoal_tac "c+(a-c) < c+(b-c)" 1);
```
``` 610 by (Full_simp_tac 1);
```
``` 611 by (subgoal_tac "c <= b" 1);
```
``` 612 by (blast_tac (claset() addIs [less_imp_le, le_trans]) 2);
```
``` 613 by (Asm_simp_tac 1);
```
``` 614 qed "diff_less_mono";
```
``` 615
```
``` 616 Goal "a+b < (c::nat) ==> a < c-b";
```
``` 617 by (dtac diff_less_mono 1);
```
``` 618 by (rtac le_add2 1);
```
``` 619 by (Asm_full_simp_tac 1);
```
``` 620 qed "add_less_imp_less_diff";
```
``` 621
```
``` 622 Goal "(i < j-k) = (i+k < (j::nat))";
```
``` 623 by (rtac iffI 1);
```
``` 624 by (case_tac "k <= j" 1);
```
``` 625 by (dtac le_add_diff_inverse2 1);
```
``` 626 by (dres_inst_tac [("k","k")] add_less_mono1 1);
```
``` 627 by (Asm_full_simp_tac 1);
```
``` 628 by (rotate_tac 1 1);
```
``` 629 by (asm_full_simp_tac (simpset() addSolver cut_trans_tac) 1);
```
``` 630 by (etac add_less_imp_less_diff 1);
```
``` 631 qed "less_diff_conv";
```
``` 632
```
``` 633 Goal "(j-k <= (i::nat)) = (j <= i+k)";
```
``` 634 by (simp_tac (simpset() addsimps [less_diff_conv, le_def]) 1);
```
``` 635 qed "le_diff_conv";
```
``` 636
```
``` 637 Goal "k <= j ==> (i <= j-k) = (i+k <= (j::nat))";
```
``` 638 by (asm_full_simp_tac
```
``` 639 (simpset() delsimps [less_Suc_eq_le]
```
``` 640 addsimps [less_Suc_eq_le RS sym, less_diff_conv,
```
``` 641 Suc_diff_le RS sym]) 1);
```
``` 642 qed "le_diff_conv2";
```
``` 643
```
``` 644 Goal "Suc i <= n ==> Suc (n - Suc i) = n - i";
```
``` 645 by (asm_full_simp_tac (simpset() addsimps [Suc_diff_le RS sym]) 1);
```
``` 646 qed "Suc_diff_Suc";
```
``` 647
```
``` 648 Goal "i <= (n::nat) ==> n - (n - i) = i";
```
``` 649 by (etac rev_mp 1);
```
``` 650 by (res_inst_tac [("m","n"),("n","i")] diff_induct 1);
```
``` 651 by (ALLGOALS (asm_simp_tac (simpset() addsimps [Suc_diff_le])));
```
``` 652 qed "diff_diff_cancel";
```
``` 653 Addsimps [diff_diff_cancel];
```
``` 654
```
``` 655 Goal "k <= (n::nat) ==> m <= n + m - k";
```
``` 656 by (etac rev_mp 1);
```
``` 657 by (res_inst_tac [("m", "k"), ("n", "n")] diff_induct 1);
```
``` 658 by (Simp_tac 1);
```
``` 659 by (simp_tac (simpset() addsimps [le_add2, less_imp_le]) 1);
```
``` 660 by (Simp_tac 1);
```
``` 661 qed "le_add_diff";
```
``` 662
```
``` 663 Goal "0<k ==> j<i --> j+k-i < k";
```
``` 664 by (res_inst_tac [("m","j"),("n","i")] diff_induct 1);
```
``` 665 by (ALLGOALS Asm_simp_tac);
```
``` 666 qed_spec_mp "add_diff_less";
```
``` 667
```
``` 668
```
``` 669 Goal "m-1 < n ==> m <= n";
```
``` 670 by (exhaust_tac "m" 1);
```
``` 671 by (auto_tac (claset(), simpset() addsimps [Suc_le_eq]));
```
``` 672 qed "pred_less_imp_le";
```
``` 673
```
``` 674 Goal "j<=i ==> i - j < Suc i - j";
```
``` 675 by (REPEAT (etac rev_mp 1));
```
``` 676 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
```
``` 677 by Auto_tac;
```
``` 678 qed "diff_less_Suc_diff";
```
``` 679
```
``` 680 Goal "i - j <= Suc i - j";
```
``` 681 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
```
``` 682 by Auto_tac;
```
``` 683 qed "diff_le_Suc_diff";
```
``` 684 AddIffs [diff_le_Suc_diff];
```
``` 685
```
``` 686 Goal "n - Suc i <= n - i";
```
``` 687 by (case_tac "i<n" 1);
```
``` 688 by (dtac diff_Suc_less_diff 1);
```
``` 689 by (auto_tac (claset(), simpset() addsimps [leI]));
```
``` 690 qed "diff_Suc_le_diff";
```
``` 691 AddIffs [diff_Suc_le_diff];
```
``` 692
```
``` 693 Goal "0 < n ==> (m <= n-1) = (m<n)";
```
``` 694 by (exhaust_tac "n" 1);
```
``` 695 by (auto_tac (claset(), simpset() addsimps le_simps));
```
``` 696 qed "le_pred_eq";
```
``` 697
```
``` 698 Goal "0 < n ==> (m-1 < n) = (m<=n)";
```
``` 699 by (exhaust_tac "m" 1);
```
``` 700 by (auto_tac (claset(), simpset() addsimps [Suc_le_eq]));
```
``` 701 qed "less_pred_eq";
```
``` 702
```
``` 703 (*In ordinary notation: if 0<n and n<=m then m-n < m *)
```
``` 704 Goal "[| 0<n; ~ m<n |] ==> m - n < m";
```
``` 705 by (subgoal_tac "0<n --> ~ m<n --> m - n < m" 1);
```
``` 706 by (Blast_tac 1);
```
``` 707 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
``` 708 by (ALLGOALS(asm_simp_tac(simpset() addsimps [diff_less_Suc])));
```
``` 709 qed "diff_less";
```
``` 710
```
``` 711 Goal "[| 0<n; n<=m |] ==> m - n < m";
```
``` 712 by (asm_simp_tac (simpset() addsimps [diff_less, not_less_iff_le]) 1);
```
``` 713 qed "le_diff_less";
```
``` 714
```
``` 715
```
``` 716
```
``` 717 (** (Anti)Monotonicity of subtraction -- by Stefan Merz **)
```
``` 718
```
``` 719 (* Monotonicity of subtraction in first argument *)
```
``` 720 Goal "m <= (n::nat) --> (m-l) <= (n-l)";
```
``` 721 by (induct_tac "n" 1);
```
``` 722 by (Simp_tac 1);
```
``` 723 by (simp_tac (simpset() addsimps [le_Suc_eq]) 1);
```
``` 724 by (blast_tac (claset() addIs [diff_le_Suc_diff, le_trans]) 1);
```
``` 725 qed_spec_mp "diff_le_mono";
```
``` 726
```
``` 727 Goal "m <= (n::nat) ==> (l-n) <= (l-m)";
```
``` 728 by (induct_tac "l" 1);
```
``` 729 by (Simp_tac 1);
```
``` 730 by (case_tac "n <= na" 1);
```
``` 731 by (subgoal_tac "m <= na" 1);
```
``` 732 by (asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1);
```
``` 733 by (fast_tac (claset() addEs [le_trans]) 1);
```
``` 734 by (dtac not_leE 1);
```
``` 735 by (asm_simp_tac (simpset() addsimps [if_Suc_diff_le]) 1);
```
``` 736 qed_spec_mp "diff_le_mono2";
```
| 12,616
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๏ปฟ C Sharp - Find n terms of even natural number and their sum
# C#: Calculate n terms of even natural number and their sum
## C# Sharp For Loop: Exercise-16 with Solution
Write a program in C# Sharp to display the n terms of even natural number and their sum.
Visual Presentation:
Sample Solution:
C# Sharp Code:
``````using System; // Importing necessary namespace
public class Exercise16 // Declaration of the Exercise16 class
{
public static void Main() // Main method, entry point of the program
{
int i, n, sum = 0; // Declaration of variables 'i' for iteration, 'n' for the number of terms, 'sum' to store the sum
Console.Write("\n\n"); // Displaying new lines
Console.Write("Calculate n terms of even natural number and their sum:\n"); // Displaying the purpose of the program
Console.Write("---------------------------------------------------------"); // Displaying a separator
Console.Write("\n\n");
Console.Write("Input number of terms : "); // Prompting the user to input the number of terms
n = Convert.ToInt32(Console.ReadLine()); // Reading the number of terms entered by the user
Console.Write("\nThe even numbers are :"); // Displaying a message for even numbers
for (i = 1; i <= n; i++) // Loop to calculate even numbers and their sum
{
Console.Write("{0} ", 2 * i); // Displaying the even number in the sequence
sum += 2 * i; // Adding the even number to the sum
}
Console.Write("\nThe Sum of even Natural Number up to {0} terms : {1} \n", n, sum); // Displaying the sum of even natural numbers
}
}
```
```
Sample Output:
```Calculate n terms of even natural number and their sum:
---------------------------------------------------------
Input number of terms : 10
The even numbers are :2 4 6 8 10 12 14 16 18 20
The Sum of even Natural Number upto 10 terms : 110
```
Flowchart:
C# Sharp Code Editor:
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What is a 100 stacked line chart
Show Connect Line on Stack Column/Bar checkbox to control connecting stacked column/bars with line or not. A new Line tab will show in Plot Properties level. The Line chart supports 100% stacking, where the last series is always rendered as a flat line on top of the value axis. You can enable the 100% stack featureย Jan 27, 2020 Stacked area charts also support 100% stacking, where the stacks of (the major axis of the chart, such as the X axis on a typical line chart).
Aug 13, 2014 Parts of whole data are often plotted as pie charts, but Prism lets you show it as a stack if you want (horizontal or vertical). When plotted this way,ย Aug 12, 2014 The problem with stacked area graphs is that of baselining. When we compare multiple lines in a line graph which are comparing from the the case for proportional stacked area graphs (where the top of the y-axis is 100%). May 17, 2013 In the Stacked Bar Chart to 100% example workbook, right-click SUM(Sales) on the Columns shelf in the Primary Setup tab, and then click Addย Mar 29, 2017 Today, we'll talk about stacked bar charts, because โ much to my a dual-axis graph with separate bars for each product and a line series forย Oct 12, 2012 Area chart - Area charts are used to represent cumulated totals using numbers or percentages (stacked area charts in this case) over time. Line
100% Stacked Line. It is similar to a stacked line chart, with the difference that the y-axis shows percentages rather than
Sep 4, 2019 In this guide, we will learn the stacked bar chart in the following steps: We will start We can achieve that by changing it into a 100% stacked bar chart. switch Rows tab in Format Lines pane -> change Grid Lines to None. Values along the Y axis range between 0 and 100%. The height of each series in a Percent Stacked Area Chart is determined by its contribution to the sum of allย Yes, in theory, one could use a stacked line chart (where line values accumulate) or a 100% stacked line chart (where lines accumulate to 100%), but a stackedย In both area charts and line graphs, data points are plotted and then A stacked area chart shows how much each part contributes to the whole amount.
You will see the different Line charts available. A Line chart has the following sub -types โ. 2-D Line charts. Line. 100% Stacked Line. Line with Markers. Stacked
Mar 29, 2017 Today, we'll talk about stacked bar charts, because โ much to my a dual-axis graph with separate bars for each product and a line series forย Oct 12, 2012 Area chart - Area charts are used to represent cumulated totals using numbers or percentages (stacked area charts in this case) over time. Line
Stacked bar chart comes under the bar chart. Two types of stacked bar charts are available. Stacked bar chart and 100% stacked bar chart. Where the stacked bar chart represents the given data directly. But 100% stacked bar chart will represent the given data as the percentage of data that contribute to a total volume in a different category.
Sep 4, 2019 In this guide, we will learn the stacked bar chart in the following steps: We will start We can achieve that by changing it into a 100% stacked bar chart. switch Rows tab in Format Lines pane -> change Grid Lines to None.
Dec 9, 2016 A stacked line graph will follow the same structure โ x-axis, independent, y-axis, dependent, plot and connect โ but will consist of more than
Area chart - Area charts are used to represent cumulated totals using numbers or percentages (stacked area charts in this case) over time. Line chart - A line chart is often used to visualize a trend in data over intervals of time โ a time series โ thus the line is often drawn chronologically.
Excel charts allow you to display data in many different formats. Lines, bars, pie pieces, two-dimensional, three-dimensional, stacked bars and different axes are ย A 100% Stacked Bar chart shows the relationship of individual segments to the whole. Each bar SEGMENT COMPARISON LINE ยท GRIDLINES. BAR. GAPS.
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Do you have an Online Class?
# 1. A firm recently purcha
1. A firm recently purchased a new facility costing \$984 thousand. The firm financed this purchase with an amortized loan at an interest rate of 8.8 percent APR, with monthly payments of \$23.9 thousand. How long will it take to pay off this loan? (Enter answer in months, accurate to two decimal places.)2. An investment is expected to produce \$2,399 at the end of each year for the next 13 years. Other investments of similar riskiness available to you are yielding 11 percent return. What is the maximum you should be willing to pay for this investment?3. You are not thrilled about spending your entire life working. So, you have decided that you will save \$9 thousand a year, starting at the end of this year, and retire as soon as you can accumulate \$1 million. If you can earn an average of 7.23 percent on your savings, how many years will pass before you get to retire?Enter answer in years, accurate to two decimal places.4. GDebi, Inc. plans to issue 4.7 percent coupon bonds, with annual coupon frequency, 18 years to maturity and \$1000 face value. If the prevailing market yield on bonds of similar riskiness and maturity is 5.4 percent, what would be the market price of GDebiโs bonds?5. How much money must you invest today, at 5.4 percent fixed annual interest rate, in order to have 10 thousand in 19 years?
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7 Interest Rate Risk
# T term structure equation 1
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Unformatted text preview: tial position ฯ S = (F0 โ Fฯ) ร 100 bonds ร no. contracts Ec 174 INTEREST RATE RISK on ร 9 ร 0.005 = โ\$225,000 (a loss) โข ฮB/B0 = โDb*(ฮry) = โ12.25 ร 0.005 = โ6.125% โข Fฯ has dropped by 6.125% to 0.93875 ร 1274.84 = \$1,196.76. โข ฯshort = (F0 โ Fฯ) ร units = (1274.84โ1196.76) ร 100ร29 = \$226,432 (a gain). โข Total net change in value is only 226,432 - 225,000 = \$1,432. Ec 174 INTEREST RATE RISK p. 12 of 17 E. Interest Rate Swaps {BKM ยง23.4} 1. Introduction to Swaps: a) A swap is an agreement between two entities (counterparties) to exchange a sequence of cash flows in each of several future periods. 1) The agreement specifies the formula for computation of the cash flows and the exact dates for their payment. 2) Typically, one cash flow stream is fixed, and the other is โfloatingโ in that it depends on the future value of some market variable like an interest or exchange rate. b) A swap arrangement is like a multi- period series of forward contracts. For example, con- sider the following โone- cash- flow fixed- for- floatingโ swap. 1) Logan agrees to buy 100 oz. of gold in one year at forward price F0 = \$600/oz. Sharon, with gold to deliver, is the counterparty with the short position. 2) Once Logan takes delivery of the gold and pays Sharon, he can sell the gold at ST. 3) Logan has exchanged cash flows with Sharon. He will get the floating cash flow \$STร100 (which depends on the market price of gold ST in one year), and she will get the fixed cash flow \$60,000 (known today). c) Three common types of swaps. 1) Interest rate swaps โ one company pays interest at a fixed rate and receives it at a float- ing rate (usually LIBOR); the counterparty does the opposite. 2) Currency swaps โ one company pays interest and principal in one currency and receives interest and principal in another, vis- ร - vis a counterparty company. 3) Credit default swaps (CDSs) โ not really swaps, but insurance policies. Company B holds a bond or loan or debt security from corporation X. B buys a default protection policy from insurance company S and makes periodic payments (insurance premiums). If X defaults on the bond payments, S pays the principal to B. 4) Swaps began about 1980; they became a major OTC derivative, with about \$300 trillion of open interest in 2008. AIG got hit hard on Notation (Interest Rate Swaps) CDSs. \$A Notional principal 2. Mechanics of Interest Rate Swaps: {Hull ยง7.1, 7.5 - 7.6} rโ Floating rate (LIBOR) rx Fixed (swap) rate a) โPlain Vanillaโ interest rate swaps. X party RECEIVING fixed CF 1) At time t = 0, parties X and F agree to: F party RECEIVING floating CF โข a dollar value of notional principal (\$A) โข a sequence of dates during the next few years for payments of cash flows โข a fixed or predetermined interest rate (rx) โข a market or floating rate (rโ) which varies over the life of the swap agreement 2) X will pay \$rโA of interest at floating or variable rate to F at regular intervals...
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## This document was uploaded on 02/18/2014 for the course ECON 174 at UCSD.
Ask a homework question - tutors are online
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# Trie and Trees
## trie
In computer science, a trie, also called digital tree and sometimes radix tree or prefix tree (as they can be searched by prefixes), is an ordered tree data structure that is used to store a dynamic set or associative array where the keys are usually strings. Unlike a binary search tree, no node in the tree stores the key associated with that node; instead, its position in the tree defines the key with which it is associated. All the descendants of a node have a common prefix of the string associated with that node, and the root is associated with the empty string. Values are normally not associated with every node, only with leaves and some inner nodes that correspond to keys of interest. For the space-optimized presentation of prefix tree, see compact prefix tree.
Trie is one of the most important data structure for autocomplete.
### Trie Problems
Various problems on Trie can be found here
### Postfix to Infix
#### Evaluate Postfix expression
Valid operators are +, -, *, /.
Each operand may be an integer or another expression.
Some examples:
["2", "1", "+", "3", "*"] -> ((2 + 1) * 3) -> 9 ["4", "13", "5", "/", "+"] -> (4 + (13 / 5)) -> 6
### Tree Problems
#### Find BST is balanced or not
Balanced tree is defined to be a tree such that no two leaf nodes differ in distance from the root by more than one.
#### Find Binary Tree is BST or not
A binary search tree (BST) is a node based binary tree data structure which has the following properties.
โข The left subtree of a node contains only nodes with keys less than the nodeโs key.
โข The right subtree of a node contains only nodes with keys greater than the nodeโs key.
โข Both the left and right subtrees must also be binary search trees.
Method 1
Perform inorder traversal on tree and store it in temporary array. By property of inorder traversal the numbers stored should be sorted sequence of itโs a BST else itโs not BST.
The only caveat is that this method require O(n) space
Method 2
#### BST - Recursive Inorder Traversal
Time complexity O(n) and space complexity is size of stack for function calls
#### BST - Iterative Inorder Traversal
Time complexity O(n) and space complexity is size of stack
#### BST - Morris Inorder Traversal
Morris Inorder Traversal run without using recursion and without extra stack space.
Morris Inorder runs in O(NlogN) time and O(1) space
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# How Many Kilograms Equal One Pound?
## Answer
One pound equals 0.45359 kilograms approximately.
1 Additional Answer
Ask.com Answer for: how many kilograms equal one pound
One pound is equal to 0.4535924 kilograms.
Convert to
Q&A Related to "How Many Kilograms Equal One Pound"
There are 2.20462262 pounds in one kilogram. That means that if a person weighs 100 pounds then they would weigh 45.35923703 kilograms. http://answers.ask.com/Reference/Other/how_many_po...
32 kilogram = 70.547923899 pound. http://wiki.answers.com/Q/What_is_32_kilogram_is_e...
1 Kilogram = 2.20462 Pounds. Thanks for using ChaCha. Please call or text http://www.chacha.com/question/what-is-the-equal-o...
81 kilograms = 178.574432 pounds. ChaCha! http://www.chacha.com/question/what-is-81-kilogram...
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# MEDICAL
WAYS TO CURB THE NEGATIVE AFFECTS OF RISING HEALTHCARE COSTS
1. ๐ 0
2. ๐ 0
3. ๐ 141
1. Huh???
The only way to curb these negative effects is to lower health care costs.
1. ๐ 0
2. ๐ 0
2. Just send more people to med school and then have an oversupply of doctors ;D
1. ๐ 0
2. ๐ 0
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kmmiles.com
Search
# 45.042 km in miles
## Result
45.042 km equals 27.9711 miles
You can also convert 45.042 km to mph.
## Conversion formula
Multiply the amount of km by the conversion factor to get the result in miles:
45.042 km ร 0.621 = 27.9711 mi
## How to convert 45.042 km to miles?
The conversion factor from km to miles is 0.621, which means that 1 km is equal to 0.621 miles:
1 km = 0.621 mi
To convert 45.042 km into miles we have to multiply 45.042 by the conversion factor in order to get the amount from km to miles. We can also form a proportion to calculate the result:
1 km โ 0.621 mi
45.042 km โ L(mi)
Solve the above proportion to obtain the length L in miles:
L(mi) = 45.042 km ร 0.621 mi
L(mi) = 27.9711 mi
The final result is:
45.042 km โ 27.9711 mi
We conclude that 45.042 km is equivalent to 27.9711 miles:
45.042 km = 27.9711 miles
## Result approximation
For practical purposes we can round our final result to an approximate numerical value. In this case forty-five point zero four two km is approximately twenty-seven point nine seven one miles:
45.042 km โ
27.971 miles
## Conversion table
For quick reference purposes, below is the kilometers to miles conversion table:
kilometers (km) miles (mi)
46.042 km 28.592082 miles
47.042 km 29.213082 miles
48.042 km 29.834082 miles
49.042 km 30.455082 miles
50.042 km 31.076082 miles
51.042 km 31.697082 miles
52.042 km 32.318082 miles
53.042 km 32.939082 miles
54.042 km 33.560082 miles
55.042 km 34.181082 miles
## Units definitions
The units involved in this conversion are kilometers and miles. This is how they are defined:
### Kilometers
The kilometer (symbol: km) is a unit of length in the metric system, equal to 1000m (also written as 1E+3m). It is commonly used officially for expressing distances between geographical places on land in most of the world.
### Miles
A mile is a most popular measurement unit of length, equal to most commonly 5,280 feet (1,760 yards, or about 1,609 meters). The mile of 5,280 feet is called land mile or the statute mile to distinguish it from the nautical mile (1,852 meters, about 6,076.1 feet). Use of the mile as a unit of measurement is now largely confined to the United Kingdom, the United States, and Canada.
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https://www.physicsforums.com/threads/factorising-this-damned-equation.360975/
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# Factorising this damned equation
โข StephenP91
In summary, Stephen is seeking help with factoring the polynomial 2x^3 - 6x^2 + 2 = 0 in Pure Core 1 Mathematics without using a calculator. He has tried using the remainder theorem and the rational root theorem with no success. He is looking for a simple factor that is not a non-integer and wants to plot the graph by finding where it crosses the x-axis. He suggests possibly factoring 2x^3 - 6x^2 into 2x^2(x-3) and then subtracting 2 from the y-coordinates to find the coordinates of the roots.
StephenP91
Well, it's only Pure Core 1 Mathematics. I am trying to factorise:
2x^3 - 6x^2 + 2 = 0
Now, you can't use a calculator. I've tried finding a factor using the remainder theorem, but I just can't find a simple one (|x<5|). I am sure they don't expect us to use a complicated number, like a non-integar. So I just need someone to help me with factorising this.
Thank you,
Stephen.
Remainder theorem?
There is the rational root theorem which gives you a short list of things to try -- and if your polynomial has any rational roots, they must appear on this list.
That said, it's "easy" to see that this polynomial doesn't have any linear factors. Can you give an argument that it doesn't have any quadratic factors?
However, I guarantee that this polynomial has a nontrivial factor... so if it's not linear, and it's not quadratic, what must it be?
P.S. by "factorize", I assume you mean to factor over the integers -- i.e. you want each factor to be a polynomial with integer coefficients.
By factorise I mean, place intro brackets so that I may find the information I am looking for. Namely where the graph crosses the X axis so that I can plot the graph.
I was thinking though. Could I just factorise 2x^3 - 6x^2 into 2x^2(x-3) and then get the points I need, then from that subtract 2 to each of the y co-ords to get the co-ords of each of the roots?
## What is factorising?
Factorising is a mathematical process in which an equation is broken down into its simplest form by identifying common factors or terms.
## Why do we need to factorise equations?
Factorising equations can help us solve them more easily, as it simplifies the equation and makes it easier to work with. It also allows us to find the roots or solutions of the equation.
## How do we factorise an equation?
To factorise an equation, we need to identify common factors or terms and then use mathematical techniques such as grouping, difference of squares, or trial and error to simplify the equation.
## What are the benefits of factorising an equation?
Factorising an equation can help us solve it more efficiently, as it reduces the amount of calculations needed. It also allows us to find the solutions or roots of the equation, which can be useful in real-world applications.
## What are some common mistakes to avoid when factorising an equation?
Some common mistakes to avoid when factorising an equation include not factoring out the greatest common factor, forgetting to include negative signs, and incorrectly grouping terms. It is also important to check that the factors are correct by expanding them back to the original equation.
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#### calculation of capacity of silos
โข Home
โข calculation of capacity of silos
##### calculation of capacity of silos - fermeminiature.fr
To determine silo capacity in "wet ton", multiply silo capacity value, dry matter,... haylage densities, pressures, and capacities in tower silos. A secondary objective of this paper is to report the silo capacities that were de termined numerically as part of the pro cedure for determining the structural de.
##### ESTIMATING SILO CAPACITIES AND SILAGE WEIGHTS*
ESTIMATING SILO CAPACITIES AND SILAGE WEIGHTS 3 two men have been kept iri the silo trampling during filling, and that the silage was allowed to settle two days, and the silo then refilled. The depth of the ยท silage and n6t ยท the height of the silo should be taken in using the table.
##### SB222 1919 Capacity of Silos and Weights of Silage
THE CAPACITY OF SILOS AND WEIGHTS OF SILAGE C. H. ECKLES, O. E. REED1, J. B. FITCH PREVIOUS INVESTIGATIONS The extensive use of silos within recent years is responsible for numerous calls for information regarding the capacity of silos and for the weights of silage under a variety of conditions.
##### calculation of capacity of silos - pizza-pietro.fr
Calculation Of Capacity Of Silos - versmarktviventra.nl If you plan to use a silo for storage space for grain, you should calculate how much space you have available in total. A silo is a relatively cylindrical shape with a half-dome on top, so you can use the formulas for the volume of a sphere and a cylinder to determine how much storage space you have in your silo.
##### cement silo volume calculation - museaalverhaal.nl
CSG Silo Volume CalculatorThis calculator is designed to give the volume of any cylindrical silo with a half sphere dome top It is assumed in this calculator that the cylinder and the dome have the same diameter Enter the radius (1/2 of the diameter) of the silo cylinder and the vertical.Hopper and silo bulk discharge flow rate calculationHopper discharge rate calculation method 1 using shear ...
##### Silo design | Hopper design | Bulk storage | Silo ...
Silo design is taken as per previous models, however, minor changes in silo height or diameter are made to obtain additional capacity. Changing product specification. Changing of silo construction material to that specified. The assumption that structural design verification ensures proper product flow.
##### Tower Silo Capacities - Ontario
This Factsheet contains estimates of tower silo capacities for alfalfa silage, whole-plant corn silage and various types of high moisture corn based on a series of analyses of silo capacity carried out at the University of Guelph and on research done at research establishments in the Netherlands and Sweden.
##### Calculation Of Capacity Of Silos - apotheekederveen.nl
calculation of capacity of silos Flat Bottom Silos Symaga. The wide range and the versatility of our flat or conical bottom silos for the needs of our customers by offering silos from 5mยณ to 25,000mยณ capacity. CALCULATION OF THE STRUCTURE Silo calculation is made according to different norm as
##### calculation of capacity of silos - mpslab.com.pl
calculation of capacity of silos . Silo Volume Calculator Kotzur. ... whole-plant corn silage and various types of high moisture corn based on a series of analyses of silo capacity carried out at the University of Guelph and on research done at research establishments in the Netherlands and Sweden.get price.
##### calculation of capacity of silos - vimoplus.nl
Calculation Of Capacity Of Silos - nautiflex.it. Calculation Of Capacity Of Silos - spo2tu.be. Calculation Of Capacity Of Silos. We are a large-scale manufacturer specializing in producing various mining machines including different types of sand and gravel equipment, milling equipment, mineral processing equipment and building materials equipment.
##### Calculation Of Capacity Of Silos - besteyeclinic.in
how do you calculate the capacity of cement silo. How you calculate the capacity of cement silo calculation of capacity of silos. calculation of capacity of silos Zenith crushing equipment is designed to Online service. Portable Cement Silos Portable Cement Silos 7SL Series. 15 Bag to 37 Ton Capacity.
##### calculation of capacity of silos - shop.waw.pl
calculation of capacity of silos calculation of capacity of silos Request for Quotation You can get the price list.FACTOR OF SAFETY FOR BEARING CAPACITY OF SOILS -Civil of safety .; Galvanized Spiral Steel Silo Assembly Steel Silo for Grain . Bidragon is a professional manufacturer of qualified galvanized spiral steel silos including grain storage silo, wheat silo, cement silo, corn silo, pellet .
##### calculation of capacity of silos - pontoise-diagnostic.fr
Bunker Silo Sizing and Management - DocuShare. and to maximize the effective use of bunker silos. These criteria include silage removal rate and bunker width, height Bunker Silo Sizing and Management Oklahoma Cooperative Extension Service. 1011- 1011-3 volume or capacity would be the same.
##### Silo total volume calculation / Silo inventory calculation
The silo has a diameter r 0. The ratio a/r 0 can then be calculated. 3.2 Calculate the height of the heap. This calculation requires to know the angle of repose ฮณ of the material. The height of the heap b can the be calculated by b = (r 0 +a).tan(ฮณ) 3.3 Calculation of the volume of the heap
##### Calculation Of Capacity Of Silos
27-02-2021ยท calculation of capacity of silos - thomigartenbau.ch Estimating vertical silo inventories In contrast to horizontal silos, it is a relatively simple matter to calculate silage volume in an upright tower: 314 x silo diameter 2 x silage depth 2 However, in vertical silos, bulk density is influenced by silage depth, moisture content and silo diameter Since these relationships are complex, it is not.
##### calculation of capacity of silos - sanctuarybeachcondo.co.za
Calculation for D12m Silo Scribd. Calculation for D12m Silo ... SILO CAPACITY CALCULATION: BG3= 9 D= 12 m m m m ... Silos Fundamentals of Theory, Be() Get Price; Tower Silo Capacities OMAFRA... tower silo capacities for alfalfa silage, wholeplant corn silage and various types of high moisture corn based on a series of analyses of silo capacity ...
##### Silo Volume Calculator - Kotzur
If you already know your silo's diameter from the manufacturer's specifications, change the option from the dropdown list to 'diameter' and enter this measurement instead. Step 2. Measure the barrel height to calculate the volume of the whole silo or, if it is not full, the height of the contents against the silo wall to calculate the volume and weight of the contents.
##### Calculation of capacity of silos - Manufacturer Of High ...
Calculation method for Design Silos and Hoppers Silos Solving flowability issues out of silos thanks to good design calculation methods, Silos design Flow of powder Silos design calculation method: according to its capacity to reach a nominal speed expressed in terms of throughput, cycle time or number of batches h Get Support
##### calculation of capacity of silos - recprojekt.pl
calculation of capacity of silos [randpic] The capacity of a silo Math Central 2013-3-9 Candonn, The volume of a cylinder is ฯ r 2 h where r is the radius, h is the height and ฯ is approximately 3.1416. Your cylindrical silo then has volume. ฯ (12/2) 2. The capacit
##### calculation of capacity of silos - domwilanow.pl
silo capacity calculation in xls. Tower Silo Capacity Calculator This spreadsheet, developed by Dr. Brian Holmes (emeritus), is designed to help estimate the tons of silage in a tower silo. Calculations are presented on a dry matter and as fed basis. chat online. Get Price
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https://conversion.org/speed/centimetre-per-minute/walking-speed
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# centimetre per minute to walking speed conversion
Conversion number between centimetre per minute [cm/min] and walking speed is 0.00011904761904762. This means, that centimetre per minute is smaller unit than walking speed.
### Contents [show][hide]
Switch to reverse conversion:
from walking speed to centimetre per minute conversion
### Enter the number in centimetre per minute:
Decimal Fraction Exponential Expression
[cm/min]
eg.: 10.12345 or 1.123e5
Result in walking speed
?
precision 0 1 2 3 4 5 6 7 8 9 [info] Decimal: Exponential:
### Calculation process of conversion value
โข 1 centimetre per minute = ((0.01/60)) / (1.4) = 0.00011904761904762 walking speed
โข 1 walking speed = (1.4) / ((0.01/60)) = 8400 centimetre per minute
โข ? centimetre per minute ร ((0.01/60)ย ("m/s"/"centimetre per minute")) / (1.4ย ("m/s"/"walking speed")) = ? walking speed
### High precision conversion
If conversion between centimetre per minute to metre-per-second and metre-per-second to walking speed is exactly definied, high precision conversion from centimetre per minute to walking speed is enabled.
Since definition contain rounded number(s) too, there is no sense for high precision calculation, but if you want, you can enable it. Keep in mind, that converted number will be inaccurate due this rounding error!
### centimetre per minute to walking speed conversion chart
Start value: [centimetre per minute] Step size [centimetre per minute] How many lines? (max 100)
visual:
centimetre per minutewalking speed
00
100.0011904761904762
200.0023809523809524
300.0035714285714286
400.0047619047619048
500.005952380952381
600.0071428571428572
700.0083333333333334
800.0095238095238095
900.010714285714286
1000.011904761904762
1100.013095238095238
Copy to Excel
## Multiple conversion
Enter numbers in centimetre per minute and click convert button.
One number per line.
Converted numbers in walking speed:
Click to select all
## Details about centimetre per minute and walking speed units:
Convert Centimetre per minute to other unit:
### centimetre per minute
Definition of centimetre per minute unit: โก 1 cm / 60 s. The speed with which the body moves 1 centimetre in 1 minute.
Convert Walking speed to other unit:
### walking speed
Definition of walking speed unit: โ 1.4 m/s. With average walking speed, you walk around 5 km in 1 hour. In more than one literature, the 1.4 m/s speeds are used in calculations. The link presented in the source has calculated this average walking speed from the average of 3 additional literature.
โ Back to Speed units
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http://ised.seedsnet.org/modules/en-math_expression/simplifying-algebraic-expressions.html
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# Simplifying Algebraic Expressionsย - Removing Brackets
Lesson Objective
Learn how to remove brackets in algebraic expressions.
Algebraic expressions may contain brackets. In order to simplify the expressions, you may need to remove the brackets.
This lesson shows you the basics that you need to know when removing brackets.
# Study Tips
Tip #1
Remember to multiply all the terms in the brackets with the term outside the brackets.
Tip #2
It is important understand why you are able to multiply terms together in order to remove the brackets in an expression.
Now, watch the following math video to know more.
# Math Video
Click play to watch
Math Video Transcript
# Practice Questionsย & More
Multiple Choice Questions (MCQ)
Now, let's try some MCQ questions to understand this lesson better.
You can start by going through the series of questions on simplifying algebraic expressions - removing brackets or pick your choice of question below.
1. Question 1 on removing brackets
2. Question 2 on removing brackets and simplifying
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https://www.sscadda.com/quant-rrb-ntpc-21at-january-2020-profit-loss-number-system-interest
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Home Quantitative Aptitude Mathematics Quiz For RRB NTPC : 21st January 2020 for Number system,...
# Mathematics Quiz For RRB NTPC : 21st January 2020 for Number system, profit & loss and Simple interest
In Today's Maths Quiz we are providing latest pattern questions based on the topics of Number system, profit & loss, and Simple interest
| Updated On February 10th, 2020 at 07:30 pm
Q1. Given: 2x โ 4 รขโฐยค 2 โ x/3 and 2(2x + 5) > 3x โ 5, then x can take which of the following values?
(a) -14
(b) 3
(c) 4
(d) 14
Q2. A painter can paint a fence in 24 hours. After 6 hours he takes a break. What fraction of the fence is yet to be painted?
(a) 0.6
(b) 0.2
(c) 0.75
(d) 0.8
Q3. If 1/6 of x โ 7/2 of 3/7 equals โ 7/4, then the value of x is
(a) -1.5
(b) 3
(c) -2.5
(d) 6
Q4. x and y are two numbers such that their mean proportion is 9 and third proportion is 243. What is the value of x and y?
(a) 3 and 9
(b) 3 and 27
(c) 6 and 27
(d) 6 and 81
Q5. If 21% of an electricity bill is discounted, Rs 1817 is still to be paid. How much was the original bill amount?
(a) Rs 1502
(b) Rs 2336
(c) Rs 2300
(d) Rs 1538
Q6. The average revenues of 7 consecutive years of a company is Rs 79 lakhs. If the average of first 4 years is Rs 74 lakhs and that of last 4 years is Rs 86 lakhs, what is the revenue for the 4th year?
(a) Rs 87 lakhs
(b) Rs 89 lakhs
(c) Rs 85 lakhs
(d) Rs 83 lakhs
Q7. A shopkeeper, sold dried apricots at the rate Rs 1210 a kg and bears a loss of 12%. Now if he decides to sell it at Rs 1331 per kg, what will be the result?
(a) 6.4 percent loss
(b) 3.2 percent gain
(c) 6.4 percent gain
(d) 3.2 percent loss
Q8. If the shopkeeper sells an item at Rs 1600 which is marked as Rs 2000, then what is the discount he is offering?
(a) 25 percent
(b) 20 percent
(c) 30 percent
(d) 10 percent
Q9. If cosecA + cotA = x, then value of x is
(a) 1/(cosecA โ cotA)
(b) 1/(secA โ tanA)
(c) 1/(secA โ cosA)
(d) 1/(sinA โ cosA)
Q10. Dalajit lent Rs 10800 to Jaabir for 3 years and Rs 7500 to Kabir for 2 years on simple interest at the same rate of interest and received Rs 1422 in all from both of them as interest. The rate of interest per annum is
(a) 3.5 percent
(b) 4 percent
(c) 3 percent
(d) 4.5 percent
#### Solutions:
##### RRB NTPC 2019 | RRB JE Classes Day 1 | Review of Previous Year Quant Papers for RRB | Sumit Sir
Important Links for RRB NTPC Recruitment 2019
RRB NTPC Previous year Cut Off | 1st & 2nd Stage Examination
RRB NTPC Recruitment 2019: Check FAQs
RRB NTPC Exam Pattern 2019 รขโฌโ Check Here
RRB NTPC Previous Year Exam Analysis
RRB NTPC Exam Syllabus
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# Background
For a prime ${\displaystyle p}$ and a positive integer ${\displaystyle n}$ let ${\displaystyle \mathbb {F} _{p^{n}}}$ be the finite field with ${\displaystyle p^{n}}$ elements. Let ${\displaystyle F}$ be a map from the finite field to itself. Such function admits a unique representation as a polynomial of degree at most ${\displaystyle p^{n}-1}$, i.e.
${\displaystyle F(x)=\sum _{j=0}^{p^{n}-1}a_{j}x^{j},a_{j}\in \mathbb {F} _{p^{n}}}$.
The function ${\displaystyle F}$ is
โข linear if ${\displaystyle F(x)=\sum _{j=0}^{n-1}a_{j}x^{p^{j}}}$,
โข affine if it is the sum of a linear function and a constant,
โข DO (Dembowski-Ostrim) polynomial if ${\displaystyle F(x)=\sum _{0\leq i\leq j,
โข quadratic if it is the sum of a DO polynomial and an affine function.
For ${\displaystyle \delta }$ a positive integer, the function ${\displaystyle F}$ is called differentially ${\displaystyle \delta }$-uniform if for any pairs ${\displaystyle a,b\in \mathbb {F} _{p^{n}}}$, with ${\displaystyle a\neq 0}$, the equation ${\displaystyle F(x+a)-F(x)=b}$ admits at most ${\displaystyle \delta }$ solutions.
A function ${\displaystyle F}$ is called planar or perfect nonlinear (PN) if ${\displaystyle \delta _{F}=1}$. Obviously such functions exist only for ${\displaystyle p}$ an odd prime. In the even case the smallest possible case for ${\displaystyle \delta }$ is two (APN function).
For planar function we have that the all the nonzero derivatives, ${\displaystyle D_{a}F(x)=F(x+a)-F(x)}$, are permutations.
## Equivalence Relations
Two functions ${\displaystyle F}$ and ${\displaystyle F'}$ from ${\displaystyle \mathbb {F} _{p^{n}}}$ to itself are called:
โข affine equivalent if ${\displaystyle F'=A_{1}\circ F\circ A_{2}}$, where ${\displaystyle A_{1},A_{2}}$ are affine permutations;
โข EA-equivalent (extended-affine) if ${\displaystyle F'=F''+A}$, where ${\displaystyle A}$ is affine and ${\displaystyle F''}$ is afffine equivalent to ${\displaystyle F}$;
โข CCZ-equivalent if there exists an affine permutation ${\displaystyle {\mathcal {L}}}$ of ${\displaystyle \mathbb {F} _{p^{n}}\times \mathbb {F} _{p^{n}}}$ such that ${\displaystyle {\mathcal {L}}(G_{F})=G_{F'}}$, where ${\displaystyle G_{F}=\lbrace (x,F(x)):x\in \mathbb {F} _{p^{n}}\rbrace }$.
CCZ-equivalence is the most general known equivalence relation for functions which preserves differential uniformity. Affine and EA-equivalence are its particular cases. For the case of quadratic planar functions the isotopic equivalence is more general than CCZ-equivalence, where two maps are isotopic equivalent if the corresponding presemifields are isotopic.
# On Presemifields and Semifields
A presemifield is a ring with left and right distributivity and with no zero divisor. A presemifield with a multiplicative identity is called a semifield. Any finite presemifield can be represented by ${\displaystyle \mathbb {S} =(\mathbb {F} _{p^{n}},+,\star )}$, for ${\displaystyle p}$ a prime, ${\displaystyle n}$ a positive integer, ${\displaystyle \mathbb {S} =(\mathbb {F} _{p^{n}},+)}$ additive group and ${\displaystyle x\star y}$ multiplication linear in each variable. Every commutative presemifield can be transformed into a commutative semifield[1].
Two presemifields ${\displaystyle \mathbb {S} _{1}=(\mathbb {F} _{p^{n}},+,\star )}$ and ${\displaystyle \mathbb {S} _{2}=(\mathbb {F} _{p^{n}},+,\circ )}$ are called isotopic if there exist three linear permutations ${\displaystyle T,M,N}$ of ${\displaystyle \mathbb {F} _{p^{n}}}$ such that ${\displaystyle T(x\star y)=M(x)\circ N(y)}$, for any ${\displaystyle x,y\in \mathbb {F} _{p^{n}}}$. If ${\displaystyle M=N}$ then they are called strongly isotopic. Each commutative presemifields of odd order defines a planar DO polynomial and viceversa:
โข given ${\displaystyle \mathbb {S} =(\mathbb {F} _{p^{n}},+,\star )}$ let ${\displaystyle F_{\mathbb {S} }(x)={\frac {1}{2}}(x\star x)}$;
โข given ${\displaystyle F}$ let ${\displaystyle \mathbb {S} _{F}=(\mathbb {F} _{p^{n}},+,\star )}$ defined by ${\displaystyle x\star y=F(x+y)-F(x)-F(y)}$.
Given ${\displaystyle \mathbb {S} =(\mathbb {F} _{p^{n}},+,\star )}$ a finite semifield, the subsets
${\displaystyle N_{l}(\mathbb {S} )=\{\alpha \in \mathbb {S} :(\alpha \star x)\star y=\alpha \star (x\star y)}$ for all ${\displaystyle x,y\in \mathbb {S} \}}$
${\displaystyle N_{m}(\mathbb {S} )=\{\alpha \in \mathbb {S} :(x\star \alpha )\star y=x\star (\alpha \star y)}$ for all ${\displaystyle x,y\in \mathbb {S} \}}$
${\displaystyle N_{r}(\mathbb {S} )=\{\alpha \in \mathbb {S} :(x\star y)\star \alpha =x\star (y\star \alpha )}$ for all ${\displaystyle x,y\in \mathbb {S} \}}$
are called left, middle and right nucleus of ${\displaystyle \mathbb {S} }$.
The set ${\displaystyle N(\mathbb {S} )=N_{l}(\mathbb {S} )\cap N_{m}(\mathbb {S} )\cap N_{r}(\mathbb {S} )}$ is called the nucleus. All these sets are finite field and, when ${\displaystyle \mathbb {S} }$ is commutative, ${\displaystyle N_{l}(\mathbb {S} )=N_{r}(\mathbb {S} )\subseteq N_{m}(\mathbb {S} )}$. The order of the different nuclei are invariant under isotopism.
## Properties
Hence two quadratic planar functions ${\displaystyle F,F'}$ are isotopic equivalent if their corresponding presemifields are isotopic. Moreover, we have:
โข ${\displaystyle F,F'}$ are CCZ-equivalent if and only if the corresponding presemifileds are strongly isotopic[2];
โข for ${\displaystyle n}$ odd, isotopic coincides with strongly isotopic;
โข if ${\displaystyle F,F'}$ are isotopic equivalent, then there exists a linear map ${\displaystyle L}$ such that ${\displaystyle F'}$ is EA-equivalent to ${\displaystyle F(x+L(x))-F(x)-F(L(x))}$;
โข any commutative presemifield of odd order can generate at most two CCZ-equivalence classes of planar DO polynomials;
โข if ${\displaystyle \mathbb {S} _{1}}$ and ${\displaystyle \mathbb {S} _{2}}$ are isotopic commutative semifields of characteristic ${\displaystyle p}$ with order of middle nuclei and nuclei ${\displaystyle p^{m}}$ and ${\displaystyle p^{k}}$ respectively, then either one of the following is satisfied:
โข ${\displaystyle m/k}$ is odd and the semifields are strongly isotopic,
โข ${\displaystyle m/k}$ is even and the semifields are strongly isotopic or the only isotopisms are of the form ${\displaystyle (\alpha \star N,N,L)}$ with ${\displaystyle \alpha \in N_{m}(\mathbb {S} _{1})}$ non-square.
# Known cases of planar functions and commutative semifields
Among the known example of planar functions, the only ones that are non-quadratic are the power functions ${\displaystyle x^{\frac {3^{t}+1}{2}}}$ defined over ${\displaystyle \mathbb {F} _{3^{n}}}$, with ${\displaystyle t}$ is odd and gcd(${\displaystyle t,n}$)=1.
In the following the list of some known infinite families of planar functions (and corresponding commutative semifields):
โข ${\displaystyle x^{2}}$ over ${\displaystyle \mathbb {F} _{p^{n}}}$ (finite field ${\displaystyle \mathbb {F} _{p^{n}}}$);
โข ${\displaystyle x^{p^{t}+1}}$ over ${\displaystyle \mathbb {F} _{p^{n}}}$ with ${\displaystyle n/gcd(t,n)}$ odd (Albert's commutative twisted fields);
โข ${\displaystyle L(t^{2}(x))+{\frac {1}{2}}x^{2}}$ over ${\displaystyle \mathbb {F} _{p^{2km}}}$ with ${\displaystyle L(x)={\frac {1}{8}}(x^{p^{k}}-x),t(x)=x^{p^{km}}-x}$ (Dickson semifields);
โข ${\displaystyle (ax)^{p^{s}+1}-(ax)^{p^{k}(p^{s}+1)}+x^{p^{k}+1}}$
โข ${\displaystyle bx^{p^{s}+1}+(bx^{p^{s}+1})^{p^{k}}+cx^{p^{k}+1}}$
over ${\displaystyle \mathbb {F} _{p^{2k}}}$ where ${\displaystyle a,b\in \mathbb {F} _{2^{2k}}^{\star },b}$ not square, ${\displaystyle c\in \mathbb {F} _{2^{2k}}\setminus \mathbb {F} _{2^{k}},gcd(k+s,2k)=gcd(k+s,k)}$ and for the first one also ${\displaystyle gcd(p^{s}+1,p^{k}+1)\neq gcd(p^{s}+1,(p^{k}+1)/2)}$. Without loss of generality it is possible to take ${\displaystyle a=1}$ and fix a value for ${\displaystyle c}$;
โข ${\displaystyle x^{p^{s}+1}-a^{p^{t}-1}x^{p^{t}+p^{2t+s}}}$ over ${\displaystyle \mathbb {F} _{p^{3t}},a}$ primitive, ${\displaystyle gcd(3,t)=1,t-s\equiv 0}$ mod ${\displaystyle 3,3t/gcd(s,3t)}$ odd;
โข ${\displaystyle x^{p^{s}+1}-a^{p^{t}-1}x^{p^{3t}+p^{t+s}}}$ over ${\displaystyle \mathbb {F} _{p^{4t}},a}$ primitive, ${\displaystyle p^{s}\equiv p^{t}\equiv 1}$ mod 4, ${\displaystyle 2t/gcd(s,2t)}$ odd;
โข ${\displaystyle a^{1-p}x^{2}+x^{2p^{m}}+a^{1-p}T(x)-T(x)^{p^{m}}}$, with ${\displaystyle T(x)=\sum _{i=0}^{k}(-1)^{i}x^{p^{2i}(p^{2}+1)}+a^{p-1}\sum _{j=0}^{k-1}(-1)^{k+j}x^{p^{2j+1}(p^{2}+1)}}$, over ${\displaystyle \mathbb {F} _{p^{2m}}}$ for ${\displaystyle a\in \mathbb {F} _{p^{2}}^{\star },m=2k+1}$.
## Cases defined for p=3
โข ${\displaystyle x^{10}\pm x^{6}-x^{2}{\mbox{ over }}\mathbb {F} _{p^{n}}{\mbox{ with }}n}$ odd (Coulter-Matthews and Ding-Yuan semifields);
โข ${\displaystyle L(t^{2}(x))+D(t(x))+{\frac {1}{2}}x^{2},{\mbox{ over }}\mathbb {F} _{3^{2k}}{\mbox{ with }}k{\mbox{ odd, }}t(x)=x^{3^{k}}-x,\beta \in \mathbb {F} _{3^{2k}}\setminus \mathbb {F} _{3^{k}},\alpha =t(\beta ),L(x)=\alpha ^{-5}x^{3}+x,D(x)=-\alpha ^{-10}x^{10}}$ (Ganley semifields);
โข ${\displaystyle L(t^{2}(x))+{\frac {1}{2}}x^{2},{\mbox{ over }}\mathbb {F} _{3^{2k}}{\mbox{ with }}k{\mbox{ odd, }}t(x)=x^{3^{k}}-x,\beta \in \mathbb {F} _{3^{2k}}\setminus \mathbb {F} _{3^{k}},\alpha =t(\beta ),L(x)=-x^{9}-\alpha x^{3}+(1-\alpha ^{4})x}$ (Cohen-Ganley semifileds);
โข ${\displaystyle L(t^{2}(x))+{\frac {1}{2}}x^{2},{\mbox{ over }}\mathbb {F} _{3^{10}}{\mbox{ with }}t(x)=x^{243}-x,\beta \in \mathbb {F} _{3^{10}}\setminus \mathbb {F} _{3^{5}},\alpha =t(\beta ),L(x)=-(\alpha ^{-53}x^{27}+\alpha ^{-18}x^{9}-x)}$ (Penttila-Williams semifileds);
โข ${\displaystyle L(t^{2}(x))+D(t(x))+{\frac {1}{2}}x^{2},{\mbox{ over }}\mathbb {F} _{3^{8}}{\mbox{ with }}t(x)=x^{9}-x,L(x)=x^{243}+x^{9},D(x)=x^{246}+x^{82}-x^{10}}$ (Coulter-Henderson-Kosick semifield);
โข ${\displaystyle x^{2}+x^{90}{\mbox{ over }}\mathbb {F} _{3^{5}}}$.
# Known cases of APN functions in odd characteristic
โข ${\displaystyle x^{3}}$ over ${\displaystyle \mathbb {F} _{p^{n}}}$, ${\displaystyle p\neq 3}$;
โข ${\displaystyle x^{p^{n}-2}}$ over ${\displaystyle \mathbb {F} _{p^{n}}}$ with ${\displaystyle p^{n}\equiv 2{\pmod {3}}}$;
โข ${\displaystyle x^{\frac {p^{n}-3}{2}}}$ over ${\displaystyle \mathbb {F} _{p^{n}}}$ with ${\displaystyle p^{n}\equiv 3,7{\pmod {20}},~p^{n}>7,~p^{n}\neq 27,n}$ odd;
โข ${\displaystyle x^{{\frac {p^{n}+1}{4}}+{\frac {p^{n}-1}{2}}}}$ over ${\displaystyle \mathbb {F} _{p^{n}}}$ with ${\displaystyle p^{n}\equiv 3{\pmod {8}}}$;
โข ${\displaystyle x^{\frac {p^{n}+1}{4}}}$ over ${\displaystyle \mathbb {F} _{p^{n}}}$ with ${\displaystyle p^{n}\equiv 7{\pmod {8}},~n>1}$;
โข ${\displaystyle x^{\frac {2p^{n}-1}{3}}}$ over ${\displaystyle \mathbb {F} _{p^{n}}}$ with ${\displaystyle p^{n}\equiv 2{\pmod {3}}}$;
โข ${\displaystyle x^{p^{m}+2}}$ over ${\displaystyle \mathbb {F} _{p^{n}}}$ with ${\displaystyle p^{m}\equiv 1{\pmod {3}},~n=2m}$;
โข ${\displaystyle x^{3^{n}-3}}$ over ${\displaystyle \mathbb {F} _{3^{n}}}$ with ${\displaystyle n>1}$ odd;
โข ${\displaystyle x^{\frac {3^{\frac {n+1}{2}}-1}{2}}}$ over ${\displaystyle \mathbb {F} _{3^{n}}}$ with ${\displaystyle n\equiv 3{\pmod {4}},~n>3}$;
โข ${\displaystyle x^{{\frac {3^{\frac {n+1}{2}}-1}{2}}+{\frac {3^{n}-1}{2}}}}$ over ${\displaystyle \mathbb {F} _{3^{n}}}$ with ${\displaystyle n\equiv 1{\pmod {4}},~n>1}$;
โข ${\displaystyle x^{\frac {3^{n+1}-1}{8}}}$ over ${\displaystyle \mathbb {F} _{3^{n}}}$ with ${\displaystyle n\equiv 3{\pmod {4}}}$;
โข ${\displaystyle x^{{\frac {3^{n+1}-1}{8}}+{\frac {3^{n}-1}{4}}}}$ over ${\displaystyle \mathbb {F} _{3^{n}}}$ with ${\displaystyle n\equiv 1{\pmod {4}}}$;
โข ${\displaystyle x^{\frac {3^{n+1}-1}{3^{L}+1}}}$ over ${\displaystyle \mathbb {F} _{3^{n}}}$, where ${\displaystyle L={\frac {n+1}{2^{\ell }}}}$ with ${\displaystyle n\equiv -1{\pmod {2^{\ell }}}}$;
โข ${\displaystyle x^{\frac {5^{\ell }+1}{2}}}$ over ${\displaystyle \mathbb {F} _{5^{n}}}$ with ${\displaystyle \gcd(2n,\ell )=1}$;
โข ${\displaystyle x^{{\frac {5^{n}-1}{4}}+{\frac {5^{\frac {n+1}{2}}-1}{2}}}}$ over ${\displaystyle \mathbb {F} _{5^{n}}}$ with ${\displaystyle n}$ odd;
โข ${\displaystyle x^{{\frac {5^{n+1}-1}{2(5^{L}+1)}}+{\frac {5^{n}-1}{4}}}}$ over ${\displaystyle \mathbb {F} _{5^{n}}}$, where ${\displaystyle L={\frac {n+1}{2^{\ell }}},~n\equiv -1{\pmod {2^{\ell }}}}$ and ${\displaystyle \ell \geq 2}$;
โข Coulter R. S., Henderson M. Commutative presemifields and semifields. Advances in Math. 217, pp. 282-304, 2008
โข Budaghyan L., Helleseth T. On Isotopism of Commutative Presemifields and CCZ-Equivalence of Functions. Special Issue on Cryptography of International Journal of Foundations of Computer Science, v. 22/6), pp- 1243-1258, 2011
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โข I
Killtech
TL;DR Summary
How does SRT fix the asymmetries between inertial frames, in particular in the twin paradox?
Originally i just wanted to look at how much analogy can be made between light and sound waves using all that math has to offer to depict them in most similar framework possible - just so as to have a different perspective to understand some things better. Anyhow, no matter how well one tries to hide the (sound) medium (and that can be done pretty well), it will always be there and not allow for a perfect equivalence of inertial frames.
In order to make comparisons with light/SRT of some specific aspects i was curious about i went back to the twin paradox and looked as some special modifications to make them stand out. In particular the twin paradox in a closed world (cylinder manifold which is geometrically perfectly flat) and luckily the internet already had something on it: https://physics.stackexchange.com/questions/353216/twin-paradox-in-closed-universe
I'm not sure if the explanation is fully correct, but it surprised me as it breaks the equivalence of frames (i picked the example specifically to understand how SRT maintains the equivalence under non trivial circumstance).
The problem can be traced back to the issue that in a closed world the one way speed of light seems to be partially measurable: send two signals around the world in opposed directions and if they don't come back at the same time there is a difference in the one way speed of light along that axis. In the example in the link this causes Betties plane of const time to twist such that in her frame her other instances are at a different age on her plane of const time (due to clock synch convention).
And I don't understand how to fix this, because all waves travel at a fixed speed i.e. independently from the source they were emitted from. So if Betty and Albert emitted signals to measure the one way speed of light from Alberts place they will travel together and an asymmetry becomes visible making Alberts frame to stand out (unless a many worlds approach is taken). So Albert's frame seems to be more at rest then Betty's.
Anyhow, so i thought maybe this is a issue unique to a closed world setup. But in all versions of the twin paradox there is a age asymmetry between the "traveling" and "home" twins. I looked at the the "out and back" twin resolution at Wikipedia: https://en.wikipedia.org/wiki/Twin_paradox#A_non_space-time_approach. It needs the 3 twins A (Albert), O (Outgoing) and B (Back) i.e. trins. So to make the example easier let's copy/clone each of the trins indefinitely and arrange them in evenly spaced grids A, O, B. Now whenever a trin passes by another the times of all clocks can be recorded. In particular of interest are the time intervals ##\Delta t_{I,J}## measured by a clock at grid ##I## for the time between two meetings with a trin from grid ##J##. Now let all trins be at that at ##x,t=0## point our initial condition. Furthermore the start of the return trip can be set to when O meets B the first time since ##t=0##. Then I gather that the resolution implies $$\frac 1 2 \Delta t_{O,A} + \frac 1 2 \Delta t_{B,A} < \Delta t_{A,O} = \Delta t_{A,B}$$
So that would seem to imply that frames O and B are still not fully equivalent to A since ##\Delta t_{A,B} \neq \Delta t_{B,A}## even without the closed world assumption. So it would seem clocks tick indeed faster in one grid then in the other but in an absolute sense (after all the ##\Delta t_{I,J}## just compare two times of the same inertial frame clock between events at its exact location - i.e. independent of any clock synch and whatever). I also considered viewing grid O as the "stay home trin" and A as the outgoing one. That needs introducing another grid C that serves the travel back trin role for O. But doing so is a bad idea because that seems to shatter logic consistency a bit (so far i could not resolve the contradictions between the ##\Delta t_{I,J}## relations of all 4 grids).
Anyhow, the closed world case provides a mean to associated the ##\Delta t_{I,J}## asymmetry with one way speed of light difference and having that, one can add more grids moving at different velocities just to probe the asymmetry. But does that not give a means of deriving the one speed of light in all directions and deduct a frame specific offset velocity vector?
So as you can see I am at a loss now since the one way speed of light is not allowed to have any measurable effect therefore all the asymmetries cannot exist. Or to put it more provocatively: a detectible offset velocity in the one way speed of light is basically an aether wind - so i expected these asymmetries only to pop up for my sound waves analogon but not for light. I don't get where the mistake in all this is and the more I try to understand SRT the less I actually do.
Homework Helper
Gold Member
Summary:: How does SRT fix the asymmetries between inertial frames, in particular in the twin paradox?
Originally i just wanted to look at how much analogy can be made between light and sound waves using all that math has to offer to depict them in most similar framework possible - just so as to have a different perspective to understand some things better. Anyhow, no matter how well one tries to hide the (sound) medium (and that can be done pretty well), it will always be there and not allow for a perfect equivalence of inertial frames.
In order to make comparisons with light/SRT of some specific aspects i was curious about i went back to the twin paradox and looked as some special modifications to make them stand out. In particular the twin paradox in a closed world (cylinder manifold which is geometrically perfectly flat) and luckily the internet already had something on it: https://physics.stackexchange.com/questions/353216/twin-paradox-in-closed-universe
I'm not sure if the explanation is fully correct, but it surprised me as it breaks the equivalence of frames (i picked the example specifically to understand how SRT maintains the equivalence under non trivial circumstance).
Here's a very old thread on the closed universe case:
Mentor
The problem can be traced back to the issue that in a closed world the one way speed of light seems to be partially measurable
The root cause is actually that in the closed world, it is no longer the case that all inertial frames are equivalent globally. One particular inertial frame has the property that its surfaces of simultaneity are closed; in all other inertial frames, the surfaces of simultaneity are not closed, and how "not closed" they are varies from frame to frame. (Roughly speaking, in one particular inertial frame, in the 1x1 dimensional case--one dimension of space and one of time--the surfaces of simultaneity are circles, while in all other inertial frames, the surfaces of simultaneity are helixes; they don't close back up on themselves, and by how much they fail to do that is a finite number that varies from frame to frame.) So there is a global geometric asymmetry between inertial frames that exactly correlates to the asymmetry in aging for twins.
i thought maybe this is a issue unique to a closed world setup.
That particular issue is unique to the closed world setup, yes.
But in all versions of the twin paradox there is a age asymmetry between the "traveling" and "home" twins
That is because in all versions of the twin paradox, there is some asymmetry between the twins. It just isn't always the same asymmetry. Different versions have different asymmetries between the twins. This can cause a great deal of confusion among people who think there should be just one "trick" that resolves all possible twin paradoxes. There isn't--at least, not unless you count the general statement that does cover all cases, which is "look at the actual curves in spacetime followed by each twin and compare their lengths", as a "trick". (Most people don't appear to want to do that because the general method looks way too much like actual work instead of a "trick" or "shortcut". )
Killtech and hutchphd
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I am at a loss now
My standard recommendation for twin paradox questions is to read the Usenet Physics FAQ article:
Different versions have different asymmetries between the twins.
Note, btw, that one of the things the Usenet Physics FAQ article will make you realize is that, not only are there different asymmetries between the twins in different twin paradox scenarios, one can even find different asymmetries in the same scenario. As the article shows, there are several different ways of analyzing the standard twin paradox that make it seem like the asymmetry is different things.
Staff Emeritus
I think that it's a dead end to expect any close analogy between sound and light in SR. My understanding is that the group structures are different, and that this shows up in the speed of light being frame independent, while the speed of sound is n ot.
However, I'm not familiar enough with the ins and outs of group theory to offer a more rigorous proof that the groups are different.
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2021 Award
That is because in all versions of the twin paradox, there is some asymmetry between the twins. It just isn't always the same asymmetry. Different versions have different asymmetries between the twins.
That is a good point. In the case of the "repeating" universe there is an asymmetry with respect to the global rest frame.
Killtech
The root cause is actually that in the closed world, it is no longer the case that all inertial frames are equivalent globally. [...]
So this is indeed considered correct. I understood the original explanation but I was not sure if it was really correct given the loss of equivalence.
That is a good point. In the case of the "repeating" universe there is an asymmetry with respect to the global rest frame.
That particular issue is unique to the closed world setup, yes.
Can you point me to a proof of that is issue is specific to a closed universe? I mean the closed world makes the asymmetry quite obvious, but this does not automatically imply it's not there without it. proof by example does not count for me as a mathematician.
That is because in all versions of the twin paradox, there is some asymmetry between the twins. It just isn't always the same asymmetry. [...]
Yeah, I know some of the asymmetries and looked at the various versions of the twin paradox just to find they did not look at what i was interested in. I understand most of the usual explanations but am not sure how to translate it into my problem. In particular: is it possible to derive whether the SoS is closed purely from local measurements (and if so how is that proofed)?
I am looking at very particular asymmerty that i cannot name any better them ##\Delta t_{B,A}<\Delta t_{A,B}## for two frame independent scalar values. In my case I am not exactly looking at a classic twin paradox, since these quantities technically only make sense using evenly spaced grids which is not the case in any version of the paradox i know of. It's derived from a version though and most of the calculations can be taken from there for a shortcut...
My standard recommendation for twin paradox questions is to read the Usenet Physics FAQ article:
It's based on the outbound/inbound twin example for a start. But there is no extension of the scenario having enough grid points for the events to occur that define ##\Delta t_{I,J}##. These values are however tied to the times the clocks in the original paradox show during certain events, so some of them can be taken from the original calculation.
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PeroK
I think that the simplest explanation of the twin paradox is to note that the elapsed time along a path through spacetime is analogous to the length of a path through Euclidean space. So asking "why is the inertial twin oldest" is analogous to asking "why is a straight line the shortest distance between two points". I usually ask what answer you would give to that question.
Can you point me to a proof of that is issue is specific to a closed universe?
In an open universe you don't loop round and return to your starting place, so there's no frame picked out by that process.
Killtech
I think that the simplest explanation of the twin paradox is to note that the elapsed time along a path through spacetime is analogous to the length of a path through Euclidean space. So asking "why is the inertial twin oldest" is analogous to asking "why is a straight line the shortest distance between two points". I usually ask what answer you would give to that question.
except that every twin is inertial in my example and the ##\Delta t_{I,J}## treat each as such so by your words finding the oldest would already get quite tricky.
In an open universe you don't loop round and return to your starting place, so there's no frame picked out by that process.
Do you even know what the word "proof" means?
weirdoguy and PeroK
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Can you point me to a proof of that is issue is specific to a closed universe?
I canโt even write down a general expression for what this โissueโ is. Can you? Without that I donโt even know what it is that you want a proof of.
robphy
Killtech
I canโt even write down a general expression for what โthis issueโ is. Can you?
Sure: apparently in a closed world each frame has a characteristic quantity: the delay between two signals sent out in opposing directions coming back around the world to their source. Unlike the closed/open SoS this one distinguishes each frame in a much finer fashion. Show that no such characteristic quantity can be constructed locally / without the usage of the closed world property.
Mentor
Can you point me to a proof of that is issue is specific to a closed universe?
It follows from the topology of the closed universe: in the 1-1 dimensional case the topology of the closed universe is ##S^1 \times R^1##, whereas the topology of standard 1-1 dimensional Minkowski spacetime is ##R^2##. The issue I described is only present in topologies that have compact components; ##S^1## is compact, but ##R^n## is not.
If you want to try to visualize this, think of a cylinder vs. a plane. On a plane, you can draw two mutually perpendicular axes in any orientation you want and they will all be equivalent. But on a cylinder, only one orientation of the axes will result in the "horizontal" axis being a closed circle (and the "vertical" axis in this case will go straight up the cylinder without winding around it). In every other orientation, the "horizontal axis" will be a helix, winding up and down the cylinder (and the "vertical" axis will also wind around the cylinder), and the "degree of winding" will be a number that varies continuously with the orientation.
The only difference between the case I just described and the spacetime case is that "rotating" the axes is done hyperbolically, so the definition of "orthogonal" changes (roughly speaking, the "rotated" axes become a narrower and narrower rhombus instead of staying square). But everything else is the same.
in a closed world each frame has a characteristic quantity: the delay between two signals sent out in opposing directions coming back around the world to their source.
This is a side effect (and not the only one) of the underlying topology difference I described above. The underlying topology difference is the fundamental property from which all others come.
Killtech
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is it possible to derive whether the SoS is closed purely from local measurements
No. It is a global property, not a local property. More generally, the underlying topology of the spacetime is a global property, not a local property.
I know some of the asymmetries and looked at the various versions of the twin paradox just to find they did not look at what i was interested in.
The FAQ article I linked to does not discuss the "closed universe" case, that's true. So you won't find anything specific to that issue in the article.
Mentor
every twin is inertial in my example
In the closed universe example, yes, all the twins can be "inertial" (as in, always at rest in the same inertial frame), but still meet more than once. That is another side effect of the underlying topology of the closed universe; in standard Minkowski spacetime with underlying topology ##R^n##, it is impossible for any two observers who are inertial in this sense to meet more than once.
Mentor
there is no extension of the scenario having enough grid points for the events to occur that define ##\Delta t_{I,J}##. These values are however tied to the times the clocks in the original paradox show during certain events, so some of them can be taken from the original calculation.
I don't know what you mean by this. Do you just mean that the article doesn't cover the closed universe scenario (which, as I posted previously, I agree it does not)? Or something else?
except that every twin is inertial in my example and the ##\Delta t_{I,J}## treat each as such so by your words finding the oldest would already get quite tricky.
One pair of twins meet half way through your experiment. Adding the elapsed time between meetups for these two will be less than the elapsed time between meetups for the third. This is just the Minkowski equivalent of the triangle inequality. Again, I'd ask you what would you consider an acceptable response to "why is the triangle inequality true in Euclidean space"?
Sure: apparently in a closed world each frame has a characteristic quantity: the delay between two signals sent out in opposing directions coming back around the world to their source. Unlike the closed/open SoS this one distinguishes each frame in a much finer fashion. Show that no such characteristic quantity can be constructed locally / without the usage of the closed world property.
A flat Minkowski space is symmetric under four translations, three rotations, and three boosts. The "cylindrical" Minkowski spacetime is not symmetric under boost in the tangential direction (not globally, anyway). That lack of symmetry is the source of your "characteristic quantity". So if you want to find such a quantity in flat spacetime you need to find a way in which flat Minkowski spacetime violates the symmetries that define it.
Presuming that SoS is surface of simultaneity, I don't see that your characteristic quantity is any different to the failure of the planes to close. In both cases, the size of the quantity increased with increasing speed relative to the preferred frame.
Dale
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the delay between two signals sent out in opposing directions coming back around the world to their source
Sure. That is easy, the two signals never return to their source in an open universe by definition. QED
I thought you wanted a general proof of some unspecified issue.
Killtech
I don't know what you mean by this. Do you just mean that the article doesn't cover the closed universe scenario (which, as I posted previously, I agree it does not)? Or something else?
Well, two similar cases actually: yes, the closed world example is one but the grid example is another.
See, I wanted to carry the closed world example to the open world scenario an make them as indistinguishable as possible. So instead of going around the world and coming back from the other side I just clone each twin and put them where each twin previously saw his own image when looking around the loop. Doing so effectively creates a grid of evenly spaced twins (when iteratively done for each twins clone):
Code:
-----> B1 -----> B2 -----> B3 ----->
------ A1 ------ A2 ------ A3 ------
Now while the outgoing twin B2 never gets to it's original home he will meet a clone of that home A3 instead. As for the twin A2 staying at home, he will never see his twin B2 again, but he will see the clone of him B1 instead. This case is different from the normal twin paradox as no twin will ever get home exactly yet the age differences between reunions ##\Delta t_{I,J}## are still available - and from the perspectives of the twins itself it's not clear if they would be able to distinguish if they are in a closed or open world within the scope of the experiment.
But if both twins frame are really equivalent, then both the outgoing twin and the home twin must measure the exact same time difference between reunions with their own clocks i.e. ##\Delta t_{A,B} = \Delta t_{B,A}## since there is no way to say which one is outgoing and which one stays. On the other hand the scenario in the closed world yields ##\Delta t_{A,B} > \Delta t_{B,A}##
In my original post I used this setup, so i can take some calculations from the inbound/outbound twin variant:
Code:
-----> O1 -----> O2 -----> O3 ----->
------ A1 ------ A2 ------ A3 ------
<----- B1 <----- B2 <----- B3 <-----
Mentor
instead of going around the world and coming back from the other side I just clone each twin and put them where each twin previously saw his own image when looking around the loop.
This is just "unrolling the cylinder" onto a flat plane. It doesn't change the underlying topology of the closed world example, since you are treating each clone as if it were the same as the twin it's cloned from, so one twin passing by repeated clones of the other twin counts as the same thing as passing the other twin repeatedly.
from the perspectives of the twins itself it's not clear if they would be able to distinguish if they are in a closed or open world within the scope of the experiment.
If the clones are actually indistinguishable, then the twins will be able to tell they are in a closed world because from their viewpoint they are meeting the same other twin again and again; as noted above, the underlying topology is the same as the closed world so the scenario will have all the same properties, and those properties are testable.
If the clones are not indistinguishable, then your scenario is not equivalent to the closed world scenario any more, so of course it will not have the same properties.
Killtech
This is just "unrolling the cylinder" onto a flat plane. It doesn't change the underlying topology of the closed world example, since you are treating each clone as if it were the same as the twin it's cloned from, so one twin passing by repeated clones of the other twin counts as the same thing as passing the other twin repeatedly.
Yes, the topologies are clearly different. "Unrolling the cylinder" onto a plane repeatedly by creating a copy of its contents obviously creates a different topology. Take the point of where one twin and its clone is. In the closed world the clone is actually the twin itself and therefore these two points are one and the same, whereas in in the open world topology they are different with quite a distance in between them. Also if there are two spatial dimensions (like on a cylinder) a geodesics will intersect itself while it cannot ever do that in a flat open world.
If the clones are actually indistinguishable, then the twins will be able to tell they are in a closed world because from their viewpoint they are meeting the same other twin again and again; as noted above, the underlying topology is the same as the closed world so the scenario will have all the same properties, and those properties are testable.
If the clones are not indistinguishable, then your scenario is not equivalent to the closed world scenario any more, so of course it will not have the same properties.
The question of indistinguishability is unclear. All the things and events used in calculating the results of the experiments are exactly the same. There are differences, yet none of those seem to be relevant for the calculation done. If the calculations are effectively the same in both cases they cannot lead to different results.
Anyhow, consider the following: for each of the twins it takes a different amount of time to meet their sibling. so the twin that ages less in between meetings will know that the time dilation actually happens on his side and not his twin - even though he is supposedly in a rest frame. This realization however only requires exchanging data on all recorded previous meetings each of the twins had when they meet at the same point. That's the case in the closed world scenario! This is obviously deeply problematic if it happened the same way for the grid twins. So there must be a big difference in the calculation somewhere.
Mentor
"Unrolling the cylinder" onto a plane repeatedly by creating a copy of its contents obviously creates a different topology.
Not if the clones are indistinguishable. Then the unrolled cylinder is indistinguishable from the rolled up cylinder, and has the same topology.
in in the open world topology they are different with quite a distance in between them.
If the clone and the twin are indistinguishable, then this is not correct.
The question of indistinguishability is unclear.
No, it isn't. "Indistinguishable" means just what it says: it means "can't be distinguished". Which in turn means that you are wrong to think of things that you say are "indistinguishable" as still being "different" in some way. You removed all the differences when you said they were indistinguishable.
All the things and events used in calculating the results of the experiments are exactly the same.
In the case where the clones are indistinguishable, yes.
There are differences, yet none of those seem to be relevant for the calculation done.
No, there are no differences; you took them all away when you said the clones were indistinguishable.
If the calculations are effectively the same in both cases they cannot lead to different results.
That's true. But it only applies to the case where you said the clones were indistinguishable. If the clones are not indistinguishable, then it's no longer true that "all the things and events used in calculating the results of the experiments are exactly the same", because being able to tell the difference between one clone and another counts as an experimental result.
for each of the twins it takes a different amount of time to meet their sibling. so the twin that ages less in between meetings will know that the time dilation actually happens on his side and not his twin - even though he is supposedly in a rest frame. This realization however only requires exchanging data on all recorded previous meetings each of the twins had when they meet at the same point. That's the case in the closed world scenario!
Yes, all of this is true for the closed world scenario. Which means it is also true for the "grid scenario" where the clones are indistinguishable, since that scenario is the same scenario as the closed world scenario; you made it that way when you said the clones were indistinguishable. "Indistinguishable" means that it is impossible for any clone of either twin to tell the difference between one meeting with a clone of the other twin and another; to him, they are all meetings with the same other twin. Which means that they are all meetings with the same other twin; you've explicitly ruled out anything that could be used to tell them apart. So you don't actually have two different scenarios; you just have one scenario that you have confused yourself about by describing it in two different ways that make you incorrectly think they're somehow different, when they're actually not.
This is obviously deeply problematic if it happened the same way for the grid twins.
For the case where the grid clones are indistinguishable, no, it's not. See above.
So there must be a big difference in the calculation somewhere.
Not for the scenario where the grid clones are indistinguishable. See above.
I suspect that what is confusing you is that, in your mind, you are not allowing the grid clones to be truly indistinguishable; you are, in your mind, reserving some property as being different between them. For example, you might be thinking of them as being "at different points in space". But if that is actually the case, then there is a way to measure it. "Different points in space" has no meaning if there is literally no observable that can tell them apart. And "indistinguishable" means that there is no observable that can tell them apart. If you are thinking of them as being at "different points in space", you are not thinking of them as being indistinguishable. That is a logical contradiction; you can't have it both ways.
Killtech and Motore
Killtech
I suspect that what is confusing you is that, in your mind, you are not allowing the grid clones to be truly indistinguishable; you are, in your mind, reserving some property as being different between them. For example, you might be thinking of them as being "at different points in space". But if that is actually the case, then there is a way to measure it. "Different points in space" has no meaning if there is literally no observable that can tell them apart. And "indistinguishable" means that there is no observable that can tell them apart. If you are thinking of them as being at "different points in space", you are not thinking of them as being indistinguishable. That is a logical contradiction; you can't have it both ways.
The twins become indistinguishable if the access to certain information is restricted or rather the twins are forbidden from conducting some other experiments.
Unfortunately math and logic is always quite a bit more tricky when done correctly. Thus to put this into more rigorous mathematical terms: the calculation of the age difference in the closed world scenario does make use of only some postulates/assumptions/properties of that setup, but far from all. This means that the calculation holds true for any setup which satisfies this specific smaller set of assumptions that is actually used. Now this limited set of assumption is the same for the case of the grid, which makes them indistinguishable/equivalent i.e. using these assumptions only will not allow you to tell the scenarios apart (they can indeed be considered as one).
A math analogy: any statement derived without using the AC (axiom of choice) will still hold true when either AC or its negation is added retroactively. So the single scenario where the original conclusion was done becomes two distinct ones which now differ but the conclusion itself remains valid for both. thus indistinguishability is relative to the chosen set of assumptions.
however, when calculating other things which make use of properties of the setups that were previously not used, the differences between these setups show - for example the closed SoS can only happen in the closed world setting for the topological reasons you rightly noted.
More interestingly let's observe a crucial feature of the age calculation: the twins age at their reunion is determined from A's frame. But it could have been done from frame B, too yielding the reverse age difference for the very same event. Although that sounds like a paradox, in the closed world case this is actually okay, because it turns out that by doing so we somehow implicitly make B have a closed SoS, rather then A. So this means that in fact both calculations were made in two different worlds/scenarios: one where A is the preferred frame and in the other it's B.
And it reveals a general problem that this method to determine the outcome of an event (which twin is older at their reunion) actually depends on the choice of frame it is done in! This is more general then the closed world scenario.
Now in the cased of the grid the topology is indeed different as no SoS can ever be closed. So when we switch between the frames A and B there is no difference. mathematically, when the cylinder is cut the border conditions that needed to be satisfied for consistency along the line the cylinder was cut open are no longer required. And therefore the previously two different spaces in the closed worlds are identical and thus one and the same for the grid in the open world scenario.
Yet doing the calculation still yields a frame dependent result for the reunion: we get A will be older then B and B older then A (or equivalently ##\Delta t_{A,B} < \Delta t_{B,A}## and ##\Delta t_{A,B} > \Delta t_{B,A}##) - a clean contradiction.
So either two different inertial frames can never be truly equivalent to begin with (even in an open world) or there is a error in the calculation that makes the result depend on the frame.
PeroK
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But it could have been done from frame B, too yielding the reverse age difference for the very same event.
Only if you make a mistake in the calculation
And it reveals a general problem that this method to determine the outcome of an event (which twin is older at their reunion) actually depends on the choice of frame it is done in!
This is not correct. The age is always given by the integral of the proper time along the path. The global topology changes the possible paths but not the calculation.
PeroK, vanhees71, Motore and 1 other person
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The twins become indistinguishable if the access to certain information is restricted or rather the twins are forbidden from conducting some other experiments.
That's not what "indistinguishable" means. Either there are observables that distinguish the clones from each other, or there aren't. Saying that there are, but you won't allow the twins to use them, is not a way of making them indistinguishable; it's just a way of confusing yourself. Stop it.
Unfortunately math and logic is always quite a bit more tricky when done correctly.
Indeed. But that doesn't excuse doing it incorrectly, as you are doing, just because it's easier.
The rest of your post is just you confusing yourself further. You should stop that.
vanhees71, Motore and Dale
Mentor
Yet doing the calculation still yields a frame dependent result for the reunion
If you're in the open world, there is no reunion. A reunion is only possible in the closed world. @Ibix already pointed this out in post #8 (and I reiterated it in post #14).
This is what comes of inventing elaborate ways of confusing yourself instead of focusing on the basics, namely the difference in topology between the two scenarios and what it implies.
Killtech
Only if you make a mistake in the calculation
This is what comes of inventing elaborate ways of confusing yourself instead of focusing on the basics
Fine, basic question then: in the following setup in a flat open world
Code:
... ------ A1 ------ A2 ------ A3 ------ ...
... -----> B1 -----> B2 -----> B3 -----> ...
which grid-vertex ages more between two meetings with the others grid vertices? those in A or in B?
A2 between meetings with B2 and B1 (##= \Delta t_{A,B}##) or
B2 between meetings with A2 and A3 (##= \Delta t_{B,A}##)?
at each meeting of vertices a picture is taken showing the state of both clocks build into each vertex (during which they are at the exact same point). Age differences can be deducted from reading the clocks in between two recorded consequent pictures.
Unlike all other twin paradox setups this one is perfectly symmetric between frames A and B (at least in an open world) and does include only the most trivial of movement.
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2021 Award
Fine, basic question then: in the following setup in a flat open world
Code:
... ------ A1 ------ A2 ------ A3 ------ ...
... -----> B1 -----> B2 -----> B3 -----> ...
which grid-vertex ages more between two meetings with the others grid vertices? those in A or in B?
This is literally just the Lorentz transform.
A2 between meetings with B2 and B1 (=ฮtA,B) or
B2 between meetings with A2 and A3 (=ฮtB,A)?
##\Delta t_{A,B}=\Delta tโ_{B,A}## (edit: assuming equal proper spacing for the A and B clocks)
Unlike all other twin paradox setups this one is perfectly symmetric between frames A and B (at least in an open world)
Yes, and as such the obvious symmetrical relationship is obtained.
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Gold Member
Unlike all other twin paradox setups this one is perfectly symmetric between frames A and B (at least in an open world) and does include only the most trivial of movement.
But there is never a reunion. And hence no way for the twins to compare their ages at a reunion. This is not a twin paradox.
Mentor
which grid-vertex ages more between two meetings with the others grid vertices? those in A or in B?
In ordinary flat Minkowski spacetime with the standard topology, it depends on how you specify the scenario. You can specify it so that the apparent aging is symmetric (which appears to be your intent), or you can specify it so it's not, by any amount you like, in either direction (A's aging less than B's, or the reverse). It all depends on how you specify the A and B observer "grids" to be spaced in spacetime.
Note, by the way, that all of the above is also true in the closed world scenario. You can find a set of "A" and "B" observers that age the same between meetings, or a set that age differently, by any amount you like, in either direction. The only difference in the closed world case is that there is one particular family of specifications that includes an observer or set of observers (which you could label the "A" set or the "B" set, that's just a matter of labeling) whose worldlines are "parallel" to the axis of the cylindrical spacetime, and whose surfaces of simultaneity are closed circles (in the 1x1 dimensional case). (There is a family of such specifications because you still have an infinite range of possibilities for the other family of observers, depending on how much less you want them to age between meetings than the family just described.) And for this particular family of specifications, the set of observers described above are always the ones who age the most between meetings.
Mentor
which grid-vertex ages more between two meetings with the others grid vertices? those in A or in B?
Note, btw, that this scenario you are discussing now has nothing to do with the standard twin paradox, because no observer ever accelerates; all observers are always inertial. The "meetings" are either with successive members of the other set of observers (an A with successive B's, or a B with successive A's) or due to at least one observer going "all the way around the universe" in the closed world case, or a combination of both. (This, btw, is a key difference from the standard twin paradox, since it makes possible scenarios where both sets of observers age the same between meetings, something which is impossible in the standard twin paradox--which seems to be a point that is confusing you. Changing scenarios again and again does not necessarily help you to understand any of them; it is better to get a thorough understanding of just one scenario before trying to ring changes on it.)
Mentor
2021 Award
or you can specify it so it's not, by any amount you like, in either direction (A's aging less than B's, or the reverse)
Hmm, I am not seeing it. What are you thinking of here?
Mentor
I am not seeing it. What are you thinking of here?
Consider the following two examples:
(1) Pick a frame. The A observers all move at speed ##v## to the left, and the B observers all move at speed ##v## to the right, in the chosen frame. Space them out the same distance from each other spatially in the chosen frame. Under these conditions, the A's will age the same between successive B meetings, as the B's age between successive A meetings. (Try it and see, if it's not already obvious from the symmetry in the construction I've just described.)
(2) Pick a frame. This frame is the A rest frame. The A at the spatial origin ##x = 0## meets one B at time ##t = 0##. He meets the next B at time ##t = T##. Draw the hyperbola of constant proper time ##T## through the ##t = T, x = 0## event. Pick the point on that hyperbola where the worldline of the next A to the right of the A at the spatial origin meets it; call this point P. Draw the worldline of the B observer that passes through the spacetime origin, ##t = 0, x = 0## (and meets the A observer there) so that it meets the next A observer at point P. Then draw the worldline of the B observer next to the left, the one that meets the A observer at ##t = T, x = 0##, with the same slope. You now have sufficient information to draw in the entire "grids" of both A and B observers, and by construction, the A's age the same between successive B meetings, as the B's age between successive A meetings.
These sorts of scenarios are not often discussed, but IMO they should be; if people had a better sense of the actual range of possibilities for scenarios like this, it would give them better tools for understanding particular individual scenarios.
Killtech and vanhees71
Mentor
Consider the following two examples
Oh, and to be clear, both of those constructions were assuming the flat, open world (i.e., standard Minkowski spacetime, 1x1, with topology ##R^2##).
In the closed world (flat metric, topology ##S^1 \times R^1##), construction #1 can still be done, but construction #2 cannot, at least not globally; it will always break down before it can be extended around the entire cylinder. (This is assuming that the chosen frame for both constructions is the preferred frame of the closed world, i.e., the one whose time axis is parallel to the axis of the cylinder and whose spatial surfaces of simultaneity are closed.)
Killtech and vanhees71
Mentor
2021 Award
Consider the following two examples:
These two examples were โsymmetricโ. I understood that. I didnโt see the asymmetric case.
I do see it now. The velocity is not particularly relevant because that is symmetric. However the proper distance between grid clocks can be made asymmetric (in my above post I was assuming equal proper spacing).
If the proper spacing between B clocks is much smaller than between A clocks then A will only age a little between B clocks and B will age a lot between A clocks.
Anyway, assuming that @Killtech was intending equal proper spacing then my comments above hold. And since he didnโt actually state it your comments hold more generally.
vanhees71
Mentor
I do see it now. The velocity is not particularly relevant because that is symmetric. However the proper distance between grid clocks can be made asymmetric (in my above post I was assuming equal proper spacing).
Yes, exactly; that spacing is a free parameter, so we can tailor it however we like to adjust the relative aging.
vanhees71 and Dale
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# Difference between revisions of "An abstract algebraic logic view of some mutiple-valued logics"
Authors: Title of the chapter: An abstract algebraic logic view of some mutiple-valued logics Title of the book: Beyond Two: Theory and Applications of Multiple-Valued Logic Editor(s): Pages: 25-58 Publisher: Physica-Verlag City: Heidelberg-Berlin-New York Year: 2003
## Abstract
Abstract Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum-Tarski process. The notions of logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainly by Blok, Pigozzi and Czelakowski, and obtains a deep theory and very nice and powerful results for the so-called protoalgebraic logics.
I will show how the idea (already explored by Wรณjckici and Nowak) of defining logics using a scheme of "preservation of degrees of truth" (as opposed to the more usual one of "preservation of truth") characterizes a wide class of logics which are not necessarily protoalgebraic and provide another fairly general framework where recent methods in Abstract Algebraic Logic (developed mainly by Jansana and myself) can give some interesting results. After the general theory is explained, I apply it to an infinite family of logics defined in this way from subalgebras of the real unit interval taken as an MV-algebra. The general theory determines the algebraic counterpart of each of these logics without having to perform any computations for each particular case, and proves some interesting properties common to all of them.
Moreover, in the finite case the logics so obtained are protoalgebraic, which implies they have a "strong version" defined from their Leibniz filters; again, the general theory helps in showing that it is the logic defined from the same subalgebra by the truth-preserving scheme, that is, the corresponding finite-valued logic in the most usual sense. However, for infinite subalgebras the obtained logic turns out to be the same for all such subalgebras and is not protoalgebraic, thus the ordinary methods do not apply. After introducing some (new) more general abstract notions for non-protoalgebraic logics I can finally show that this logic too has a strong version, and that it coincides with the ordinary infinite-valued logic of Lukasiewicz.
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{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
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http://www.primidi.com/electric_field/quantitative_definition
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# Electric Field - Quantitative Definition
Quantitative Definition
Consider a point charge q with position (x,y,z). Now suppose the charge is subject to a force due to other charges. Since this force varies with the position of the charge and by Coloumb's Law it is defined at all points in space, is a continuous function of the charge's position (x,y,z). This suggests that there is some property of the space that causes the force which is exerted on the charge q. This property is called the electric field and it is defined by
Notice that the magnitude of the electric field has units of Force/Charge. Mathematically, the E field can be thought of as a function that associates a vector with every point in space. Each such vector's magnitude is proportional to how much force a charge at that point would "feel" if it were present and this force would have the same direction as the electric field vector at that point. It is also important to note that the electric field defined above is caused by a configuration of other electric charges. This means that the charge q in the equation above is not the charge that is creating the electric field, but rather, being acted upon by it. This definition does not give a means of computing the electric field caused by a group of charges.
From the definition, the direction of the electric field is the same as the direction of the force it would exert on a positively charged particle, and opposite the direction of the force on a negatively charged particle. Since like charges repel and opposites attract, the electric field is directed away from positive charges and towards negative charges.
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{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
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https://www.logicmatters.net/2015/11/01/gently-into-november/
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# Gently into November
I have just putย an update for the Gentle Introduction to category theory online. Things have been moving pretty slowly (pressure of other interests, a planned chapter not really working out, spending far too much time revising earlier chapters, etc.); but weโve inched forward toย 157 fun-packed pages. Again, this version is stopping far short of a natural break point. But Iโll not be able to return to this for a few weeks, so I thought I should at least make available the best version of the earlier chapters I now have, and I can addย three new chapters. (I note the newly added Theorem 63 which corrects a terrible blunder in some remarks in an earlier version!ย No promisesย thatย there arenโt more blunders to be found.)
### 17 thoughts on โGently into Novemberโ
1. There is an obvious garble in (F8) in ยง12.2: it should read โAnd List_arw takes a function f : X โ Y to the function List(f): List(X) โ List(Y) which sends the list โฆโ or some such.
2. Finally, let me get to some comments on Sec. 8.3 and subobjects. (Iโm separating points since theyโre different โthreadsโ). Most participants found it much harder, also because of how much material it contains โ having a pullback snuck in didnโt help, but also the section contains no actual Definition marked as such.
Also, you give the presentation of subobjects from topos theory, but I find it a bit sad that you donโt give the standard definition of subobjects of Z, namely as monics into X -> Z (up to a certain equivalence relation) โ I like the discussion from Barr and Wells, Category theory for computing science, Sec. 2.8.8.
Generalizing from inclusions to arbitrary monics, with this equivalence relation, seems already noteworthy. And this would already be useful for the later example of pullbacks in Set (Sec. 9.5 (b)) โ you give an intuition for the pullback X รZ Y for the special case where X, Y are subsets of Z. This intuition generalizes easily to other monics: a pullback of two subobjects is the generalization of their intersection.
Now, it is true that for topoi you can add subobjects classifiers, the truth arrow, characteristic arrows and so on, but that applies to fewer categories (that is, only topoi) and Iโve usually seen it come afterwards, while subobjects exist in most categories. (Still, if you were to come to topos theory, Iโd be happy; in my last approach I got stuck on topoi for modeling intuitionistic logic, using Goldblatt).
1. Thanks once more โฆ. this is very helpful. Iโll rethink the presentation here. (And by the way, I agree that Barr and Wells, Category Theory for Computing Science, is a pretty terrific book!)
3. Sec. 10.1, Th. 49: I had trouble understanding the last proof step, namely โRearranging part of this diagram, and unpacking again the composites f โฆ c, g โฆ c, this is just to say that there is a unique arrow v : C โ E making this cut-down diagram commute:โ.
In my head, I also had to โunpackโ q = f โ e.
Also, I had trouble getting the proof structure in this step: The diagram commutes iff this rearranged diagram commutes (not clear from cut-down; I missed โthis is just to sayโ, maybe thatโs just my fault). So, v is the unique arrow making the earlier diagram commute iff it is the unique arrow making the later diagram commute. Because of the โonly ifโ direction, we get that v is the mediating arrow.
On Chap. 10: seminar colleagues somehow didnโt find 10.2 very illuminating.
In 10.4, I wondered if Theorem 47 (on pullbacks) could be extended or had been forgotten โ from nLab it seems not, I wonder why.
1. (1) Yes, something seems amiss with the presentation at the end of Th 49. This needs tidying as you suggest.
(2) Iโll have another look at ยง10.2 then!
(3) Theorem 47 isnโt exactly forgotten, but perhaps the remark at the end of 10.1 needs about it not needing a separate proof given proofs of the other results needs to be made more emphatically.
4. Time for non-typos.
Chap. 7, theorem 24.
Other examples include all categories with a null object and at least another non-isomorphic object X: if 0 = 1, 0 ร X = 1 ร X = X != 1. (Observed by a seminar colleague, Iโm sorry I forget who).
In the given proof, you claim that โBut trivially the wedge factors uniquely through itselfโ, but Iโm not sure how trivial is โuniquelyโ. However, it might be trivial once youโve fully spelled out the given category, and we pick the free one given the constraint (alternatively, we might just compute the set of all valid equations as the closure of the given ones, but Iโm not sure how). The set of arrows is luckily finite, since the only loop is between f and p, and pโฆf = 1 while (f โ p)^2 = f โ p (precisely, f โ p โ f โ p = f โ (p โ f) โ p = f โ 1 โ p = f โ p).
Then, I guess we can verify that f โ p != id, q โ f โ p != q, so f โ p is not a mediating arrow, as required.
1. First, Iโm not sure now why I gave the (supposed) โbrute forceโ counterexample rather than the nice general proof you mention. I certainly agree: I should use the nice proof.
Second, I fear just I may just have got myself in a tangle thinking through the supposed counterexample (I rather suspect I just muddled up p.f with f.p at a crucial point). Best deleted, then!
Very many thanks, then, for spotting this.
5. Another typo. The statement of Theorem 59 (2) should have C^1 is iso to C (not 1!) โ which is fortunately what the ensuing proof proves. [Though Iโll rewrite the proof to separate out the quick-and-easy key idea from checking that it works.]
6. Hi, Iโm a member of the study group Paolo Giarrusso mentioned, in the University of Tรผbingen. Weโre reading about limits (chapters 8 and 9) this week. Here are some typos. Is it better to report them each week, or to collect them and report everything at once?
โ Theorem 43, end of the first paragraph of its proof: u = \alpha_C should be u = \alpha_I; otherwise itโs not type-correct.
โ Theorem 45, first line of the first paragraph of its proof: The cone X Y should be X X instead. Ditto for the same cone on the first line of the second paragraph.
1. Thanks for this (what annoying typos)! Oh yes, please send the corrections week-by-week, both for me (Iโll be putting an update online in a couple of weeks, I hope) and for others.
7. Other comment: something looks wrong at page 48, where footnote 2 is referred to:
H = (H, โ)).ยฒ
Not only thereโs a double closed parenthesis, but Iโm guessing thereโs text missing there, where you probably announce youโre going to stop distinguishing \mathcal{G} from G.
8. One question on the proof of Theorem 16. It seems that point 1 proves that, if f = g, then f \compose xโ = g \compose xโ, with the assumption that f \compose x = g \compose x. But does that need such an elaborate proof, given equality is, well, equality?
1. Thanks! While at nitpicking: Def. 25 doesnโt literally introduce โpoint elementsโ, but the term appears in Thm. 18, and right before Def. 30.
Also, FYI: Iโm reading the early chapters because weโre currently using your notes in a small study group (CS students) ;-) โ Iโll watch for feedback, but we just started, so not much yet (other than people liking the precise explanation of duality).
1. Thanks for this too! I do hope that your study group finds the notes helpful, and Iโll be delighted to get any feedback that you have. In particular, Iโd like to know about any sections which people find less clear than they should be.
(As you know, the Gentle Intro is very much work in progress, but the story can be continued in the older Notes available from the categories page here, where I also I say how old and new fit together.)
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https://hg.ucc.asn.au/dropbear/file/8bba51a55704/libtommath/bn_mp_lcm.c
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### view libtommath/bn_mp_lcm.c @ 1470:8bba51a55704
Update to libtommath v1.0.1
author Matt Johnston Thu, 08 Feb 2018 23:11:40 +0800 60fc6476e044 f52919ffd3b1
line wrap: on
line source
```#include <tommath_private.h>
#ifdef BN_MP_LCM_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
*
* LibTomMath is a library that provides multiple-precision
* integer arithmetic as well as number theoretic functionality.
*
* The library was designed directly after the MPI library by
* Michael Fromberger but has been written from scratch with
*
* The library is free for all purposes without any express
* guarantee it works.
*
* Tom St Denis, [emailย protected], http://libtom.org
*/
/* computes least common multiple as |a*b|/(a, b) */
int mp_lcm (mp_int * a, mp_int * b, mp_int * c)
{
int res;
mp_int t1, t2;
if ((res = mp_init_multi (&t1, &t2, NULL)) != MP_OKAY) {
return res;
}
/* t1 = get the GCD of the two inputs */
if ((res = mp_gcd (a, b, &t1)) != MP_OKAY) {
goto LBL_T;
}
/* divide the smallest by the GCD */
if (mp_cmp_mag(a, b) == MP_LT) {
/* store quotient in t2 such that t2 * b is the LCM */
if ((res = mp_div(a, &t1, &t2, NULL)) != MP_OKAY) {
goto LBL_T;
}
res = mp_mul(b, &t2, c);
} else {
/* store quotient in t2 such that t2 * a is the LCM */
if ((res = mp_div(b, &t1, &t2, NULL)) != MP_OKAY) {
goto LBL_T;
}
res = mp_mul(a, &t2, c);
}
/* fix the sign to positive */
c->sign = MP_ZPOS;
LBL_T:
mp_clear_multi (&t1, &t2, NULL);
return res;
}
#endif
/* ref: \$Format:%D\$ */
/* git commit: \$Format:%H\$ */
/* commit time: \$Format:%ai\$ */
```
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https://www.wolfram.com/mathematica/new-in-10/symbolic-geometry/mesh-regions.html.zh?footer=lang
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# ็ฝๆ ผๅบๅ
In[1]:= XMeshRegion[{{0}, {1}, {2}, {3}}, {Line[{1, 2}], Line[{3, 4}]}, MeshCellLabel -> 0 -> "Index"]
Out[1]=
In[2]:= XDelaunayMesh[RandomReal[{0, 1}, {6, 1}]]
Out[2]=
In[3]:= XMeshRegion[{{0, 0}, {1, 0}, {0, 1}, {1, 1}}, Triangle[{{1, 2, 3}, {3, 2, 4}}], MeshCellLabel -> 0 -> "Index"]
Out[3]=
In[4]:= XDelaunayMesh[RandomReal[{0, 1}, {25, 2}]]
Out[4]=
In[5]:= XMeshRegion[{{0, 0, 0}, {2, 0, 0}, {2, 2, 0}, {0, 2, 0}, {1, 1, 2}}, {Tetrahedron[{1, 2, 3, 5}], Tetrahedron[{1, 3, 4, 5}]}, MeshCellLabel -> 0 -> "Index"]
Out[5]=
In[6]:= XDelaunayMesh[RandomReal[{0, 1}, {25, 3}]]
Out[6]=
In[7]:= XBoundaryMeshRegion[{{0}, {1}}, Point[{1, 2}], MeshCellLabel -> 0 -> "Index"]
Out[7]=
In[8]:= XConvexHullMesh[RandomReal[{0, 1}, {6, 1}]]
Out[8]=
In[9]:= XBoundaryMeshRegion[{{0, 0}, {1, 0}, {0, 1}}, Line[{1, 2, 3, 1}], MeshCellLabel -> 0 -> "Index"]
Out[9]=
In[10]:= XConvexHullMesh[RandomReal[{0, 1}, {25, 2}]]
Out[10]=
In[11]:= XBoundaryMeshRegion[{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, Triangle[{{1, 2, 3}, {1, 2, 4}, {2, 3, 4}, {1, 3, 4}}], MeshCellLabel -> 0 -> "Index"]
Out[11]=
In[12]:= XConvexHullMesh[RandomReal[{0, 1}, {25, 3}]]
Out[12]=
## Mathematica
Questions? Comments? Contact a Wolfram expertย ยป
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https://www.888casino.com/blog/side-bets/card-counting-the-tiger-7-ox-6-and-kill-the-oxtiger-baccarat-side-bets
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When the patent on EZ Baccarat and its Dragon 7 side bet expired in September, 2013 (see this post), there was a bit of a land rush. Bally (Shuffle Master) was the first to get in, creating "Fortune Baccarat" with the Fortune 7 side bet. Others have followed.ย This post considers Dai Bacc, which is a new entry to the same intellectual property landscape.
The main game of Dai Bacc is simply EZ/Fortune Baccarat. This is theย non-commission version of baccarat where wagers on Banker push when the Banker wins with a three-card total of 7. For convenience, here is the combinatorial analysis for the Banker bet, showing a house edge of 1.0183%.
Dai Bacc comes with three side bets, described as follows:
Tiger 7. This is simply the Dragon/Fortune 7 bet. It pays 40-to-1 when the Banker hand wins with a three-card total of 7.
Ox 6. This side bet appears to be new. It pays 40-to-1 if the Player hand wins with a three-card total of 6. Recall that EZ Baccarat has the "Panda" side bet, which pays 25-to-1 if the Player hand wins with a three-card total of 8. So, the Ox side bet replaces the still-patented Panda.
Kill the Ox/Tiger. This side bet pays 30-to-1 if one of the following happens:
โข The Banker hand is a three-card total of 7 and the Banker hand does not win, or,
โข The Player hand is a three-card total of 6 and the Player hand does not win.
Note that if the hand ends with the Banker having a three-card total of 7 and the Player having a three-card total of 6, then both the Tiger 7 and Kill the Ox/Tiger bet win. In other words, despite its name, these wagers are not mutually exclusive. Well, perhaps one can say "the tiger killed the ox."
I am going to cover card counting each of these three side bets in this post. The reader who is used to the themes behind card counting these sorts of side bets will immediately recognize that the key to each of these is getting a three-card hand. In other words, 8's and 9's are very important cards.
On to the results.
## Card Counting the Tiger 7 Side Bet
The Tiger 7 side bet wins when the Banker hand has a winning three-card total of 7. This side bet is simply the Dragon/Fortune 7 that I analyze in full detail in this post. As a reminder, here is the combinatorial analysis for the Tiger 7 side bet:
In particular:
โข The house edge is 7.611%.
โข The standard deviation is 6.085.
โข The hit frequency is 2.253%.
The card counting system for the Tiger 7 side bet is the well-known "Dragon count":
โข A, 2, 3, T = 0
โข 4, 5, 6, 7 = -1
โข 8, 9 = -2
I re-ran a fresh one hundred million (100,000,000) eight-deck shoes for this post. Here are the results fromย this simulation:
Although this side bet would never be considered a top-level opportunity, it is surprising just how much action it gets internationally. Players are counting it everywhere.
## Card Counting the Ox 6 Side Bet
The Ox 6 side bet wins when the Player hand has a winning three-card total of 6. The following table gives the combinatorial analysis for the Ox 6 side bet:
In particular:
โข The house edge is 10.338%.
โข The standard deviation is 6.000.
โข The hit frequency is 2.187%.
A heuristic argument makes the card counting system we're going to present obvious.
First, an excess of 7's, 8's and 9's leads to two-card initial totals that are standing hands or naturals. Without drawing a third card, there is no way to win the Ox 6 bet. Also, if the Player hand does draw, there is no way to get a three-card total of 6 if the drawn third-card is a 7, 8 or 9. It follows that 7's, 8's and 9's are the bad cards for the Ox 6 side bet.
Second, to win the Ox 6 side bet, the Player hand must draw to a 6. Because the Player hand draws whenever the initial two-card total is 0, 1, 2, 3, 4 or 5, it follows that an excess of 6, 5, 4, 3, 2, A helps with this draw. In other words, A, 2, 3, 4, 5, 6 are the good cards for the Ox 6 side bet.
The following table gives the effect-of-removals together with a card counting system:
As shown in this table, the card counting system we propose for the Ox 6 side bet is:
โข A, 2, 3, 4, 5, 6 = -1
โข 7, 8, 9 = +2
โข T = 0
This card counting system has a solid betting correlation of 0.980. It is very similar to the Tiger count, with three additional indices for A, 2 and 7. It's easy to imagine how a single card counter could co-card count both the Tiger and Ox side bets.
The following table gives the results of a simulation of one hundred million (100,000,000) eight-deck shoes using the card counting system above to count the Ox 6 side bet:
The win-rate for the Ox 6 side bet is comparable to the Tiger 7 (0.642 vs. 0.737 units per 100 hands). The player who counts both simultaneously is winning about 1.379 units per 100 hands, which is quite respectable.
## Card Counting the Kill the Ox/Tiger Side Bet
The final side bet offered with Dai Bacc is the somewhat awkward "Kill the Ox/Tiger" (Kill) side bet. In an austere logical format, winning this side bet can be expressed as follows:
The Kill bet WINS when (the Player gets a non-winning three-card total of 6)ย ORย (the Banker gets a non-winning three-card total of 7).
If you're a math geek, you might recall De Morgan's law that states that the following are logically equivalent propositions:
โข not-(Aย OR B)
โข (not-A) AND (not-B)
Let
โข A = "The Player gets a non-winning three-card total of 6."
โข B = "The Banker gets a non-winning three-card total of 7."
Then winning the Kill bet amounts to the proposition "A OR B." So, losing the Kill bet is equivalent to not-(A OR B). Using De Morgan's law, this is equivalent to:
The Kill bet LOSES when (the Player hand is NOTย a three-card non-winning 6) AND (the Banker hand is NOTย a three-card non-winning 7).
I hope this helps explain the Kill bet, but probably NOT.ย At any rate, here is the combinatorial analysis for the Kill side bet:
In particular:
โข The house edge is 13.756%.
โข The standard deviation is 5.098.
โข The hit frequency is 2.782%.
Card counting the Kill bet is just as intuitive as counting the Tiger 7 or Ox 6: the player wants the hand to end quickly with a big two-card hand (8's and 9's are good cards) and doesn't want drawing cards that lead to big totals (2, 3, 4, 5, 6, 7 are bad cards).
The following table gives the effect-of-removals together with a card counting system for the Kill side bet:
The card counting system we propose for the Kill side bet is:
โข 2, 3, 4, 5, 6, 7 = -1
โข 8, 9, T = +1
โข A = 0
The betting correlation for this system is 0.977. This is remarkably good given the simplicity of the system. There are very few instances of a level-1 count working out for a side bet.
The following table gives the results of a simulation of one hundred million (100,000,000)ย eight-deck shoes using the card counting system above to count the Kill side bet:
Given the high initial house edge forย the Kill side bet, it is surprising just how well this works out for the counter.
## Co-Card Counting the Tiger 7, Ox 6 and Kill the Ox/Tiger Side Bets
The following table summarizes the results above for these three side bets:
It does not seem reasonable for a single counter to keep track of all three wagers, even with a score card. A team approach may be the strongest attempt.
Imagine three counters playing together at the same table. One of them counts the Tiger 7, the second counts the Ox 6 and the last team member counts the Kill side bet. When the counter for any one of these side bets makes a wager on his bet, the other two teammates also make that wager. It follows that each of teammates individually will win at a rate of:
0.737 + 0.642 + 0.550 = 1.929 units per 100 hands
Together, the team of three counters will win at a rate of 3*1.929 = 5.757 units per 100 hands.
For example, if a \$100 maximum wager is permitted on each of these three side bets, then this team will win at a rate of about \$576 per 100 hands. If these players can play multiple spots or the maximums are higher, it's even better. On the other hand, a \$25 maximum will be a snooze fest.
As far as its overall vulnerability, Dai Bacc is not in the same league of vulnerability as Super Pay Egalite or UR Way Egalite, or the similarly designed ZooBac. It isn't even as good as the recently analyzed Tie Max side bet. However, Dai Bacc is a solid opportunity and casinos that offer Dai Bacc should keep a careful eye on the game.
Received his Ph.D. in Mathematics from the University of Arizona in 1983. Eliot has been a Professor of both Mathematics and Computer Science. Eliot retired from academia in 2009. Eliot Jacobson
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Bitmoji Classroom TutorialEyebrow Tutorial for BeginnersVoluptuous PythonBeehive Minecraft
# Watch Lisp Artificial Intelligence Tutorial
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ูุฉ ุงูุฅุฑุดุงุฏูุฉ ุญูู Lisp Artificial Intelligence Tutorial LISP A Artificial Intelligence Programming Language (Lec. no 4) ุจูุงุณุทุฉ Arif Hasan. ุงุญุตู ุนูู ุงูุญู ูู 04:24 ุฏูููุฉ. ุชุงุฑูุฎ ุงููุดุฑ 2019-04-08 04:16:33 ูุงุณุชูุงู
14,010 ร ุนุฏุฏ ุงูุฒูุงุฑุงุช ุ lisp+artificial+intelligence+tutorial
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# The Cohen-Macaulay binomial edge ideals of graphs with n โค 12
In the paper $$S_2$$-condition and Cohen-Macaulay binomial edge ideals by Alberto Lerda, Carla Mascia, Giancarlo Rinaldo and Francesco Romeo, (section 5), we present the following
Theorem
Let $$G$$ be a graph on $$[n]$$, with $$n\leq 12$$. The following conditions are equivalent:
1. $$S/J_G$$ is Cohen-Macaulay;
2. $$S/J_G$$ satisfies $$S_2$$-condition;
3. $$G$$ is accessible;
4. $$G$$ is Strongly unmixed.
In this file we give the set of graphs satisfying the equivalent conditions of the Theorem in nauty format. Moreover we provide the implementation in C++ of the algorithms computing conditions (2), (3) and (4) downloadable from here.
We refer to the article for an in-depth description of the notation and results. In this page you will find a description of the algorithm and an implementation with an example of computation. In this section we also provide an example of computation necessary for the proof of Proposition 2.9.
## Proof
Fixed the number of vertices $$n$$, the proof can be synthesized in 5 steps:
1. compute all non isomorphic graphs on $$[n]$$;
2. keep only the graphs which are indecomposable and unmixed;
3. keep only the ones that are accessible;
4. keep only the ones that are strongly unmixed;
5. verify that the graphs obtained from step 3 and 4 are the same.
In the next sections we will discuss the main structure of the algorithms we implemented:
1. Some details on the tools
2. Unmixedness
3. Accessibility
4. Strongly unmixedness
5. $$S_2$$
## Some details on the tools
The first tools we used is nauty, which generates all non isomorphic graphs with a fixed number of vertices and outputs them in a text format that we can read in our own tools.
We build a collection of utilities which follows the Unix philosophy
Make each program do one thing well
Expect the output of every program to become the input to another, as yet unknown, program
(The Art of Unix Programming)
There are multiple tools isS2, isAccessible, isUnmixed, etc each of which expect input and generate output in the format of nauty, this way we improve composability, we can combine those tools in any order (using Unix pipelines).
E.g. suppose that we are interested in all the unmixed graphs with 9 vertices, we can simply run:
nauty-geng 9 -c | isUnmixed
Altough the previous command works, nowadays it would be interesting to be able to use multiple cores to solve the problem. GNU parallel lets us run a program in parallel without having to deal with pthread,fork,C++ threads or similar explicitly. Using parallel the previous code becomes
nauty-geng 9 -c | parallel --pipe -j8 --N50000 isUnmixed
Letโs analyze the options of parallel
โข --pipe means that the input is read from the pipe (and not from a file)
โข -j8 means that 8 is the maximum number of jobs that will run in parallel
โข --N50000 means that 50000 graphs will be sent to each process
โข isUnmixed is the command we want to run
## Unmixedness
For the unmixedness we implemented Algorithm 1 described in the paper Cohen-Macaulay binomial edge ideals of small deviation. This is the most critical part of the sequence of tools, it is rather slow and it gets a lot of graphs as input (from all the non isomorphic connected graphs we have only filtered away the undecomposable one).
In particular, the main problem is that we have to visit the graph for each subset of vertices to find the cutsets. Obviously, we visit the graph only one time for a fixed subset of vertices, we memorize the result and reuse it every time we need it. Moreover, we donโt compute all the cutset at the begininng, every time we find a cutset we verify the property $$c(T)=|T|+1$$ and we stop as soon as possibile.
## Accessibility
To verify the accessibility, again, we have to iterate over each subset to find the cutsets. As well as the unmixedness, we implemented the accessibility using memoization, we keep a vector where each index correspond to a subset of vertices (in C++ this can be implemented easily thanks to bitwise operations) and iterate over them in inclusion order (which is the usual ordering between numbers, removing a vertex from a subset of vertices means removing a 1 in the bitmask, which gives a smaller number). We use the fact that the property of being an accessible system is defined in an inductive way.
## Strongly unmixedness
The implementation of strongly unmixedness, thank to recursion, follows exactly the definition. It could have been optimized more, but for what we need it is enough.
First of all, we verify whether the connected components are complete graphs. To do this we look at the number of edges in each connected component and we compare it with the number of edges of the complete graph ($$q=p(p-1)/2$$). If this is true we return true.
Then, we verify whether $$J_G$$ is unmixed, if this is not true we return false
Finally we have to look for a cutvertex such that $$G\setminus\{v\}$$, $$G_\bar{v}\setminus\{v\}$$ and $$G_\bar{v}$$ are strongly unmixed. And in this step we use recursion.
## $$S_2$$
We can summarize the algorithm in the following steps
1. Iterate over the dimension of the faces from $$-1$$ (the empty set face) to the dimension of the complex-1
2. For each size we generate all faces which belong to the complex
3. For each face we generate the link
4. Finally, we check if the link is connected looking at the 1-skeleton
We didinโt need $$S_2$$ for the purpose of the proof, it will be useful for the research. It is rather difficult to understand why a given ideal is not $$S_2$$, for this reason a computer program is useful.
## Implementation and the case n=7
In this section we give the results of our computation by the routines that we implemented in C++ (using g++ of gcc ver. 7.5) whose sources are downloadable from here.
The program in C++ has been developed under Linux on Ubuntu distribution (tested on versions 18.04). It uses standard library and nauty package and with some effort can work on any platforms. To use it you have to download, and extract by
tar -xzvf s2binomials.tgz
move to the folder
cd s2binomials
and then compile the c++ program by
make
Moreover the program use an external one that is nauty-geng (that generates graphs up to isomorphisms). So it expects to have the binary file with this name in the system.
.
As an example we focus on the set of connected graphs with 7 vertices. We verify
1. The equivalence between accessible, $$S_2$$ and strongly unmixed ones
2. A minimal face with respect to the dimension of an unmixed graph foe which the corresponding link is not connected (Proposition 2.9).
We show the input and the output in the console. We first generate the graphs related to unmixed binomial edge ideals.
scripts/generate_unmixed.sh 7
The output file is outs/g7/g7unmixed.nty and it contains 23 graphs in nauty format.
Now we filter graphs in the file with respect to the 3 conditions: accessible, $$S_2$$ and strongly unmixed. The 3 files obtained are:outs/g7/g7accessible.nty, outs/g7/g7s2.nty and outs/g7/g7stronglyUnmixed.nty. Then we compare the 3 files and we observe that are identical. Here is the input
scripts/compare_accessible_s2_stronglyunmixed.sh 7
and the corresponding output
For each class of graphs the first line contains the time needed to complete the computation while the second one shows the number of graphs which have been found.
Accessible:
0:00.03 real,0.01 user,0.00 sys
15
S2:
0:00.09 real,0.09 user,0.00 sys
15
Strongly unmixed:
0:00.01 real,0.00 user,0.00 sys
15
File differences (there should be no differences)
We observe that there are 15 accessible/S2/strongly unmixed graphs and the 3 files obtained are identical since thre is no extra output.
Now, for all graphs that are in g7unmixed.nty that are not $$S_2$$ we compute a minimal face with respect to the dimension whose corresponding link is not connected. The input is
.
scripts/checkS2.sh 7
and the corresponding output is
--------------
The face
x: 3 4 5
y: 3 4 5
of dimension 5 has a disconnected link with facets
x:
y:1 2
x: 6 7
y:
with graph G {1--5;2--5;1--6;2--6;3--6;1--7;2--7;4--7}
--------------
The face
x: 6
y:1 2 3 4 6
of dimension 5 has a disconnected link with facets
x:1 2
y:
x: 7
y: 5
with graph G {1--5;2--5;3--6;4--6;5--6;1--7;2--7;3--7;5--7;6--7}
--------------
The face
x: 5
y:1 2 3 4 5
of dimension 5 has a disconnected link with facets
x: 3 4
y:
x: 6 7
y:
with graph G {1--5;2--5;3--6;4--6;5--6;1--7;2--7;3--7;4--7;5--7;6--7}
--------------
The face
x: 5
y:1 2 3 4 5
of dimension 5 has a disconnected link with facets
x: 3 4
y:
x: 6 7
y:
with graph G {1--5;2--5;1--6;2--6;3--6;4--6;5--6;1--7;2--7;3--7;4--7;5--7;6--7}
--------------
The face
x: 3 4
y:1 2 3 4
of dimension 5 has a disconnected link with facets
x:1 2
y:
x: 5 6
y:
with graph G {1--5;2--5;3--5;1--6;2--6;3--6;5--6;1--7;2--7;3--7;4--7;5--7;6--7}
--------------
The face
x: 4
y:1 2 3 4
of dimension 4 has a disconnected link with facets
x:1 2 3
y:
x: 5 6 7
y:
with graph G {1--5;2--5;3--5;4--5;1--6;2--6;3--6;4--6;5--6;1--7;2--7;3--7;4--7;5--7;6--7}
--------------
The face
x: 5
y:1 2 3 4 5
of dimension 5 has a disconnected link with facets
x: 3 4
y:
x: 6 7
y:
with graph G {1--4;2--5;1--6;2--6;3--6;4--6;5--6;1--7;2--7;3--7;4--7;5--7;6--7}
--------------
The face
x: 5
y:1 2 3 4 5
of dimension 5 has a disconnected link with facets
x: 2 3
y:
x: 6 7
y:
with graph G {1--4;1--5;4--5;1--6;2--6;3--6;4--6;5--6;1--7;2--7;3--7;4--7;5--7;6--7}
As expected the graphs are $$23-15=8$$. Morever the 6th is (See Example 2.10 and Figure 1):
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# How Many Grams Are in a Tablespoon: Grams to Tablespoon and Tablespoon to Grams
## Conversion Formulas
Convert tablespoons to grams and grams to tablespoons using our conversion chart and conversion calculator, and learn the relationship between these two units.
Conversion of tablespoons to grams and grams to tablespoons is an easy one, but one must be aware that the tablespoons are used to measure the volume, while the grams are used to measure the weight.
In order to convert tablespoons to grams and vice versa, we must make an assumption - tablespoons are used to measure volume, and to convert volume to weight, we must assume that the tablespoons are used to measure the volume of water, which has a density of ~1g/milliliter.
That being said, we can write:
1 US Tablespoon = 1/2 US Fluid Ounces = 14.7868 milliliters = 14.7868 grams
1 Imp Tablespoon = 0.625 Imp Fluid Ounces = 17.7582 milliliters = 17.7582 grams
It must be emphasized that if You have a liquid that is less dense than the water (oil, gas/petrol, etc.) or denser than the water (salty water, for example), You also have to take the density of the liquid into account.
## Tablespoons to Grams and Grams to Tablespoons Conversion Examples
Here are a few conversion examples that may be helpful:
### How many tablespoons are in 100 grams:
V(US tbsp) = M(grams)ย / 14.7868 = 100 / 14.7868 = 6.76278 US tbsp
V(Imp tbsp) = M(grams) / 17.7582 =ย 100 / 17.7582 = 5.63120 Imp tbsp
So, if You have 100g (of water), that would be 6.767278 US tablespoons or 5.63120 Imp tablespoons.
etc.
## Tablespoons to Grams and Grams to Tablespoons Conversion Calculator
In order to convert tablespoons to grams and grams to tablespoons, feel free to use this conversion calculator - write the value that You have and click 'Calculate' to convert it:
US Tablespoons:
Grams:
Grams:
US Tablespoons:
### Imp. Tablespoons to Grams
Imp. Tablespoons:
Grams:
### Grams to Imp. Tablespoons
Grams:
Imp. Tablespoons:
## Tablespoons to Grams and Grams to Tablespoons Conversion Charts
Here are some tablespoons to grams and grams to tablespoons quick conversion charts to aid You with units' conversions:
Grams US Tablespoons IMP Tablespoons 1 0.06762 0.056312 5 0.33813 0.281560 10 0.67627 0.563120 14.7868 1.0 0.832674 15 1.01441 0.844680 20 1.35255 1.126240 25 1.69069 1.407800 29.5736 2 1.665348 44.3604 3 2.498023 50 3.38139 2.815600 59.1472 4 3.330697 73.934 5 4.163372 100 6.76278 5.631201 147.868 10 8.326744 250 16.9069 14.07800 500 33.8139 28.15600 1000 67.6278 56.31201
For more values, please, use the conversion calculators.
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# Null space of 3x3 matrix
#### junglebeast
There are 2 issues I want to talk about in this post.
(1) General algorithm for gauss-jordan elimination computation of null space
(2) Closed form solution to 3x3 null space
Following the example here,
I thought a general algorithm to compute the null space would be to
1) augment with 0 vector on the right
2) compute gauss-jordan elimination
3) take 2nd to last column, and fill in extra elements with 1's to get the null space
This works in the example provided there. However, on the next example, step 3 needs to be changed...
1, 0, 1
2, 1, 3
1, 1, 2
which has a null space of 1, 1, -1
using gauss-jordan elimination, the closest I can get is
1, 0.5, 1.5, 0
0, 1, 1, 0
0, 0, 0, 0
x1 = -0.5 x2 - 1.5 x3
x2 = -x3
[1, 1, -1]
This gives me the right null space, but step #3 of my above method clearly wasn't right..how can I generalize step 3 into a straight-forward algorithm?
Now onto my second issue.
I found a method to compute the eigenvector corresponding to an eigenvalue of a 3x3 matrix closed form. It is simply:
a1*a5 - a2*(a4-e)
a1*a2 - a5*(a0-e)
(a0-e)*(a4-e) - a1*a1
Removing the 'e's, this is essentially a short cut to get the null space. However it doesn't seem to work for non-symmetric matrices. I feel like there should be a similar method that works for non-symmetric 3x3's...which could be used to avoid the SVD method in this case
Last edited by a moderator:
Related Linear and Abstract Algebra News on Phys.org
#### fresh_42
Mentor
2018 Award
Algorithms to solve linear equation systems are well known and are part of basic computer science courses. The fact that there might be examples with faster algorithms is well known, too, and we cannot reason upon examples. This doesn't allow a generalization which would be needed to talk about an algorithm.
If you are interested in the subject, then you might want to read about the improvements on the matrix exponent: https://en.wikipedia.org/wiki/Strassen_algorithm#Asymptotic_complexity
"Null space of 3x3 matrix"
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## 65 Reputation
1 years, 196 days
## Help regarding solution...
Maple 2019
Can anyone help me to find the correct solution up to 5 decimal points and how to implement the newton method on the same expression rather than applying fsolve command.pls find mt attachment.
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Maple
I have an implicit relation in lambda and x, so I am unable to find the value of x in terms of lambda so I generate a set of data points for different values ofย ย x=0,0.1...0.9 and obtained the value of lambda. so my question is how to write a polynomial in terms of lambda for generated data by using curve fitting and how do I checked that the obtained polynomial is correct? pls have look at my maple sheet
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Maple
Hi everyone, I am trying to dsolve a ode but could not get the answer, could anyone please help me or guide me what's wrong with my worksheet
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Maple 2019
Hi, i am using solve and solve command to find the root but when i used fsolve command to separate only real root, could separate all roots, can anyone correct me, please
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# Building A Web Based Business โ 4 Top Tips
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uppose it costs the university \$6200 to offer a section of a class (regardless of the number of students enrolled so this is a fixed cost)
The enrollment in a course offered by the College of Business is random and is described by the foll Show more The enrollment in a course offered by the College of Business is random and is described by the following probability distribution: there is a 9% chance of 18 students 22% chance of having 28 students 34% chance of having 38 students and otherwise there are 49 students. Suppose it costs the university \$6200 to offer a section of a class (regardless of the number of students enrolled so this is a fixed cost) and tuition is \$340 per student per class. What is the expected profit or loss for the University on this class? Show less
PLACE THIS ORDER OR A SIMILAR ORDER WITH BEST NURSING TUTORS TODAY AND GET AN AMAZING DISCOUNT
The post uppose it costs the university \$6200 to offer a section of a class (regardless of the number of students enrolled so this is a fixed cost) appeared first on BEST NURSING TUTORS .
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``` how to find potential on a perticular point in d complex electric circuit??
perticularly whn resistors and capacitors both r connected
```
5 years ago
Share
``` Dear student,
Potential difference between two points in circuit is the energy lost by the charge in being transferred from one point to another. For example, potential difference between A and B is found with following formula;
VAB=VB-VA=โฮต-โi.R
This formula shows the energy lost by charge moving from point A to point B.
If the direction of the current and current passing through the resistor are the same then we take i.R as "+", if they are in opposite directions then we take i.R as "-".
VAB=VB-VA= -ฮต2-(+i.R1+i.R2)
VCB=VB-VC= -ฮต3-(-i.R3)
All the best.
Winย exciting giftsย by answering the questions on Discussion Forum. So help discuss any query on askiitians forum and become an Elite Expert League askiitian.
Sagar Singh
B.Tech, IIT Delhi
```
5 years ago
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More Questions On Electric Current
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### Exponential map transform
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โข 3f
โข Posts: 1984
#### Exponential map transform
ยซ on: August 05, 2018, 05:02:17 AM ยป
Not much activity here, UF seems to be unpopular here. Here's a UF transformation formula implementing the "exponential map": https://fractalforums.org/fractal-movie-gallery/19/emndl-vs-nanomb-mandelbrot-set-zooms/1691/msg8575#new.
Output sample is random zoom past a few minibrots and making a tree.
Code: [Select]
ExponentialMap {;; Vertical exponential transform.; Make narrow high window (say 100X800), zoom, put point of interest at bottom.; Turn on. Tweak with the 2 controls, and select appropriate width (depends on image, ; narrower for deeper zooms).;global:ย if (4 * #height < 3 * #width)ย ย pixeldim = 3/#magn/#heightย elseย ย pixeldim = 4/#magn/#widthย endifย w = #width * pixeldimย h = #height * pixeldimย cc = #centerย c0 = cc - 1i/2 * h +0i*w -w/4 * @shย b = w/h*log(#magn) *@bย a = 1i*w/btransform:ย ย if @isonย ย ย c = #pixelย ย ย dc = a*(exp(-1i*b/w*(c-c0)))ย ย ย #pixel = c0+dcย ย endifdefault:ย title = "Exponential map"ย param isonย ย caption = "On"ย ย default = falseย endparamย float param bย ย caption = "vert. control"ย ย default = 1.2ย endparamย float param shย ย caption = "hor. shift"ย ย default = 0ย endparam}
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https://www.lmfdb.org/ArtinRepresentation/15.133249137678121328919445504.240.a.h
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crawl-data/CC-MAIN-2020-45/segments/1603107917390.91/warc/CC-MAIN-20201031092246-20201031122246-00302.warc.gz
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# Properties
Label 15.133...504.240.a.h Dimension $15$ Group $\PSL(2,16)$ Conductor $1.332\times 10^{26}$ Root number $1$ Indicator $1$
# Related objects
## Basic invariants
Dimension: $15$ Group: $\PSL(2,16)$ Conductor: $$133\!\cdots\!504$$$$\medspace = 2^{30} \cdot 137^{8}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 17.1.133249137678121328919445504.1 Galois orbit size: $8$ Smallest permutation container: 240 Parity: even Determinant: 1.1.1t1.a.a Projective image: $\PSL(2,16)$ Projective field: Galois closure of 17.1.133249137678121328919445504.1
## Defining polynomial
$f(x)$ $=$ $x^{17} - 3 x^{16} - 4 x^{14} + 12 x^{13} + 24 x^{12} + 12 x^{11} - 28 x^{10} - 90 x^{9} - 74 x^{8} + 116 x^{6} + 132 x^{5} + 72 x^{4} + 28 x^{3} + 12 x^{2} + 5 x + 1$.
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 269.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $x^{15} + 2 x^{6} + 8 x^{5} + 15 x^{4} + 9 x^{3} + 7 x^{2} + 18 x + 18$
Roots:
$r_{ 1 }$ $=$ $2 + 19\cdot 23 + 21\cdot 23^{2} + 9\cdot 23^{3} + 17\cdot 23^{4} + 5\cdot 23^{5} + 20\cdot 23^{6} + 4\cdot 23^{7} + 7\cdot 23^{8} + 5\cdot 23^{9} + 4\cdot 23^{10} + 13\cdot 23^{11} + 3\cdot 23^{12} + 2\cdot 23^{13} + 2\cdot 23^{14} + 7\cdot 23^{15} + 16\cdot 23^{16} + 13\cdot 23^{17} + 12\cdot 23^{18} + 3\cdot 23^{19} + 22\cdot 23^{20} + 9\cdot 23^{22} + 20\cdot 23^{24} + 9\cdot 23^{25} + 18\cdot 23^{26} + 3\cdot 23^{27} + 19\cdot 23^{28} + 23^{29} + 7\cdot 23^{30} + 22\cdot 23^{31} + 8\cdot 23^{32} + 19\cdot 23^{33} + 7\cdot 23^{34} + 4\cdot 23^{35} + 12\cdot 23^{37} + 18\cdot 23^{38} + 14\cdot 23^{39} + 10\cdot 23^{40} + 11\cdot 23^{41} + 10\cdot 23^{42} + 18\cdot 23^{43} + 11\cdot 23^{44} + 18\cdot 23^{45} + 3\cdot 23^{46} + 17\cdot 23^{47} + 3\cdot 23^{48} + 22\cdot 23^{49} + 22\cdot 23^{50} + 16\cdot 23^{51} + 15\cdot 23^{52} + 10\cdot 23^{53} + 19\cdot 23^{54} + 4\cdot 23^{55} + 9\cdot 23^{56} + 13\cdot 23^{57} + 22\cdot 23^{58} + 21\cdot 23^{59} + 7\cdot 23^{60} + 4\cdot 23^{61} + 14\cdot 23^{62} + 18\cdot 23^{63} + 15\cdot 23^{64} + 9\cdot 23^{65} + 10\cdot 23^{66} + 18\cdot 23^{67} + 6\cdot 23^{68} + 4\cdot 23^{69} + 20\cdot 23^{70} + 21\cdot 23^{71} + 7\cdot 23^{72} + 14\cdot 23^{73} + 20\cdot 23^{74} + 8\cdot 23^{75} + 6\cdot 23^{76} + 10\cdot 23^{77} + 20\cdot 23^{78} + 10\cdot 23^{79} + 23^{80} + 2\cdot 23^{81} + 14\cdot 23^{82} + 4\cdot 23^{83} + 14\cdot 23^{84} + 4\cdot 23^{85} + 22\cdot 23^{86} + 4\cdot 23^{87} + 9\cdot 23^{88} + 3\cdot 23^{89} + 13\cdot 23^{90} + 6\cdot 23^{91} + 4\cdot 23^{92} + 3\cdot 23^{93} + 20\cdot 23^{94} + 15\cdot 23^{95} + 9\cdot 23^{96} + 20\cdot 23^{97} + 3\cdot 23^{98} + 23^{99} + 11\cdot 23^{100} + 12\cdot 23^{101} + 6\cdot 23^{102} + 19\cdot 23^{103} + 22\cdot 23^{104} + 15\cdot 23^{105} + 18\cdot 23^{106} + 9\cdot 23^{107} + 14\cdot 23^{108} + 4\cdot 23^{109} + 4\cdot 23^{110} + 10\cdot 23^{111} + 11\cdot 23^{112} + 23^{113} + 6\cdot 23^{114} + 3\cdot 23^{115} + 10\cdot 23^{116} + 22\cdot 23^{117} + 19\cdot 23^{118} + 18\cdot 23^{119} + 23^{120} + 20\cdot 23^{121} + 19\cdot 23^{122} + 23^{123} + 4\cdot 23^{124} + 22\cdot 23^{125} + 5\cdot 23^{126} + 19\cdot 23^{127} + 22\cdot 23^{128} + 22\cdot 23^{129} + 10\cdot 23^{130} + 17\cdot 23^{131} + 19\cdot 23^{132} + 10\cdot 23^{133} + 21\cdot 23^{134} + 6\cdot 23^{135} + 17\cdot 23^{136} + 5\cdot 23^{137} + 23^{138} + 19\cdot 23^{139} + 8\cdot 23^{140} + 19\cdot 23^{141} + 22\cdot 23^{142} + 8\cdot 23^{143} + 15\cdot 23^{144} + 2\cdot 23^{145} + 19\cdot 23^{146} + 17\cdot 23^{147} + 13\cdot 23^{148} + 10\cdot 23^{149} + 8\cdot 23^{150} + 16\cdot 23^{151} + 6\cdot 23^{152} + 8\cdot 23^{153} + 6\cdot 23^{154} + 12\cdot 23^{155} + 20\cdot 23^{156} + 5\cdot 23^{157} + 17\cdot 23^{158} + 22\cdot 23^{159} + 21\cdot 23^{160} + 6\cdot 23^{161} + 12\cdot 23^{162} + 8\cdot 23^{163} + 2\cdot 23^{164} + 10\cdot 23^{165} + 20\cdot 23^{166} + 10\cdot 23^{167} + 8\cdot 23^{168} + 8\cdot 23^{169} + 2\cdot 23^{170} + 16\cdot 23^{171} + 2\cdot 23^{172} + 7\cdot 23^{173} + 18\cdot 23^{174} + 20\cdot 23^{175} + 20\cdot 23^{176} + 18\cdot 23^{177} + 14\cdot 23^{178} + 5\cdot 23^{179} + 11\cdot 23^{180} + 23^{181} + 12\cdot 23^{182} + 22\cdot 23^{183} + 21\cdot 23^{184} + 6\cdot 23^{185} + 12\cdot 23^{186} + 8\cdot 23^{187} + 16\cdot 23^{188} + 9\cdot 23^{189} + 22\cdot 23^{190} + 6\cdot 23^{191} + 15\cdot 23^{192} + 20\cdot 23^{193} + 6\cdot 23^{194} + 13\cdot 23^{195} + 18\cdot 23^{196} + 17\cdot 23^{197} + 8\cdot 23^{198} + 12\cdot 23^{199} + 19\cdot 23^{202} + 18\cdot 23^{203} + 12\cdot 23^{204} + 23^{205} + 11\cdot 23^{206} + 20\cdot 23^{207} + 20\cdot 23^{208} + 11\cdot 23^{209} + 5\cdot 23^{210} + 19\cdot 23^{211} + 2\cdot 23^{212} + 8\cdot 23^{213} + 10\cdot 23^{214} + 5\cdot 23^{215} + 21\cdot 23^{216} + 21\cdot 23^{217} + 3\cdot 23^{218} + 9\cdot 23^{219} + 19\cdot 23^{220} + 22\cdot 23^{221} + 5\cdot 23^{222} + 19\cdot 23^{223} + 13\cdot 23^{224} + 20\cdot 23^{225} + 13\cdot 23^{226} + 20\cdot 23^{227} + 16\cdot 23^{228} + 20\cdot 23^{229} + 23^{230} + 4\cdot 23^{231} + 6\cdot 23^{232} + 16\cdot 23^{233} + 10\cdot 23^{234} + 11\cdot 23^{235} + 23^{236} + 22\cdot 23^{237} + 14\cdot 23^{238} + 21\cdot 23^{239} + 9\cdot 23^{240} + 10\cdot 23^{241} + 4\cdot 23^{242} + 4\cdot 23^{243} + 7\cdot 23^{244} + 15\cdot 23^{245} + 15\cdot 23^{246} + 19\cdot 23^{247} + 3\cdot 23^{248} + 20\cdot 23^{249} + 12\cdot 23^{250} + 5\cdot 23^{251} + 21\cdot 23^{252} + 14\cdot 23^{253} + 4\cdot 23^{254} + 6\cdot 23^{255} + 19\cdot 23^{256} + 17\cdot 23^{257} + 12\cdot 23^{258} + 18\cdot 23^{259} + 8\cdot 23^{260} + 15\cdot 23^{261} + 22\cdot 23^{262} + 4\cdot 23^{263} + 17\cdot 23^{264} + 6\cdot 23^{265} + 8\cdot 23^{266} + 22\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 2 }$ $=$ $6 + 22\cdot 23 + 7\cdot 23^{2} + 22\cdot 23^{3} + 22\cdot 23^{4} + 16\cdot 23^{5} + 9\cdot 23^{6} + 14\cdot 23^{7} + 6\cdot 23^{8} + 21\cdot 23^{9} + 4\cdot 23^{10} + 7\cdot 23^{11} + 4\cdot 23^{12} + 4\cdot 23^{13} + 2\cdot 23^{14} + 6\cdot 23^{15} + 23^{16} + 21\cdot 23^{17} + 14\cdot 23^{18} + 7\cdot 23^{19} + 11\cdot 23^{20} + 21\cdot 23^{21} + 11\cdot 23^{22} + 20\cdot 23^{23} + 19\cdot 23^{24} + 2\cdot 23^{25} + 4\cdot 23^{26} + 21\cdot 23^{27} + 8\cdot 23^{28} + 16\cdot 23^{29} + 2\cdot 23^{30} + 5\cdot 23^{31} + 11\cdot 23^{32} + 7\cdot 23^{33} + 8\cdot 23^{34} + 19\cdot 23^{35} + 8\cdot 23^{36} + 7\cdot 23^{37} + 20\cdot 23^{38} + 17\cdot 23^{39} + 10\cdot 23^{40} + 5\cdot 23^{41} + 19\cdot 23^{42} + 16\cdot 23^{43} + 22\cdot 23^{44} + 7\cdot 23^{45} + 19\cdot 23^{46} + 7\cdot 23^{47} + 13\cdot 23^{48} + 6\cdot 23^{49} + 11\cdot 23^{50} + 13\cdot 23^{51} + 12\cdot 23^{52} + 11\cdot 23^{53} + 8\cdot 23^{54} + 10\cdot 23^{55} + 2\cdot 23^{56} + 18\cdot 23^{57} + 14\cdot 23^{58} + 17\cdot 23^{59} + 19\cdot 23^{60} + 7\cdot 23^{61} + 13\cdot 23^{62} + 5\cdot 23^{63} + 21\cdot 23^{64} + 6\cdot 23^{65} + 20\cdot 23^{66} + 8\cdot 23^{67} + 20\cdot 23^{68} + 22\cdot 23^{69} + 11\cdot 23^{70} + 19\cdot 23^{71} + 8\cdot 23^{72} + 11\cdot 23^{73} + 20\cdot 23^{74} + 22\cdot 23^{75} + 11\cdot 23^{76} + 7\cdot 23^{77} + 19\cdot 23^{78} + 6\cdot 23^{79} + 21\cdot 23^{80} + 3\cdot 23^{81} + 6\cdot 23^{82} + 3\cdot 23^{83} + 11\cdot 23^{84} + 20\cdot 23^{86} + 22\cdot 23^{87} + 3\cdot 23^{88} + 14\cdot 23^{89} + 18\cdot 23^{90} + 21\cdot 23^{91} + 11\cdot 23^{92} + 21\cdot 23^{93} + 8\cdot 23^{94} + 9\cdot 23^{95} + 22\cdot 23^{96} + 18\cdot 23^{97} + 23^{98} + 6\cdot 23^{99} + 8\cdot 23^{100} + 15\cdot 23^{101} + 17\cdot 23^{102} + 12\cdot 23^{103} + 16\cdot 23^{104} + 4\cdot 23^{105} + 23^{106} + 7\cdot 23^{107} + 22\cdot 23^{108} + 15\cdot 23^{109} + 4\cdot 23^{110} + 4\cdot 23^{111} + 16\cdot 23^{112} + 10\cdot 23^{113} + 19\cdot 23^{114} + 22\cdot 23^{115} + 3\cdot 23^{116} + 9\cdot 23^{117} + 2\cdot 23^{118} + 7\cdot 23^{119} + 21\cdot 23^{120} + 18\cdot 23^{121} + 14\cdot 23^{122} + 4\cdot 23^{123} + 22\cdot 23^{124} + 18\cdot 23^{125} + 7\cdot 23^{126} + 6\cdot 23^{127} + 5\cdot 23^{128} + 18\cdot 23^{129} + 14\cdot 23^{130} + 3\cdot 23^{131} + 2\cdot 23^{132} + 3\cdot 23^{133} + 8\cdot 23^{134} + 13\cdot 23^{135} + 17\cdot 23^{136} + 15\cdot 23^{137} + 3\cdot 23^{138} + 22\cdot 23^{139} + 9\cdot 23^{141} + 19\cdot 23^{142} + 7\cdot 23^{143} + 18\cdot 23^{144} + 18\cdot 23^{145} + 21\cdot 23^{146} + 7\cdot 23^{147} + 15\cdot 23^{148} + 13\cdot 23^{149} + 6\cdot 23^{150} + 3\cdot 23^{151} + 7\cdot 23^{152} + 22\cdot 23^{153} + 22\cdot 23^{154} + 13\cdot 23^{155} + 23^{156} + 19\cdot 23^{157} + 9\cdot 23^{158} + 11\cdot 23^{159} + 8\cdot 23^{160} + 6\cdot 23^{161} + 13\cdot 23^{162} + 9\cdot 23^{163} + 10\cdot 23^{164} + 23^{165} + 17\cdot 23^{166} + 21\cdot 23^{167} + 6\cdot 23^{168} + 5\cdot 23^{169} + 6\cdot 23^{170} + 18\cdot 23^{171} + 4\cdot 23^{172} + 18\cdot 23^{173} + 11\cdot 23^{174} + 15\cdot 23^{175} + 13\cdot 23^{176} + 23^{177} + 6\cdot 23^{178} + 5\cdot 23^{179} + 6\cdot 23^{180} + 16\cdot 23^{181} + 18\cdot 23^{182} + 23^{183} + 14\cdot 23^{184} + 19\cdot 23^{185} + 22\cdot 23^{186} + 23^{187} + 17\cdot 23^{188} + 2\cdot 23^{189} + 15\cdot 23^{190} + 18\cdot 23^{191} + 2\cdot 23^{192} + 16\cdot 23^{193} + 3\cdot 23^{194} + 17\cdot 23^{195} + 23^{196} + 2\cdot 23^{197} + 22\cdot 23^{198} + 13\cdot 23^{199} + 7\cdot 23^{200} + 5\cdot 23^{201} + 5\cdot 23^{202} + 6\cdot 23^{203} + 18\cdot 23^{204} + 16\cdot 23^{205} + 4\cdot 23^{206} + 12\cdot 23^{207} + 19\cdot 23^{208} + 12\cdot 23^{209} + 17\cdot 23^{210} + 9\cdot 23^{211} + 3\cdot 23^{212} + 10\cdot 23^{213} + 10\cdot 23^{214} + 5\cdot 23^{215} + 6\cdot 23^{216} + 18\cdot 23^{217} + 21\cdot 23^{218} + 22\cdot 23^{219} + 18\cdot 23^{220} + 18\cdot 23^{221} + 5\cdot 23^{222} + 10\cdot 23^{223} + 21\cdot 23^{224} + 21\cdot 23^{225} + 14\cdot 23^{226} + 9\cdot 23^{227} + 19\cdot 23^{228} + 6\cdot 23^{229} + 3\cdot 23^{230} + 10\cdot 23^{232} + 7\cdot 23^{233} + 4\cdot 23^{234} + 13\cdot 23^{235} + 20\cdot 23^{236} + 11\cdot 23^{237} + 22\cdot 23^{238} + 8\cdot 23^{239} + 10\cdot 23^{240} + 11\cdot 23^{241} + 12\cdot 23^{242} + 2\cdot 23^{243} + 2\cdot 23^{244} + 3\cdot 23^{245} + 20\cdot 23^{246} + 23^{247} + 20\cdot 23^{248} + 8\cdot 23^{249} + 23^{250} + 13\cdot 23^{251} + 15\cdot 23^{252} + 3\cdot 23^{253} + 3\cdot 23^{255} + 10\cdot 23^{256} + 21\cdot 23^{257} + 6\cdot 23^{258} + 19\cdot 23^{259} + 4\cdot 23^{260} + 2\cdot 23^{261} + 15\cdot 23^{262} + 17\cdot 23^{263} + 21\cdot 23^{264} + 11\cdot 23^{265} + 20\cdot 23^{266} + 19\cdot 23^{267} + 12\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 3 }$ $=$ $9 a^{14} + 11 a^{13} + 20 a^{12} + 15 a^{11} + 15 a^{10} + 21 a^{9} + 8 a^{8} + a^{7} + 10 a^{6} + 10 a^{5} + 21 a^{4} + 21 a^{3} + 3 a^{2} + 19 a + 20 + \left(2 a^{14} + 8 a^{13} + 5 a^{12} + 6 a^{11} + 18 a^{10} + 15 a^{9} + 3 a^{8} + 3 a^{7} + 15 a^{6} + 20 a^{5} + 13 a^{4} + 4 a^{3} + 8 a^{2} + 21 a\right)\cdot 23 + \left(20 a^{14} + 9 a^{13} + 13 a^{12} + 3 a^{11} + 10 a^{10} + 10 a^{9} + 15 a^{8} + 8 a^{7} + 5 a^{6} + 13 a^{5} + 2 a^{4} + 12 a^{3} + 15 a^{2} + 16 a + 12\right)\cdot 23^{2} + \left(15 a^{13} + 9 a^{12} + 7 a^{11} + 17 a^{10} + 3 a^{9} + 16 a^{8} + 6 a^{7} + 19 a^{6} + 16 a^{5} + 5 a^{2} + 3 a + 13\right)\cdot 23^{3} + \left(19 a^{14} + 10 a^{13} + 6 a^{12} + 19 a^{11} + 16 a^{10} + 5 a^{9} + 7 a^{8} + 11 a^{7} + 7 a^{6} + 9 a^{5} + 12 a^{4} + 15 a^{3} + 22 a^{2} + 20 a + 1\right)\cdot 23^{4} + \left(12 a^{14} + 20 a^{13} + 17 a^{12} + 10 a^{11} + 14 a^{10} + 8 a^{9} + 19 a^{8} + 20 a^{6} + 18 a^{5} + 10 a^{4} + 17 a^{3} + 12 a^{2} + 6 a + 20\right)\cdot 23^{5} + \left(15 a^{14} + 21 a^{13} + 6 a^{12} + 11 a^{11} + 19 a^{10} + 17 a^{9} + 18 a^{8} + a^{7} + 3 a^{6} + 18 a^{5} + 17 a^{4} + 4 a^{3} + 21 a^{2} + 16 a + 17\right)\cdot 23^{6} + \left(3 a^{13} + 3 a^{12} + 13 a^{11} + 9 a^{10} + 4 a^{8} + 2 a^{7} + 19 a^{6} + 16 a^{5} + 21 a^{4} + 11 a^{3} + 4 a^{2} + 7 a + 9\right)\cdot 23^{7} + \left(22 a^{14} + 9 a^{13} + 6 a^{11} + 13 a^{10} + 21 a^{9} + 5 a^{8} + 9 a^{7} + 13 a^{6} + 11 a^{5} + 7 a^{3} + 17 a^{2} + 17 a + 9\right)\cdot 23^{8} + \left(14 a^{14} + 3 a^{13} + 16 a^{12} + 4 a^{11} + 20 a^{10} + 16 a^{9} + 7 a^{8} + 14 a^{6} + 9 a^{4} + 13 a^{3} + 18 a^{2} + 15 a + 1\right)\cdot 23^{9} + \left(22 a^{14} + 12 a^{13} + 15 a^{12} + 10 a^{11} + 12 a^{10} + 18 a^{9} + 11 a^{8} + 5 a^{7} + 19 a^{6} + 4 a^{5} + 12 a^{4} + 19 a^{3} + 11 a^{2} + 2 a + 17\right)\cdot 23^{10} + \left(14 a^{14} + 2 a^{13} + 9 a^{12} + 18 a^{11} + 20 a^{10} + 7 a^{9} + 5 a^{8} + 13 a^{7} + 22 a^{6} + 22 a^{5} + 3 a^{4} + 7 a^{3} + 6 a^{2} + 22 a + 8\right)\cdot 23^{11} + \left(a^{14} + 4 a^{13} + 20 a^{12} + 18 a^{11} + 2 a^{10} + 22 a^{9} + 2 a^{8} + 16 a^{7} + 22 a^{6} + 8 a^{5} + 8 a^{4} + 18 a^{3} + 8 a^{2} + 17 a + 17\right)\cdot 23^{12} + \left(6 a^{14} + 6 a^{13} + 11 a^{12} + 4 a^{9} + 17 a^{8} + 7 a^{7} + 21 a^{6} + 18 a^{5} + 8 a^{4} + 21 a^{2} + 12 a + 8\right)\cdot 23^{13} + \left(12 a^{14} + 10 a^{13} + 18 a^{12} + 3 a^{11} + 11 a^{10} + 20 a^{8} + 16 a^{7} + a^{6} + 10 a^{5} + 21 a^{3} + 10 a^{2} + 5 a + 18\right)\cdot 23^{14} + \left(21 a^{14} + 20 a^{13} + 15 a^{12} + 16 a^{11} + 9 a^{10} + 12 a^{9} + 8 a^{8} + a^{7} + 21 a^{6} + 4 a^{5} + 5 a^{4} + 7 a^{3} + 20 a^{2} + 10 a + 20\right)\cdot 23^{15} + \left(18 a^{14} + 6 a^{13} + 14 a^{12} + 15 a^{11} + 9 a^{10} + 17 a^{9} + 3 a^{8} + 7 a^{7} + 15 a^{6} + 3 a^{5} + 3 a^{4} + 21 a^{3} + 19 a^{2} + 11 a + 16\right)\cdot 23^{16} + \left(15 a^{14} + 19 a^{13} + a^{12} + 9 a^{11} + 9 a^{10} + 11 a^{9} + 15 a^{8} + 12 a^{7} + 18 a^{6} + a^{5} + 4 a^{4} + 2 a^{3} + 13 a^{2} + 11 a + 12\right)\cdot 23^{17} + \left(12 a^{14} + 14 a^{13} + 15 a^{12} + 8 a^{11} + 14 a^{10} + 8 a^{9} + 11 a^{8} + 18 a^{7} + 13 a^{6} + 21 a^{5} + 2 a^{4} + a^{3} + 22 a^{2} + 8 a + 14\right)\cdot 23^{18} + \left(7 a^{14} + 10 a^{13} + 7 a^{12} + 15 a^{11} + 8 a^{10} + 10 a^{9} + 21 a^{8} + 17 a^{7} + 4 a^{6} + 6 a^{5} + 10 a^{4} + 22 a^{3} + 15 a^{2} + 20 a + 5\right)\cdot 23^{19} + \left(10 a^{14} + 7 a^{13} + 8 a^{12} + 18 a^{11} + 9 a^{9} + 12 a^{8} + 15 a^{7} + 17 a^{6} + 22 a^{5} + 2 a^{4} + 17 a^{3} + 6 a^{2} + 14 a + 12\right)\cdot 23^{20} + \left(a^{14} + 7 a^{13} + 11 a^{12} + 16 a^{11} + 21 a^{10} + 20 a^{7} + 15 a^{6} + 15 a^{4} + 9 a^{3} + 4 a^{2} + 16 a + 6\right)\cdot 23^{21} + \left(15 a^{14} + 21 a^{13} + 19 a^{12} + 16 a^{11} + 6 a^{10} + 13 a^{9} + 12 a^{8} + 20 a^{7} + 20 a^{6} + 12 a^{5} + 11 a^{4} + a^{3} + 2 a^{2} + 17 a + 17\right)\cdot 23^{22} + \left(6 a^{14} + 17 a^{13} + 9 a^{12} + 16 a^{10} + 2 a^{9} + 3 a^{8} + 22 a^{7} + 8 a^{6} + 16 a^{5} + 8 a^{4} + 12 a^{3} + a^{2} + 6 a + 4\right)\cdot 23^{23} + \left(13 a^{14} + 6 a^{13} + 13 a^{12} + 9 a^{11} + 6 a^{10} + 12 a^{9} + 22 a^{8} + 18 a^{7} + 11 a^{6} + 18 a^{5} + a^{4} + 2 a^{3} + 7 a^{2} + 3 a + 8\right)\cdot 23^{24} + \left(a^{14} + 16 a^{13} + 12 a^{12} + 22 a^{11} + 14 a^{10} + 3 a^{9} + 14 a^{8} + 19 a^{7} + 12 a^{6} + 8 a^{5} + 4 a^{4} + a^{3} + 19 a^{2} + 11\right)\cdot 23^{25} + \left(21 a^{14} + 16 a^{13} + 20 a^{12} + a^{11} + 22 a^{10} + 13 a^{9} + 10 a^{8} + 2 a^{7} + 6 a^{6} + 12 a^{5} + 15 a^{4} + 18 a^{3} + 20 a^{2} + 10 a + 5\right)\cdot 23^{26} + \left(7 a^{14} + 12 a^{13} + 6 a^{11} + 18 a^{10} + 18 a^{9} + 11 a^{8} + 16 a^{7} + 12 a^{6} + 19 a^{5} + 16 a^{4} + 6 a^{3} + 7 a^{2} + 15 a + 5\right)\cdot 23^{27} + \left(a^{14} + 19 a^{13} + 15 a^{12} + 20 a^{11} + 5 a^{10} + 18 a^{9} + 22 a^{8} + 11 a^{7} + 22 a^{6} + 10 a^{5} + 3 a^{4} + 16 a^{3} + 12 a^{2} + 9 a + 21\right)\cdot 23^{28} + \left(16 a^{14} + 13 a^{13} + 6 a^{12} + 4 a^{11} + 6 a^{10} + 2 a^{9} + 2 a^{8} + a^{7} + 4 a^{6} + 22 a^{5} + 20 a^{4} + 14 a^{3} + 5 a^{2} + 3 a + 7\right)\cdot 23^{29} + \left(9 a^{14} + 22 a^{13} + 16 a^{12} + 9 a^{11} + 4 a^{10} + 8 a^{9} + 16 a^{8} + 18 a^{7} + 3 a^{6} + 21 a^{5} + 5 a^{3} + 8 a^{2} + 21 a + 12\right)\cdot 23^{30} + \left(18 a^{14} + 6 a^{13} + 15 a^{12} + 19 a^{11} + 9 a^{10} + 18 a^{9} + 4 a^{8} + 7 a^{7} + 22 a^{6} + 5 a^{5} + 8 a^{4} + 8 a^{3} + 12 a^{2} + 21 a + 11\right)\cdot 23^{31} + \left(22 a^{14} + a^{13} + 5 a^{12} + 16 a^{11} + 18 a^{10} + 18 a^{9} + 14 a^{8} + 18 a^{7} + 4 a^{6} + 16 a^{5} + 12 a^{4} + 19 a^{3} + 11 a^{2} + 7 a\right)\cdot 23^{32} + \left(17 a^{14} + 21 a^{13} + 15 a^{12} + 10 a^{10} + 19 a^{9} + 6 a^{8} + 21 a^{7} + 9 a^{6} + 14 a^{5} + 9 a^{4} + 17 a^{3} + 14 a^{2} + 13 a + 11\right)\cdot 23^{33} + \left(21 a^{14} + 11 a^{13} + 6 a^{12} + 18 a^{11} + 17 a^{10} + 11 a^{9} + 5 a^{8} + 14 a^{7} + 3 a^{6} + 21 a^{5} + 12 a^{4} + 5 a^{3} + 22 a^{2} + 14 a + 13\right)\cdot 23^{34} + \left(4 a^{14} + 18 a^{13} + 2 a^{12} + 20 a^{11} + 6 a^{10} + 12 a^{9} + 11 a^{8} + 4 a^{7} + 6 a^{6} + 2 a^{5} + 7 a^{4} + 12 a^{3} + a^{2} + 18 a + 8\right)\cdot 23^{35} + \left(20 a^{14} + 3 a^{13} + 3 a^{12} + 20 a^{11} + 11 a^{10} + 16 a^{9} + 19 a^{8} + 10 a^{7} + 18 a^{6} + 21 a^{5} + 21 a^{4} + 11 a^{3} + 10 a + 1\right)\cdot 23^{36} + \left(a^{14} + 17 a^{13} + 8 a^{12} + 17 a^{11} + 22 a^{10} + 3 a^{9} + 3 a^{8} + 17 a^{7} + 2 a^{6} + 20 a^{5} + 13 a^{3} + 6 a^{2} + 3 a + 16\right)\cdot 23^{37} + \left(19 a^{14} + a^{13} + 2 a^{12} + 11 a^{10} + 14 a^{9} + 7 a^{8} + 16 a^{7} + 2 a^{6} + 22 a^{5} + 3 a^{4} + 9 a^{3} + 2 a^{2} + 6 a + 8\right)\cdot 23^{38} + \left(15 a^{14} + 20 a^{13} + 9 a^{11} + 14 a^{10} + 10 a^{9} + 3 a^{8} + 9 a^{7} + 12 a^{6} + 4 a^{5} + 17 a^{4} + a^{3} + 16 a^{2} + 9 a + 2\right)\cdot 23^{39} + \left(11 a^{13} + 7 a^{12} + 7 a^{11} + 3 a^{10} + 5 a^{9} + 15 a^{8} + 5 a^{7} + 5 a^{6} + 13 a^{5} + 2 a^{4} + 18 a^{3} + 4 a^{2} + 6 a + 22\right)\cdot 23^{40} + \left(9 a^{14} + 3 a^{13} + 2 a^{12} + 21 a^{11} + 13 a^{10} + 6 a^{9} + 20 a^{8} + 8 a^{7} + 8 a^{6} + 16 a^{5} + 4 a^{4} + 21 a^{3} + a^{2} + 12 a + 6\right)\cdot 23^{41} + \left(21 a^{14} + 14 a^{13} + 13 a^{11} + 16 a^{10} + 2 a^{9} + 4 a^{8} + 11 a^{7} + 12 a^{6} + 11 a^{4} + 14 a^{3} + 2 a^{2} + 18 a + 21\right)\cdot 23^{42} + \left(5 a^{14} + 13 a^{13} + 9 a^{12} + 8 a^{11} + 10 a^{10} + 9 a^{9} + 7 a^{8} + 7 a^{7} + 17 a^{6} + 5 a^{5} + 16 a^{4} + 15 a^{3} + 14 a^{2} + 6 a + 14\right)\cdot 23^{43} + \left(6 a^{13} + 8 a^{12} + 17 a^{10} + 19 a^{9} + 18 a^{8} + 19 a^{7} + 10 a^{6} + 18 a^{5} + 18 a^{4} + 11 a^{3} + 4 a^{2} + 14 a\right)\cdot 23^{44} + \left(22 a^{14} + a^{13} + 19 a^{12} + 17 a^{11} + 4 a^{10} + 16 a^{9} + 18 a^{8} + 15 a^{7} + 20 a^{6} + 16 a^{5} + 7 a^{4} + 7 a^{3} + 12 a^{2} + 18 a + 12\right)\cdot 23^{45} + \left(10 a^{14} + 7 a^{13} + 3 a^{12} + 14 a^{11} + a^{10} + 16 a^{9} + 12 a^{7} + 3 a^{6} + 17 a^{5} + 20 a^{4} + 10 a^{3} + 17 a^{2} + 2 a + 21\right)\cdot 23^{46} + \left(11 a^{14} + 22 a^{13} + 22 a^{12} + 4 a^{11} + 20 a^{10} + 18 a^{9} + 13 a^{8} + 9 a^{7} + 7 a^{6} + a^{5} + 21 a^{4} + a^{3} + 16 a^{2} + 11 a + 17\right)\cdot 23^{47} + \left(5 a^{14} + 20 a^{13} + 7 a^{12} + 15 a^{11} + 3 a^{10} + 16 a^{9} + 14 a^{8} + 15 a^{7} + 19 a^{6} + 18 a^{5} + 3 a^{4} + 15 a^{3} + 15 a^{2} + 20 a + 20\right)\cdot 23^{48} + \left(21 a^{14} + 18 a^{13} + 5 a^{12} + 14 a^{11} + 18 a^{10} + 13 a^{8} + 12 a^{7} + 14 a^{6} + 22 a^{5} + 12 a^{4} + a^{3} + 2 a^{2} + 9 a + 18\right)\cdot 23^{49} + \left(5 a^{14} + a^{13} + 18 a^{12} + 21 a^{11} + 22 a^{10} + 18 a^{9} + 8 a^{8} + 6 a^{7} + 12 a^{6} + 16 a^{5} + 22 a^{4} + 20 a^{3} + 3 a^{2} + 8 a + 2\right)\cdot 23^{50} + \left(18 a^{14} + 5 a^{13} + 12 a^{12} + 2 a^{11} + 7 a^{10} + 15 a^{9} + 8 a^{8} + a^{7} + 11 a^{5} + 13 a^{4} + 6 a^{3} + 19 a^{2} + 18 a + 20\right)\cdot 23^{51} + \left(14 a^{14} + 8 a^{13} + 5 a^{12} + 9 a^{11} + 13 a^{10} + 22 a^{9} + 13 a^{8} + 21 a^{7} + 17 a^{6} + a^{5} + 12 a^{4} + 2 a^{3} + 18 a + 7\right)\cdot 23^{52} + \left(19 a^{14} + 7 a^{13} + 5 a^{12} + 8 a^{11} + 8 a^{10} + 15 a^{9} + 5 a^{8} + 8 a^{7} + 3 a^{6} + 14 a^{5} + 2 a^{4} + 15 a^{3} + a^{2} + 10 a + 22\right)\cdot 23^{53} + \left(4 a^{14} + 2 a^{13} + 8 a^{12} + 7 a^{11} + 7 a^{9} + 9 a^{8} + 13 a^{7} + 16 a^{6} + 17 a^{5} + 5 a^{4} + 7 a^{3} + 11 a^{2} + 2 a + 2\right)\cdot 23^{54} + \left(7 a^{14} + 9 a^{13} + 10 a^{12} + 8 a^{11} + 8 a^{10} + 22 a^{9} + 9 a^{8} + 15 a^{7} + 3 a^{6} + 17 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11 a^{7} + 18 a^{6} + 8 a^{5} + 21 a^{4} + 3 a^{3} + 14 a^{2} + 2 a + 22\right)\cdot 23^{61} + \left(22 a^{14} + 20 a^{13} + 21 a^{12} + 15 a^{11} + 19 a^{10} + 9 a^{8} + 8 a^{7} + 8 a^{6} + 16 a^{5} + 21 a^{4} + a^{3} + 12 a^{2} + 17 a + 15\right)\cdot 23^{62} + \left(20 a^{14} + 14 a^{13} + 13 a^{12} + 6 a^{11} + 5 a^{10} + 4 a^{9} + 3 a^{8} + 22 a^{7} + 22 a^{6} + 3 a^{5} + 5 a^{4} + 14 a^{3} + 4 a^{2} + 7 a + 12\right)\cdot 23^{63} + \left(21 a^{14} + 7 a^{13} + 16 a^{12} + 21 a^{11} + 8 a^{10} + 12 a^{8} + 21 a^{7} + 2 a^{6} + 5 a^{5} + 19 a^{4} + 21 a^{3} + 22 a^{2} + 16 a + 6\right)\cdot 23^{64} + \left(20 a^{14} + 11 a^{12} + 14 a^{11} + 12 a^{10} + 8 a^{9} + 11 a^{8} + 3 a^{7} + 11 a^{6} + 20 a^{5} + 7 a^{4} + a^{2} + 20 a + 22\right)\cdot 23^{65} + \left(13 a^{14} + 2 a^{13} + 14 a^{12} + 18 a^{11} + 12 a^{10} + 13 a^{9} + 2 a^{8} + 8 a^{7} + 16 a^{6} + 18 a^{5} + 9 a^{4} + a^{3} + 12 a^{2} + 19\right)\cdot 23^{66} + \left(21 a^{14} + 20 a^{13} + 7 a^{12} + 22 a^{11} + 8 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a + 22\right)\cdot 23^{78} + \left(21 a^{14} + 6 a^{13} + a^{12} + 12 a^{11} + 21 a^{10} + 22 a^{9} + 2 a^{8} + 20 a^{7} + 2 a^{6} + a^{5} + 3 a^{4} + 11 a^{3} + 6 a^{2} + 21 a + 17\right)\cdot 23^{79} + \left(12 a^{14} + 6 a^{13} + 3 a^{12} + 9 a^{11} + 10 a^{10} + 19 a^{9} + 11 a^{8} + 7 a^{7} + 10 a^{6} + 15 a^{5} + 6 a^{4} + 14 a^{3} + 4 a^{2} + 16 a + 8\right)\cdot 23^{80} + \left(a^{14} + 7 a^{13} + 8 a^{12} + 22 a^{11} + 12 a^{10} + 3 a^{9} + 11 a^{8} + 14 a^{7} + 4 a^{6} + 21 a^{5} + 8 a^{4} + 6 a^{3} + 3 a^{2} + 19 a + 15\right)\cdot 23^{81} + \left(10 a^{14} + 19 a^{13} + 8 a^{12} + 17 a^{11} + 14 a^{10} + 15 a^{9} + a^{8} + a^{7} + 8 a^{6} + a^{5} + 11 a^{4} + 17 a^{3} + 14 a^{2} + a + 20\right)\cdot 23^{82} + \left(20 a^{14} + a^{13} + 15 a^{12} + 11 a^{10} + 12 a^{9} + 12 a^{8} + 19 a^{7} + 4 a^{6} + 11 a^{5} + 14 a^{4} + a^{3} + 13 a^{2} + 3 a + 19\right)\cdot 23^{83} + \left(5 a^{14} + 20 a^{13} + 21 a^{12} + 5 a^{11} + 8 a^{10} + 7 a^{9} + 3 a^{8} + 22 a^{7} + 9 a^{6} + 10 a^{5} + 3 a^{4} + 22 a^{3} + 8 a^{2} + 12 a + 1\right)\cdot 23^{84} + \left(5 a^{14} + 14 a^{13} + 22 a^{12} + 17 a^{11} + 12 a^{10} + 15 a^{9} + 9 a^{8} + 5 a^{7} + 8 a^{6} + 13 a^{5} + 20 a^{4} + 21 a^{3} + 12 a^{2} + 5 a + 16\right)\cdot 23^{85} + \left(3 a^{14} + 12 a^{13} + a^{12} + 9 a^{11} + 13 a^{10} + 15 a^{9} + 10 a^{8} + 3 a^{7} + 17 a^{6} + 16 a^{5} + 21 a^{4} + 12 a^{3} + 21 a^{2} + 12 a + 15\right)\cdot 23^{86} + \left(20 a^{14} + 20 a^{13} + 3 a^{12} + 20 a^{11} + 21 a^{10} + 9 a^{9} + 22 a^{8} + 4 a^{7} + 6 a^{6} + 12 a^{5} + 5 a^{4} + 12 a^{3} + 20 a^{2} + 13 a\right)\cdot 23^{87} + \left(4 a^{14} + 9 a^{13} + 14 a^{12} + 11 a^{11} + 5 a^{10} + 16 a^{9} + 14 a^{8} + 10 a^{7} + 21 a^{6} + 4 a^{5} + 10 a^{4} + 5 a^{3} + 6 a^{2} + 17 a + 14\right)\cdot 23^{88} + \left(13 a^{14} + 9 a^{13} + 7 a^{12} + 18 a^{10} + 13 a^{9} + 15 a^{8} + a^{7} + 3 a^{6} + 21 a^{5} + 19 a^{4} + 15 a^{3} + 9 a^{2} + 14 a + 4\right)\cdot 23^{89} + \left(5 a^{14} + 4 a^{13} + 5 a^{12} + 14 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\left(2 a^{14} + 12 a^{13} + 11 a^{12} + 5 a^{11} + 10 a^{10} + 16 a^{9} + 18 a^{8} + 2 a^{7} + 10 a^{6} + 11 a^{5} + 4 a^{4} + 15 a^{2} + 12 a + 14\right)\cdot 23^{96} + \left(14 a^{14} + 4 a^{13} + 17 a^{12} + 7 a^{11} + 19 a^{10} + 17 a^{9} + 4 a^{8} + 10 a^{7} + 18 a^{6} + 7 a^{5} + 13 a^{4} + 20 a^{2} + 3 a + 13\right)\cdot 23^{97} + \left(9 a^{14} + 2 a^{13} + 15 a^{12} + 12 a^{11} + 5 a^{10} + 16 a^{9} + 11 a^{8} + a^{7} + 18 a^{6} + a^{5} + 12 a^{4} + 11 a^{3} + 21 a^{2} + 10 a + 8\right)\cdot 23^{98} + \left(4 a^{14} + 4 a^{13} + 20 a^{12} + 19 a^{11} + 18 a^{10} + 11 a^{9} + 10 a^{8} + 10 a^{7} + 17 a^{6} + a^{5} + 11 a^{4} + 22 a^{3} + 15 a^{2} + 8 a\right)\cdot 23^{99} + \left(14 a^{14} + 5 a^{13} + 21 a^{12} + 20 a^{10} + 12 a^{9} + 11 a^{8} + 9 a^{7} + 6 a^{6} + 13 a^{5} + 4 a^{4} + 20 a^{3} + 14 a^{2} + 16 a + 11\right)\cdot 23^{100} + \left(a^{14} + 16 a^{13} + 20 a^{12} + 19 a^{11} + 17 a^{10} + 13 a^{9} + 16 a^{8} + 15 a^{7} + 18 a^{6} + 9 a^{5} + a^{4} + 8 a^{3} + 21 a^{2} + 22 a + 8\right)\cdot 23^{101} + \left(15 a^{14} + 22 a^{13} + 16 a^{12} + 9 a^{11} + 8 a^{9} + 14 a^{8} + 8 a^{7} + 7 a^{6} + 20 a^{5} + 15 a^{4} + 6 a^{3} + 22 a^{2} + 16 a\right)\cdot 23^{102} + \left(12 a^{14} + 5 a^{13} + 11 a^{12} + a^{11} + 9 a^{10} + 17 a^{9} + 17 a^{8} + 17 a^{7} + 7 a^{6} + 7 a^{5} + 7 a^{4} + 7 a^{3} + 14 a^{2} + 22 a + 2\right)\cdot 23^{103} + \left(17 a^{14} + 22 a^{13} + 15 a^{12} + a^{11} + 15 a^{10} + 9 a^{9} + 14 a^{8} + 16 a^{7} + 10 a^{6} + 6 a^{5} + 5 a^{4} + 20 a^{3} + 15 a^{2} + 3 a + 8\right)\cdot 23^{104} + \left(20 a^{14} + 9 a^{13} + 18 a^{12} + 19 a^{11} + 10 a^{10} + 20 a^{9} + 17 a^{8} + 15 a^{7} + 3 a^{6} + 6 a^{5} + 17 a^{4} + 15 a^{3} + 9 a^{2} + 14 a + 8\right)\cdot 23^{105} + \left(16 a^{14} + 11 a^{12} + 5 a^{11} + 12 a^{10} + 13 a^{9} + 22 a^{8} + 18 a^{6} + 4 a^{5} + 21 a^{4} + 11 a^{3} + 21 a^{2} + 15 a + 2\right)\cdot 23^{106} + \left(11 a^{14} + 3 a^{13} + 2 a^{12} + 15 a^{11} + 2 a^{10} + 21 a^{9} + 11 a^{8} + 11 a^{7} + 4 a^{6} + 20 a^{5} + 3 a^{4} + 4 a^{3} + 6 a^{2} + 16 a + 21\right)\cdot 23^{107} + \left(18 a^{14} + 9 a^{13} + 20 a^{12} + 21 a^{11} + 12 a^{10} + 16 a^{9} + 6 a^{8} + 5 a^{7} + 13 a^{6} + 8 a^{5} + 6 a^{4} + 5 a^{3} + 10 a^{2} + 22 a + 3\right)\cdot 23^{108} + \left(11 a^{14} + 16 a^{13} + 10 a^{12} + 4 a^{11} + 22 a^{10} + 17 a^{9} + 4 a^{8} + 17 a^{7} + 5 a^{5} + 13 a^{4} + 15 a^{3} + 18 a^{2} + 21 a + 3\right)\cdot 23^{109} + \left(20 a^{14} + 15 a^{13} + a^{12} + 7 a^{11} + 21 a^{10} + 4 a^{9} + 11 a^{8} + 15 a^{7} + 7 a^{6} + 12 a^{5} + 12 a^{3} + 13 a^{2} + 2\right)\cdot 23^{110} + \left(5 a^{14} + 22 a^{13} + 5 a^{12} + 20 a^{11} + 14 a^{10} + 6 a^{9} + 8 a^{8} + 8 a^{7} + 20 a^{6} + 2 a^{5} + 21 a^{4} + 14 a^{2} + 18 a\right)\cdot 23^{111} + \left(2 a^{14} + 13 a^{13} + 3 a^{12} + 6 a^{11} + 22 a^{10} + 20 a^{9} + 2 a^{8} + 19 a^{7} + 16 a^{6} + 3 a^{5} + 18 a^{4} + 2 a^{3} + 15 a^{2} + 17 a\right)\cdot 23^{112} + \left(10 a^{14} + 9 a^{13} + 11 a^{12} + 11 a^{11} + 16 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a^{4} + 9 a^{3} + 8 a^{2} + 3 a + 15\right)\cdot 23^{124} + \left(a^{14} + 11 a^{13} + 2 a^{12} + 14 a^{11} + 6 a^{10} + 18 a^{9} + 14 a^{8} + 11 a^{7} + 20 a^{6} + 6 a^{5} + a^{4} + 20 a^{3} + 3 a^{2} + 18 a + 12\right)\cdot 23^{125} + \left(19 a^{14} + 3 a^{13} + 18 a^{12} + 8 a^{11} + 14 a^{10} + 5 a^{9} + 9 a^{8} + 20 a^{7} + 20 a^{6} + a^{5} + 14 a^{4} + 15 a^{3} + 16 a^{2} + 21 a + 3\right)\cdot 23^{126} + \left(7 a^{14} + 2 a^{13} + 22 a^{12} + 12 a^{11} + 5 a^{10} + 7 a^{9} + 9 a^{8} + a^{7} + 10 a^{6} + 4 a^{5} + 20 a^{4} + a^{3} + 19 a^{2} + 3 a + 17\right)\cdot 23^{127} + \left(6 a^{14} + 12 a^{13} + 15 a^{12} + 16 a^{11} + 5 a^{10} + 16 a^{8} + 19 a^{7} + 9 a^{6} + 18 a^{5} + 10 a^{4} + 9 a^{3} + 22 a^{2} + 12\right)\cdot 23^{128} + \left(3 a^{14} + 17 a^{13} + 14 a^{12} + 19 a^{11} + 4 a^{10} + 12 a^{9} + 15 a^{8} + 6 a^{7} + 14 a^{6} + 13 a^{5} + 8 a^{4} + 2 a^{3} + 6 a^{2} + 6 a + 1\right)\cdot 23^{129} + \left(21 a^{14} + 15 a^{13} + 21 a^{12} + 17 a^{11} + 2 a^{10} + 11 a^{9} + 13 a^{8} + 19 a^{7} + 7 a^{6} + 12 a^{5} + 13 a^{4} + 7 a^{3} + 18 a^{2} + 11 a + 11\right)\cdot 23^{130} + \left(7 a^{14} + 20 a^{13} + 10 a^{12} + 5 a^{11} + 7 a^{10} + 16 a^{9} + 18 a^{8} + 4 a^{7} + 5 a^{6} + 4 a^{5} + 9 a^{4} + 3 a^{3} + 6 a^{2} + 15\right)\cdot 23^{131} + \left(13 a^{14} + 20 a^{13} + 2 a^{12} + 19 a^{11} + 18 a^{10} + 12 a^{9} + 14 a^{8} + 14 a^{7} + 17 a^{6} + 20 a^{5} + 12 a^{4} + 20 a^{3} + 2 a^{2} + 11 a + 9\right)\cdot 23^{132} + \left(9 a^{14} + 21 a^{13} + 11 a^{12} + 17 a^{11} + 20 a^{10} + 16 a^{8} + 19 a^{7} + 9 a^{6} + 7 a^{5} + 12 a^{4} + 17 a^{3} + 8 a^{2} + 19 a + 1\right)\cdot 23^{133} + \left(17 a^{14} + 17 a^{13} + 13 a^{11} + 8 a^{10} + a^{9} + 15 a^{8} + a^{7} + 8 a^{6} + 6 a^{5} + 10 a^{4} + 10 a^{3} + 20 a^{2} + 17 a + 19\right)\cdot 23^{134} + \left(9 a^{14} + 6 a^{13} + 3 a^{12} + 9 a^{11} + 17 a^{10} + 11 a^{9} + 5 a^{8} + 6 a^{7} + 9 a^{6} + 19 a^{5} + 6 a^{4} + a^{3} + 7 a^{2} + 13 a + 13\right)\cdot 23^{135} + \left(4 a^{14} 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a^{2} + 13 a + 17\right)\cdot 23^{141} + \left(19 a^{14} + 7 a^{13} + 20 a^{12} + 21 a^{11} + 20 a^{10} + 3 a^{9} + 10 a^{8} + 10 a^{7} + 18 a^{6} + 11 a^{5} + 16 a^{4} + 10 a^{3} + 8 a^{2} + 13 a + 2\right)\cdot 23^{142} + \left(19 a^{14} + 13 a^{13} + 12 a^{12} + 2 a^{11} + 9 a^{10} + 18 a^{9} + 20 a^{8} + 2 a^{7} + 12 a^{6} + 11 a^{5} + 10 a^{4} + 18 a^{3} + 18 a^{2} + 13 a + 9\right)\cdot 23^{143} + \left(22 a^{14} + 12 a^{13} + 8 a^{12} + a^{11} + a^{10} + 20 a^{9} + 10 a^{8} + a^{6} + 20 a^{5} + 5 a^{4} + 6 a^{3} + 3 a^{2} + 14 a + 17\right)\cdot 23^{144} + \left(22 a^{14} + 3 a^{13} + 15 a^{12} + 11 a^{11} + 2 a^{10} + 11 a^{9} + 17 a^{8} + a^{7} + 10 a^{6} + 13 a^{5} + a^{4} + 21 a^{3} + 21 a^{2} + 14 a + 13\right)\cdot 23^{145} + \left(19 a^{14} + 8 a^{12} + 2 a^{11} + 2 a^{10} + 10 a^{9} + 5 a^{8} + 17 a^{7} + 5 a^{6} + 4 a^{5} + 18 a^{4} + 14 a^{3} + 18 a^{2} + 10 a + 7\right)\cdot 23^{146} + \left(19 a^{14} + 15 a^{13} + 13 a^{12} + 2 a^{11} + 12 a^{10} + 15 a^{9} + 5 a^{8} + 14 a^{7} + 7 a^{6} + 16 a^{5} + 8 a^{4} + 21 a^{3} + 15 a^{2} + 22 a + 12\right)\cdot 23^{147} + \left(4 a^{14} + 2 a^{13} + 12 a^{11} + 20 a^{10} + 14 a^{9} + 17 a^{8} + 17 a^{7} + 8 a^{5} + 7 a^{4} + 4 a^{3} + 12 a^{2} + 2 a + 10\right)\cdot 23^{148} + \left(13 a^{14} + 11 a^{13} + 21 a^{12} + 18 a^{11} + 3 a^{10} + 9 a^{9} + 11 a^{8} + 20 a^{7} + 4 a^{6} + 15 a^{4} + 15 a^{3} + 2 a^{2} + a + 17\right)\cdot 23^{149} + \left(13 a^{14} + a^{13} + 22 a^{12} + 15 a^{11} + 21 a^{10} + 11 a^{9} + 7 a^{8} + 9 a^{7} + 13 a^{6} + 13 a^{5} + a^{4} + 10 a^{3} + 15 a^{2} + 16 a\right)\cdot 23^{150} + \left(19 a^{14} + 3 a^{13} + 22 a^{12} + 8 a^{11} + 7 a^{10} + 7 a^{9} + a^{8} + 5 a^{7} + 22 a^{6} + 9 a^{5} + 2 a^{4} + 3 a^{3} + 19 a^{2} + 17 a + 4\right)\cdot 23^{151} + \left(16 a^{14} + 17 a^{13} + 20 a^{12} + 20 a^{10} + 7 a^{9} + 6 a^{8} + 21 a^{7} + 5 a^{6} + 13 a^{5} + 20 a^{4} + 9 a^{3} + 21 a^{2} + 10 a + 10\right)\cdot 23^{152} + \left(16 a^{14} + a^{13} + 16 a^{12} + 17 a^{11} + a^{10} + 11 a^{9} + 14 a^{8} + 16 a^{7} + 14 a^{6} + 3 a^{5} + 8 a^{4} + 21 a^{3} + 17 a^{2} + 10\right)\cdot 23^{153} + \left(10 a^{14} + 13 a^{13} + 13 a^{11} + 2 a^{10} + 21 a^{9} + 16 a^{8} + 13 a^{7} + 7 a^{6} + 22 a^{5} + 18 a^{4} + 15 a^{3} + 20 a^{2} + 10 a + 15\right)\cdot 23^{154} + \left(15 a^{14} + a^{12} + 19 a^{11} + 17 a^{10} + a^{9} + 13 a^{8} + 11 a^{7} + 17 a^{6} + 15 a^{5} + 17 a^{4} + 18 a^{3} + 21 a^{2} + 10\right)\cdot 23^{155} + \left(3 a^{14} + 21 a^{13} + 2 a^{12} + 2 a^{11} + 12 a^{10} + 16 a^{9} + 13 a^{8} + 2 a^{7} + a^{6} + 9 a^{5} + 21 a^{4} + 13 a^{3} + 12 a^{2} + 17 a + 14\right)\cdot 23^{156} + \left(14 a^{14} + 5 a^{13} + 6 a^{12} + 6 a^{11} + 16 a^{10} + 8 a^{9} + 3 a^{8} + 7 a^{7} + 5 a^{6} + 17 a^{5} + 22 a^{4} + 19 a^{3} + 10 a^{2} + 21 a + 3\right)\cdot 23^{157} + \left(16 a^{14} + 5 a^{13} + 11 a^{12} + 11 a^{11} + 9 a^{10} + 2 a^{9} + a^{8} + 21 a^{7} + 14 a^{6} + 20 a^{5} + 21 a^{4} + 3 a^{3} + 12 a^{2} + 2 a + 3\right)\cdot 23^{158} + \left(18 a^{14} + 7 a^{13} + 15 a^{12} + 4 a^{11} + 16 a^{10} + 2 a^{9} + 5 a^{8} + 7 a^{7} + 13 a^{6} + 17 a^{5} + 15 a^{4} + 14 a^{3} + 20 a + 9\right)\cdot 23^{159} + \left(22 a^{14} + 19 a^{13} + 3 a^{12} + 22 a^{11} + 13 a^{10} + 2 a^{9} + 22 a^{8} + 6 a^{7} + 22 a^{6} + 12 a^{5} + 19 a^{4} + 12 a^{3} + 10 a^{2} + 2 a + 4\right)\cdot 23^{160} + \left(a^{14} + 15 a^{13} + 2 a^{12} + 9 a^{11} + 2 a^{10} + 8 a^{9} + 19 a^{8} + 9 a^{7} + 4 a^{5} + a^{4} + 5 a^{2} + 8 a + 15\right)\cdot 23^{161} + \left(19 a^{14} + 20 a^{13} + 6 a^{12} + 19 a^{11} + 20 a^{10} + a^{9} + 6 a^{8} + 21 a^{7} + 10 a^{6} + 5 a^{5} + 14 a^{4} + 22 a^{3} + 11 a^{2} + 2 a + 13\right)\cdot 23^{162} + \left(17 a^{14} + 8 a^{13} + 5 a^{12} + a^{11} + 17 a^{10} + 16 a^{9} + 18 a^{8} + 21 a^{7} + 13 a^{6} + 16 a^{5} + 14 a^{4} + 19 a^{3} + 14 a^{2} + 17 a + 17\right)\cdot 23^{163} + \left(6 a^{14} + 10 a^{13} + 3 a^{12} + 21 a^{11} + 15 a^{10} + 22 a^{9} + 3 a^{8} + 9 a^{7} + 20 a^{6} + a^{5} + 7 a^{4} + 17 a^{3} + 14 a^{2} + 19 a + 2\right)\cdot 23^{164} + \left(13 a^{14} + 9 a^{12} + 10 a^{10} + 5 a^{9} + 5 a^{8} + 8 a^{7} + 5 a^{6} + 22 a^{5} + 21 a^{4} + 9 a^{3} + 19 a^{2} + 10 a + 8\right)\cdot 23^{165} + \left(10 a^{14} + 21 a^{13} + 21 a^{12} + 15 a^{11} + 15 a^{10} + 3 a^{9} + 22 a^{8} + 7 a^{7} + 19 a^{6} + a^{5} + 5 a^{4} + 3 a^{3} + 12 a^{2} + 14 a + 6\right)\cdot 23^{166} + \left(16 a^{14} + 3 a^{13} + 19 a^{12} + 8 a^{11} + 2 a^{10} + 7 a^{9} + 20 a^{8} + 8 a^{7} + 12 a^{6} + 14 a^{5} + 5 a^{4} + 12 a^{3} + 2 a^{2} + 12 a + 15\right)\cdot 23^{167} + \left(11 a^{14} + 3 a^{13} + 13 a^{12} + 15 a^{11} + 21 a^{10} + 8 a^{9} + 16 a^{8} + 7 a^{7} + 5 a^{6} + a^{5} + 17 a^{4} + 8 a^{3} + 15 a^{2} + 17 a + 11\right)\cdot 23^{168} + \left(2 a^{14} + 16 a^{13} + 17 a^{12} + 18 a^{11} + 15 a^{10} + 9 a^{9} + 17 a^{8} + 19 a^{7} + 9 a^{6} + 7 a^{5} + 9 a^{4} + 12 a^{3} + 15 a^{2} + 9 a + 21\right)\cdot 23^{169} + \left(8 a^{14} + 18 a^{12} + 14 a^{11} + 19 a^{9} + 5 a^{8} + 15 a^{7} + 8 a^{6} + 15 a^{5} + 15 a^{4} + 10 a^{3} + 19 a^{2} + 11 a + 4\right)\cdot 23^{170} + \left(21 a^{14} + a^{13} + 16 a^{12} + 20 a^{11} + 18 a^{10} + 10 a^{9} + a^{8} + 4 a^{7} + 5 a^{6} + 8 a^{4} + 21 a^{3} + 4 a^{2} + 5 a + 1\right)\cdot 23^{171} + \left(8 a^{14} + 20 a^{13} + 20 a^{12} + 9 a^{11} + 6 a^{10} + 13 a^{9} + 8 a^{8} + 8 a^{7} + 17 a^{6} + 11 a^{5} + 11 a^{4} + 18 a^{3} + 12 a^{2} + 9 a + 5\right)\cdot 23^{172} + \left(18 a^{14} + 10 a^{13} + 6 a^{12} + 4 a^{11} + 3 a^{10} + 9 a^{9} + 15 a^{8} + 4 a^{7} + 16 a^{6} + 3 a^{5} + 6 a^{4} + 8 a^{3} + 5 a^{2} + 19 a + 19\right)\cdot 23^{173} + \left(20 a^{14} + 9 a^{13} + 7 a^{12} + a^{11} + 16 a^{10} + a^{9} + 4 a^{8} + 11 a^{7} + 7 a^{6} + 2 a^{5} + 8 a^{4} + 11 a^{3} + 7 a^{2} + 19 a + 4\right)\cdot 23^{174} + \left(15 a^{14} + 3 a^{13} + 16 a^{12} + 19 a^{11} + 12 a^{8} + 3 a^{7} + 11 a^{6} + 17 a^{5} + 3 a^{4} + 15 a^{3} + 18 a^{2} + 21 a + 12\right)\cdot 23^{175} + \left(15 a^{14} + a^{13} + 6 a^{12} + 16 a^{11} + 16 a^{10} + 6 a^{9} + 6 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+ 5 a^{13} + 11 a^{12} + a^{11} + 6 a^{10} + 16 a^{9} + 20 a^{8} + 12 a^{7} + 3 a^{6} + a^{5} + 20 a^{4} + 3 a^{3} + 8 a + 20\right)\cdot 23^{182} + \left(4 a^{14} + 20 a^{13} + 16 a^{12} + 7 a^{11} + 18 a^{10} + 21 a^{9} + 18 a^{8} + 17 a^{6} + 19 a^{5} + 21 a^{4} + 20 a^{3} + 10 a^{2} + 20 a + 8\right)\cdot 23^{183} + \left(11 a^{14} + 18 a^{13} + 22 a^{12} + 2 a^{11} + 15 a^{10} + 21 a^{9} + 10 a^{7} + 15 a^{6} + 12 a^{5} + 17 a^{4} + 8 a^{3} + 4 a^{2} + 11 a + 15\right)\cdot 23^{184} + \left(9 a^{14} + 16 a^{13} + 22 a^{12} + 17 a^{11} + 3 a^{10} + 22 a^{9} + 3 a^{8} + 14 a^{7} + 20 a^{6} + 14 a^{5} + 16 a^{4} + 6 a^{3} + 12 a^{2} + 21 a + 2\right)\cdot 23^{185} + \left(6 a^{14} + 15 a^{13} + 3 a^{12} + 13 a^{10} + 19 a^{9} + 19 a^{8} + 2 a^{7} + a^{6} + a^{5} + 7 a^{4} + 15 a^{3} + 12 a + 16\right)\cdot 23^{186} + \left(16 a^{14} + 22 a^{13} + 5 a^{12} + 12 a^{11} + 7 a^{10} + 19 a^{9} + 22 a^{7} + 6 a^{6} + 4 a^{5} + 19 a^{4} + 10 a^{3} + 13 a^{2} + 3 a + 21\right)\cdot 23^{187} 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\left(20 a^{14} + 4 a^{12} + 4 a^{10} + 15 a^{9} + 12 a^{5} + 5 a^{4} + 10 a^{3} + 12 a^{2} + 7 a + 8\right)\cdot 23^{211} + \left(3 a^{14} + 2 a^{13} + 17 a^{12} + 7 a^{11} + 2 a^{10} + 21 a^{9} + 17 a^{8} + 15 a^{7} + 3 a^{5} + 3 a^{4} + 10 a^{3} + 3 a^{2} + 13 a + 12\right)\cdot 23^{212} + \left(14 a^{14} + 21 a^{13} + 14 a^{11} + 16 a^{10} + 8 a^{9} + 21 a^{8} + 16 a^{7} + 19 a^{6} + a^{5} + 17 a^{4} + 22 a^{3} + 7 a^{2} + 2 a + 1\right)\cdot 23^{213} + \left(13 a^{14} + 15 a^{13} + 13 a^{12} + a^{11} + 5 a^{10} + 6 a^{9} + 6 a^{8} + 12 a^{7} + 21 a^{6} + 8 a^{5} + 14 a^{4} + 19 a^{3} + a^{2} + 13 a + 9\right)\cdot 23^{214} + \left(14 a^{14} + 20 a^{13} + 11 a^{12} + 22 a^{11} + 4 a^{10} + 14 a^{9} + 5 a^{8} + 14 a^{7} + 19 a^{6} + 13 a^{4} + 22 a^{3} + 5 a^{2} + 17 a + 4\right)\cdot 23^{215} + \left(11 a^{14} + 13 a^{13} + 9 a^{12} + 10 a^{11} + 12 a^{10} + 15 a^{9} + 22 a^{8} + 17 a^{6} + 2 a^{5} + 8 a^{4} + 3 a^{3} + 8 a^{2} + 10 a\right)\cdot 23^{216} + \left(5 a^{14} + 14 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11\right)\cdot 23^{228} + \left(6 a^{14} + 8 a^{13} + 14 a^{12} + 20 a^{11} + 10 a^{10} + a^{9} + 18 a^{8} + 18 a^{7} + 5 a^{6} + 20 a^{5} + 16 a^{4} + 12 a^{3} + 15 a^{2} + 16 a + 7\right)\cdot 23^{229} + \left(18 a^{14} + 8 a^{13} + 17 a^{12} + 16 a^{11} + 14 a^{10} + 5 a^{9} + 6 a^{8} + 22 a^{7} + a^{6} + 11 a^{5} + 13 a^{4} + 4 a^{3} + 7 a^{2} + 3\right)\cdot 23^{230} + \left(22 a^{14} + 7 a^{13} + 18 a^{12} + a^{11} + a^{10} + 11 a^{9} + 13 a^{8} + 6 a^{7} + 14 a^{6} + 5 a^{5} + 2 a^{4} + 16 a^{3} + 15 a^{2} + 19 a + 4\right)\cdot 23^{231} + \left(22 a^{13} + 8 a^{11} + 12 a^{10} + 13 a^{9} + 7 a^{8} + 4 a^{7} + 19 a^{6} + 6 a^{5} + 18 a^{4} + 4 a^{3} + a^{2} + 13 a + 1\right)\cdot 23^{232} + \left(6 a^{14} + 5 a^{13} + 7 a^{12} + 16 a^{11} + 16 a^{10} + 10 a^{9} + 14 a^{8} + 18 a^{7} + 11 a^{6} + 6 a^{5} + 5 a^{4} + 20 a^{3} + 2 a^{2} + 13 a + 9\right)\cdot 23^{233} + \left(6 a^{14} + 8 a^{13} + 14 a^{12} + 7 a^{11} + 4 a^{10} + 13 a^{9} + 21 a^{8} + 6 a^{7} + 7 a^{6} + 7 a^{5} + 17 a^{4} + 20 a^{3} + 6 a^{2} + 17 a + 12\right)\cdot 23^{234} + \left(6 a^{14} + 9 a^{13} + 18 a^{12} + 10 a^{11} + 15 a^{10} + 15 a^{9} + 17 a^{8} + 17 a^{7} + 17 a^{6} + 18 a^{5} + 9 a^{4} + 5 a^{3} + 9 a^{2} + 2 a\right)\cdot 23^{235} + \left(7 a^{14} + 14 a^{13} + 10 a^{12} + 5 a^{11} + 12 a^{10} + 11 a^{9} + 9 a^{8} + 15 a^{7} + 19 a^{6} + 2 a^{5} + 22 a^{4} + 9 a^{2} + 11 a + 18\right)\cdot 23^{236} + \left(14 a^{13} + 9 a^{12} + a^{11} + 9 a^{10} + 11 a^{9} + 11 a^{8} + 8 a^{7} + 17 a^{6} + 13 a^{5} + 15 a^{4} + a^{3} + 12 a^{2} + 5 a + 6\right)\cdot 23^{237} + \left(6 a^{14} + 2 a^{13} + 20 a^{12} + 15 a^{11} + 16 a^{10} + 20 a^{9} + 14 a^{8} + 6 a^{7} + 21 a^{5} + a^{4} + 19 a^{3} + 4 a^{2} + 15 a + 13\right)\cdot 23^{238} + \left(6 a^{14} + 19 a^{13} + 21 a^{12} + 9 a^{11} + 20 a^{10} + 17 a^{9} + 5 a^{8} + 7 a^{7} + 16 a^{6} + 8 a^{5} + 16 a^{3} + 20 a^{2} + 18 a + 4\right)\cdot 23^{239} + \left(a^{14} + 20 a^{13} + 18 a^{12} + 16 a^{11} + a^{10} + 9 a^{9} + 22 a^{8} + 22 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a^{13} + 12 a^{12} + 13 a^{11} + 18 a^{10} + 14 a^{9} + 7 a^{8} + 3 a^{7} + 3 a^{6} + 15 a^{5} + 20 a^{4} + 6 a^{3} + 12 a^{2} + 21 a + 1\right)\cdot 23^{252} + \left(17 a^{14} + 17 a^{13} + 16 a^{12} + 19 a^{11} + 16 a^{10} + 7 a^{9} + 4 a^{8} + 4 a^{7} + 16 a^{6} + 2 a^{5} + 18 a^{4} + 17 a^{3} + a^{2} + 11 a + 22\right)\cdot 23^{253} + \left(5 a^{14} + 18 a^{12} + 9 a^{11} + 5 a^{10} + 7 a^{9} + 20 a^{8} + 18 a^{7} + 4 a^{6} + 11 a^{5} + 5 a^{4} + 16 a^{3} + 15 a^{2} + 17 a + 19\right)\cdot 23^{254} + \left(17 a^{13} + 5 a^{12} + 13 a^{11} + 20 a^{10} + 2 a^{9} + 15 a^{8} + 17 a^{7} + 16 a^{6} + 18 a^{5} + 13 a^{4} + 3 a^{3} + 17 a^{2} + 3 a + 20\right)\cdot 23^{255} + \left(10 a^{14} + 15 a^{13} + 9 a^{12} + 9 a^{11} + 12 a^{10} + a^{9} + 21 a^{8} + 16 a^{7} + a^{6} + 19 a^{5} + 10 a^{4} + 17 a^{3} + 18 a^{2} + a + 3\right)\cdot 23^{256} + \left(6 a^{14} + 17 a^{13} + 4 a^{12} + 13 a^{11} + 7 a^{10} + 6 a^{9} + 11 a^{8} + 11 a^{7} + 17 a^{6} + 9 a^{4} + 4 a^{3} + 9 a^{2} + 5 a + 4\right)\cdot 23^{257} + \left(13 a^{14} + 6 a^{13} + 22 a^{11} + 17 a^{10} + 17 a^{9} + 17 a^{8} + 17 a^{7} + 16 a^{6} + a^{5} + 21 a^{4} + 8 a^{3} + 11 a^{2} + 13 a\right)\cdot 23^{258} + \left(12 a^{14} + 15 a^{13} + 21 a^{12} + 16 a^{11} + 19 a^{10} + 9 a^{9} + 8 a^{8} + a^{7} + 15 a^{6} + 20 a^{5} + 19 a^{4} + 22 a^{3} + 19 a^{2} + 22 a + 13\right)\cdot 23^{259} + \left(13 a^{14} + 5 a^{13} + 15 a^{12} + 3 a^{10} + 5 a^{9} + 2 a^{8} + 19 a^{7} + 16 a^{5} + 16 a^{4} + 8 a^{3} + 8 a^{2} + 21 a + 7\right)\cdot 23^{260} + \left(6 a^{14} + 13 a^{13} + 3 a^{12} + 9 a^{11} + 7 a^{10} + 12 a^{9} + 16 a^{8} + 13 a^{7} + 22 a^{6} + 7 a^{5} + 13 a^{4} + 12 a^{3} + 21 a^{2} + 15 a + 5\right)\cdot 23^{261} + \left(18 a^{14} + 10 a^{13} + 11 a^{12} + 9 a^{11} + 4 a^{10} + 4 a^{9} + 8 a^{8} + 7 a^{7} + 15 a^{6} + 22 a^{5} + 17 a^{4} + 22 a^{3} + 9 a^{2} + 11 a + 18\right)\cdot 23^{262} + \left(8 a^{14} + 11 a^{13} + 9 a^{12} + 15 a^{11} + 2 a^{10} + a^{9} + 11 a^{8} + 2 a^{7} + 10 a^{6} + 8 a^{5} + a^{4} + 19 a^{3} + 8 a^{2} + 20 a + 12\right)\cdot 23^{263} + \left(21 a^{14} + 5 a^{13} + 12 a^{12} + 17 a^{11} + 9 a^{10} + 12 a^{9} + 7 a^{8} + 4 a^{7} + 20 a^{6} + 11 a^{5} + 21 a^{4} + 7 a^{3} + 22 a^{2} + 5 a + 17\right)\cdot 23^{264} + \left(18 a^{14} + 8 a^{13} + 16 a^{12} + 11 a^{11} + 20 a^{10} + 7 a^{9} + 12 a^{8} + 12 a^{7} + 19 a^{6} + 9 a^{5} + 16 a^{4} + 8 a^{3} + 14 a^{2} + 3 a + 4\right)\cdot 23^{265} + \left(8 a^{14} + 5 a^{13} + 15 a^{11} + 2 a^{10} + 18 a^{9} + 11 a^{8} + 6 a^{7} + 7 a^{6} + 22 a^{5} + 8 a^{4} + 15 a^{3} + 7 a^{2} + 10 a + 10\right)\cdot 23^{266} + \left(11 a^{14} + 3 a^{13} + 8 a^{12} + 2 a^{11} + 17 a^{10} + 21 a^{9} + 17 a^{8} + 6 a^{6} + 22 a^{5} + 21 a^{4} + 18 a^{3} + 2 a^{2} + 9 a + 9\right)\cdot 23^{267} + \left(12 a^{14} + 12 a^{13} + 9 a^{12} + 19 a^{11} + 18 a^{10} + 10 a^{9} + 11 a^{8} + 15 a^{7} + 7 a^{6} + 6 a^{5} + a^{4} + 9 a^{3} + 17 a^{2} + 10 a + 3\right)\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 4 }$ $=$ $2 a^{14} + 4 a^{13} + 17 a^{12} + 20 a^{11} + 4 a^{10} + 18 a^{9} + 15 a^{7} + 21 a^{6} + 12 a^{5} + 14 a^{3} + a^{2} + 13 a + 12 + \left(6 a^{14} + 11 a^{13} + 9 a^{12} + 18 a^{11} + 22 a^{10} + 15 a^{9} + 12 a^{8} + 15 a^{7} + 22 a^{6} + 5 a^{4} + 21 a^{3} + 16 a^{2} + 19 a + 22\right)\cdot 23 + \left(2 a^{14} + 16 a^{13} + 14 a^{12} + 15 a^{11} + 21 a^{10} + 4 a^{9} + 2 a^{8} + 11 a^{7} + 9 a^{6} + 3 a^{5} + 12 a^{4} + 11 a^{3} + 8 a^{2} + 19\right)\cdot 23^{2} + \left(16 a^{14} + 15 a^{13} + 6 a^{12} + 16 a^{11} + 5 a^{10} + 20 a^{9} + 19 a^{8} + 20 a^{7} + 9 a^{6} + 4 a^{5} + 12 a^{4} + 17 a^{3} + 12 a^{2} + 11 a + 3\right)\cdot 23^{3} + \left(10 a^{14} + 9 a^{13} + 10 a^{12} + a^{11} + 13 a^{10} + 11 a^{9} + 6 a^{8} + 6 a^{7} + 16 a^{6} + 9 a^{5} + 11 a^{4} + 9 a^{3} + 15 a^{2} + 6 a + 15\right)\cdot 23^{4} + \left(10 a^{14} + 21 a^{13} + 17 a^{12} + 19 a^{11} + 13 a^{10} + 17 a^{9} + a^{8} + 12 a^{7} + 3 a^{6} + 11 a^{5} + 12 a^{4} + 21 a^{3} + 21 a^{2} + 11 a + 21\right)\cdot 23^{5} + \left(21 a^{14} + 13 a^{13} + 6 a^{12} + 18 a^{11} + 6 a^{10} + 16 a^{9} + 14 a^{8} + 17 a^{7} + 4 a^{6} + 15 a^{5} + 20 a^{4} + 11 a^{3} + 10 a^{2} + 3 a + 14\right)\cdot 23^{6} + \left(12 a^{14} + 21 a^{13} + 14 a^{12} + 8 a^{11} + 12 a^{10} + 2 a^{9} + 14 a^{8} + 22 a^{7} + 11 a^{6} + 13 a^{5} + 14 a^{4} + 7 a^{3} + 20 a^{2} + 5 a + 11\right)\cdot 23^{7} + \left(15 a^{14} + 20 a^{13} + 8 a^{12} + 8 a^{11} + 12 a^{10} + 9 a^{9} + 19 a^{7} + 7 a^{6} + 3 a^{5} + 2 a^{4} + 16 a^{3} + 20 a^{2} + 13 a + 15\right)\cdot 23^{8} + \left(12 a^{14} + 5 a^{13} + 20 a^{12} + 14 a^{11} + 2 a^{10} + 2 a^{9} + 11 a^{8} + 3 a^{7} + 19 a^{6} + 2 a^{5} + 19 a^{4} + 20 a^{3} + 21 a^{2} + a + 14\right)\cdot 23^{9} + \left(19 a^{14} + 9 a^{13} + 10 a^{12} + 6 a^{11} + 16 a^{10} + 8 a^{9} + 5 a^{8} + 20 a^{7} + 13 a^{6} + 4 a^{5} + 5 a^{4} + 21 a^{3} + 8 a^{2} + 4 a + 20\right)\cdot 23^{10} + \left(18 a^{14} + 14 a^{13} + 22 a^{12} + 16 a^{11} + 3 a^{10} + 7 a^{9} + 9 a^{8} + 3 a^{7} + 22 a^{6} + 8 a^{5} + 16 a^{4} + 19 a^{3} + 19 a^{2} + 4 a + 20\right)\cdot 23^{11} + \left(4 a^{14} + 20 a^{13} + 17 a^{12} + a^{11} + 19 a^{10} + 7 a^{9} + 3 a^{8} + 7 a^{7} + 15 a^{6} + 13 a^{5} + 15 a^{4} + 8 a^{3} + 17 a^{2} + 17 a + 10\right)\cdot 23^{12} + \left(2 a^{14} + 10 a^{13} + 3 a^{12} + 8 a^{11} + 3 a^{10} + 19 a^{9} + 21 a^{8} + 14 a^{7} + 18 a^{6} + a^{5} + 6 a^{4} + 17 a^{3} + 16 a^{2} + 16 a + 9\right)\cdot 23^{13} + \left(5 a^{14} + 6 a^{13} + 2 a^{12} + 11 a^{11} + 21 a^{10} + 2 a^{9} + 18 a^{8} + 5 a^{7} + 10 a^{6} + 4 a^{5} + a^{4} + 18 a^{3} + 16 a^{2} + 7 a + 16\right)\cdot 23^{14} + \left(16 a^{14} + 12 a^{13} + 20 a^{12} + 20 a^{11} + 3 a^{10} + 14 a^{9} + 19 a^{8} + 7 a^{7} + 4 a^{6} + 6 a^{5} + 5 a^{4} + 4 a^{3} + 21 a^{2} + 5 a + 22\right)\cdot 23^{15} + \left(14 a^{14} + 2 a^{13} + 21 a^{12} + 12 a^{11} + 9 a^{10} + 3 a^{9} + 10 a^{8} + 20 a^{7} + 14 a^{6} + 11 a^{5} + 19 a^{4} + 3 a^{3} + 17 a^{2} + 1\right)\cdot 23^{16} + \left(13 a^{14} + 7 a^{13} + 19 a^{12} + 7 a^{11} + 20 a^{10} + 9 a^{9} + 10 a^{8} + 13 a^{7} + 21 a^{6} + 22 a^{4} + a^{3} + 3 a^{2} + 2 a + 20\right)\cdot 23^{17} + \left(18 a^{14} + 10 a^{12} + 10 a^{11} + 17 a^{10} + 18 a^{8} + 5 a^{7} + 12 a^{6} + 10 a^{5} + a^{4} + 5 a^{3} + 14 a^{2} + 14 a + 12\right)\cdot 23^{18} + \left(7 a^{14} + 15 a^{13} + 3 a^{12} + 17 a^{11} + 8 a^{10} + 18 a^{9} + 4 a^{8} + 22 a^{7} + 15 a^{6} + 17 a^{5} + 11 a^{4} + 8 a^{3} + 8 a^{2} + 8\right)\cdot 23^{19} + \left(21 a^{14} + 16 a^{13} + 5 a^{12} + 16 a^{11} + 16 a^{10} + 7 a^{9} + 22 a^{8} + 2 a^{7} + 22 a^{6} + 22 a^{5} + 3 a^{4} + 20 a^{3} + 22 a^{2} + 6 a + 13\right)\cdot 23^{20} + \left(2 a^{14} + 2 a^{13} + 18 a^{12} + 8 a^{11} + 21 a^{10} + 3 a^{9} + 15 a^{8} + 11 a^{7} + 16 a^{6} + 15 a^{5} + 21 a^{4} + 14 a^{3} + 16 a^{2} + 12 a + 12\right)\cdot 23^{21} + \left(7 a^{14} + 9 a^{13} + 14 a^{12} + 12 a^{11} + 15 a^{10} + 20 a^{9} + 10 a^{8} + 20 a^{7} + 13 a^{6} + 11 a^{5} + 5 a^{4} + 4 a^{3} + 2 a^{2} + 9 a + 19\right)\cdot 23^{22} + \left(7 a^{14} + 4 a^{13} + 18 a^{12} + 8 a^{11} + 6 a^{10} + 21 a^{9} + 15 a^{8} + 11 a^{7} + 3 a^{6} + 11 a^{5} + 21 a^{4} + 6 a^{3} + 18 a^{2} + 10 a + 8\right)\cdot 23^{23} + \left(a^{14} + 7 a^{13} + 18 a^{12} + 14 a^{11} + 3 a^{10} + 4 a^{9} + 11 a^{8} + a^{7} + 22 a^{6} + 10 a^{5} + 13 a^{4} + 9 a^{3} + 6 a^{2} + 1\right)\cdot 23^{24} + \left(10 a^{14} + 15 a^{13} + 18 a^{12} + 22 a^{11} + 17 a^{10} + 17 a^{9} + 15 a^{8} + 5 a^{7} + a^{6} + 21 a^{5} + 22 a^{4} + 19 a^{3} + 2 a^{2} + 7 a + 3\right)\cdot 23^{25} + \left(2 a^{14} + 17 a^{13} + 12 a^{12} + 18 a^{11} + 19 a^{10} + 13 a^{9} + 10 a^{8} + 19 a^{7} + 4 a^{6} + 18 a^{5} + 7 a^{4} + a^{3} + 5 a^{2} + 8 a + 22\right)\cdot 23^{26} + \left(11 a^{14} + 22 a^{13} + 21 a^{12} + 21 a^{11} + 19 a^{10} + 14 a^{9} + 19 a^{8} + 3 a^{7} + 13 a^{6} + 3 a^{5} + 4 a^{4} + 9 a^{3} + 17 a^{2} + 10 a + 22\right)\cdot 23^{27} + \left(8 a^{14} + 22 a^{13} + 13 a^{12} + 19 a^{10} + 22 a^{9} + 7 a^{7} + 15 a^{6} + 3 a^{5} + 4 a^{4} + 4 a^{3} + 10 a^{2} + 9 a + 17\right)\cdot 23^{28} + \left(2 a^{14} + 12 a^{13} + 14 a^{12} + 7 a^{11} + 8 a^{10} + a^{9} + 6 a^{8} + 18 a^{7} + 16 a^{6} + 9 a^{5} + 8 a^{4} + 8 a^{3} + 5 a^{2} + 18 a + 7\right)\cdot 23^{29} + \left(12 a^{14} + 21 a^{13} + 5 a^{12} + 15 a^{11} + a^{10} + 3 a^{9} + 10 a^{8} + 15 a^{7} + 20 a^{6} + 12 a^{5} + 4 a^{4} + 8 a^{3} + 11 a^{2} + 9 a + 15\right)\cdot 23^{30} + \left(8 a^{14} + 21 a^{13} + 10 a^{12} + 7 a^{11} + 3 a^{10} + 3 a^{9} + 11 a^{8} + 14 a^{7} + 10 a^{6} + 16 a^{5} + 7 a^{4} + 9 a^{3} + 6 a^{2} + 15\right)\cdot 23^{31} + \left(15 a^{14} + 7 a^{13} + 9 a^{12} + 12 a^{11} + 9 a^{10} + 11 a^{9} + 3 a^{8} + 13 a^{7} + 16 a^{6} + 16 a^{5} + 22 a^{4} + 16 a^{3} + 2 a^{2} + 6 a + 9\right)\cdot 23^{32} + \left(19 a^{14} + 18 a^{13} + 5 a^{12} + 22 a^{11} + 16 a^{10} + 13 a^{9} + 18 a^{8} + 16 a^{7} + 12 a^{6} + 17 a^{5} + 9 a^{4} + 18 a^{2} + 19 a + 2\right)\cdot 23^{33} + \left(22 a^{14} + 14 a^{13} + 18 a^{12} + 18 a^{11} + 9 a^{10} + 2 a^{9} + 7 a^{8} + 9 a^{7} + 3 a^{6} + 8 a^{5} + 3 a^{4} + 15 a^{3} + 6 a^{2} + 19 a + 15\right)\cdot 23^{34} + \left(15 a^{14} + 2 a^{13} + 8 a^{12} + 22 a^{11} + 19 a^{10} + 6 a^{9} + 2 a^{8} + 10 a^{7} + 3 a^{6} + 11 a^{5} + 4 a^{4} + 13 a^{3} + 8 a^{2} + 21 a + 4\right)\cdot 23^{35} + \left(20 a^{14} + 12 a^{13} + 19 a^{12} + 21 a^{11} + 8 a^{10} + 11 a^{9} + 15 a^{8} + 20 a^{7} + 20 a^{6} + a^{4} + 10 a^{3} + 17 a + 10\right)\cdot 23^{36} + \left(7 a^{14} + 22 a^{12} + 7 a^{11} + a^{10} + 14 a^{9} + 5 a^{8} + 16 a^{6} + 8 a^{5} + 22 a^{4} + 12 a^{3} + 2 a^{2} + 17 a\right)\cdot 23^{37} + \left(9 a^{14} + 21 a^{13} + 19 a^{12} + 6 a^{11} + 10 a^{10} + 20 a^{9} + 6 a^{8} + 10 a^{7} + 6 a^{6} + 9 a^{5} + 4 a^{4} + 8 a^{3} + 10 a^{2} + 10 a + 19\right)\cdot 23^{38} + \left(22 a^{14} + 22 a^{13} + 5 a^{12} + 7 a^{11} + 8 a^{9} + 13 a^{8} + 11 a^{7} + 2 a^{6} + 5 a^{5} + 17 a^{4} + 12 a^{3} + 4 a^{2} + 6 a + 16\right)\cdot 23^{39} + \left(14 a^{14} + 6 a^{13} + 15 a^{12} + a^{10} + 19 a^{9} + 12 a^{8} + 10 a^{7} + 20 a^{6} + 4 a^{5} + 8 a^{4} + 14 a^{3} + 7 a^{2} + 11 a + 3\right)\cdot 23^{40} + \left(16 a^{14} + 20 a^{13} + 13 a^{12} + 10 a^{11} + 7 a^{10} + 6 a^{9} + 8 a^{8} + 20 a^{7} + 9 a^{6} + 11 a^{5} + 19 a^{4} + 8 a^{3} + a^{2} + 9 a + 15\right)\cdot 23^{41} + \left(4 a^{14} + 22 a^{13} + 6 a^{12} + 12 a^{11} + a^{10} + 10 a^{9} + 5 a^{8} + 10 a^{7} + 4 a^{6} + 15 a^{5} + 16 a^{4} + 8 a^{3} + 11 a^{2} + 10 a + 22\right)\cdot 23^{42} + \left(22 a^{14} + 13 a^{13} + 2 a^{12} + 4 a^{11} + 20 a^{10} + 17 a^{9} + 4 a^{7} + 14 a^{6} + 8 a^{5} + 7 a^{4} + 17 a^{3} + 16 a^{2} + 10 a + 8\right)\cdot 23^{43} + \left(13 a^{14} + 15 a^{13} + 16 a^{12} + a^{10} + 5 a^{9} + 3 a^{8} + 6 a^{7} + 3 a^{6} + 20 a^{5} + 12 a^{4} + a^{3} + 6 a^{2} + 11 a + 20\right)\cdot 23^{44} + \left(10 a^{14} + 2 a^{13} + 5 a^{12} + 14 a^{9} + 6 a^{8} + 12 a^{6} + 20 a^{5} + 7 a^{4} + 6 a^{3} + 17 a^{2} + 13 a + 22\right)\cdot 23^{45} + \left(11 a^{14} + 2 a^{13} + 2 a^{12} + 20 a^{11} + 15 a^{10} + 17 a^{9} + a^{8} + 22 a^{7} + 22 a^{6} + 14 a^{5} + 5 a^{4} + 20 a^{3} + 12 a^{2} + 19 a + 2\right)\cdot 23^{46} + \left(12 a^{13} + 16 a^{12} + 2 a^{11} + 15 a^{10} + 16 a^{9} + 10 a^{8} + 13 a^{7} + 20 a^{6} + 7 a^{5} + 6 a^{4} + 12 a^{3} + 18 a^{2} + 12 a + 9\right)\cdot 23^{47} + \left(15 a^{13} + 18 a^{12} + 5 a^{11} + 7 a^{10} + 10 a^{9} + 7 a^{8} + 13 a^{7} + 13 a^{6} + 15 a^{5} + 18 a^{4} + 9 a^{3} + 6 a^{2} + 22 a + 22\right)\cdot 23^{48} + \left(13 a^{14} + 16 a^{13} + a^{12} + 21 a^{11} + 19 a^{10} + a^{9} + 9 a^{8} + 12 a^{7} + 10 a^{6} + 8 a^{5} + 17 a^{4} + 5 a^{3} + a + 21\right)\cdot 23^{49} + \left(16 a^{14} + 5 a^{13} + 2 a^{12} + 12 a^{11} + 11 a^{10} + 19 a^{9} + a^{8} + 21 a^{7} + 22 a^{5} + 9 a^{4} + 20 a^{3} + 12 a^{2} + 4\right)\cdot 23^{50} + \left(20 a^{14} + 14 a^{13} + 17 a^{12} + 18 a^{11} + a^{10} + 6 a^{9} + 2 a^{8} + 21 a^{7} + 9 a^{6} + 9 a^{5} + 11 a^{4} + 10 a^{3} + 10 a^{2} + 9 a + 3\right)\cdot 23^{51} + \left(4 a^{14} + 10 a^{13} + 13 a^{12} + 17 a^{11} + 14 a^{10} + 10 a^{9} + 6 a^{8} + 4 a^{7} + 4 a^{6} + 10 a^{5} + 11 a^{4} + 19 a^{3} + 17 a^{2} + 2 a + 1\right)\cdot 23^{52} + \left(19 a^{14} + 7 a^{13} + 18 a^{11} + 7 a^{10} + 5 a^{8} + 2 a^{7} + 5 a^{6} + 5 a^{5} + 12 a^{4} + 11 a^{3} + 6 a^{2} + 10 a + 22\right)\cdot 23^{53} + \left(10 a^{14} + 13 a^{13} + 20 a^{12} + 15 a^{11} + 13 a^{10} + 3 a^{9} + 12 a^{8} + 6 a^{7} + 19 a^{6} + 12 a^{5} + 2 a^{4} + 8 a^{3} + 11 a^{2} + 19 a + 12\right)\cdot 23^{54} + \left(8 a^{14} + 9 a^{13} + 13 a^{12} + 10 a^{11} + 12 a^{10} + 6 a^{9} + 4 a^{8} + 17 a^{7} + 21 a^{6} + 15 a^{5} + 14 a^{3} + 2 a + 20\right)\cdot 23^{55} + \left(14 a^{14} + 8 a^{13} + a^{12} + 11 a^{11} + 17 a^{10} + 2 a^{9} + 7 a^{8} + 22 a^{7} + 3 a^{6} + 6 a^{5} + 19 a^{4} + 7 a^{3} + 14 a^{2} + 9 a + 8\right)\cdot 23^{56} + \left(17 a^{13} + 8 a^{11} + 6 a^{10} + 14 a^{9} + 20 a^{8} + 8 a^{7} + 18 a^{6} + 4 a^{5} + 8 a^{4} + 22 a^{3} + 8 a^{2} + 17 a + 15\right)\cdot 23^{57} + \left(17 a^{14} + 6 a^{13} + 16 a^{12} + 18 a^{11} + 13 a^{10} + 14 a^{8} + 7 a^{7} + 18 a^{6} + 4 a^{5} + 10 a^{4} + 15 a^{3} + 13 a + 9\right)\cdot 23^{58} + \left(17 a^{14} + 3 a^{13} + 22 a^{12} + 10 a^{11} + 3 a^{10} + 7 a^{9} + 6 a^{8} + 8 a^{7} + 16 a^{6} + 11 a^{5} + 15 a^{4} + 7 a^{3} + 14 a^{2} + 8 a + 19\right)\cdot 23^{59} + \left(a^{14} + 21 a^{13} + 22 a^{12} + 22 a^{11} + 22 a^{10} + 20 a^{9} + 13 a^{8} + 4 a^{7} + 9 a^{6} + 3 a^{5} + 21 a^{4} + 22 a^{3} + 21 a + 19\right)\cdot 23^{60} + \left(10 a^{14} + 7 a^{13} + 22 a^{12} + 3 a^{11} + 20 a^{10} + 14 a^{9} + 11 a^{8} + 10 a^{7} + 14 a^{6} + 9 a^{5} + 7 a^{4} + 15 a^{3} + 2 a^{2} + 6 a + 18\right)\cdot 23^{61} + \left(15 a^{14} + 9 a^{13} + 3 a^{12} + 10 a^{11} + 12 a^{10} + 13 a^{9} + 22 a^{8} + 4 a^{7} + 13 a^{6} + 19 a^{5} + 18 a^{4} + 11 a^{3} + 9 a^{2} + 12 a + 18\right)\cdot 23^{62} + \left(11 a^{14} + 16 a^{13} + 6 a^{12} + 22 a^{11} + 15 a^{10} + 20 a^{9} + 22 a^{7} + 15 a^{6} + 17 a^{5} + 14 a^{4} + 18 a^{3} + 3 a^{2} + 14 a + 16\right)\cdot 23^{63} + \left(15 a^{14} + 18 a^{13} + 17 a^{12} + 10 a^{11} + 21 a^{10} + 7 a^{9} + 4 a^{8} + 14 a^{7} + 4 a^{6} + a^{5} + 10 a^{4} + 17 a^{3} + 12 a^{2} + 12 a + 5\right)\cdot 23^{64} + \left(21 a^{14} + 20 a^{13} + 18 a^{12} + 5 a^{11} + 7 a^{10} + 21 a^{9} + 12 a^{8} + 7 a^{7} + 7 a^{6} + 21 a^{5} + 14 a^{3} + 17 a^{2} + 14 a + 4\right)\cdot 23^{65} + \left(6 a^{14} + 11 a^{13} + 21 a^{12} + 7 a^{11} + 9 a^{10} + 6 a^{9} + 7 a^{8} + 15 a^{7} + 19 a^{6} + 15 a^{5} + 22 a^{4} + 17 a^{3} + 21 a^{2} + 3 a\right)\cdot 23^{66} + \left(13 a^{14} + 6 a^{13} + 15 a^{12} + 22 a^{11} + a^{10} + 21 a^{9} + 20 a^{8} + 15 a^{7} + 21 a^{6} + 20 a^{5} + 6 a^{4} + 21 a^{3} + 12 a^{2} + 5 a + 9\right)\cdot 23^{67} + \left(4 a^{14} + a^{13} + 10 a^{12} + 12 a^{11} + 22 a^{10} + 5 a^{9} + 20 a^{8} + 22 a^{7} + a^{6} + 4 a^{5} + 2 a^{4} + 5 a^{3} + 21 a^{2} + 12 a + 16\right)\cdot 23^{68} + \left(9 a^{14} + 12 a^{13} + 2 a^{12} + 11 a^{11} + 10 a^{10} + 6 a^{9} + 9 a^{8} + 19 a^{7} + 3 a^{6} + 22 a^{5} + 11 a^{4} + 2 a^{3} + 11 a^{2} + 14 a + 19\right)\cdot 23^{69} + \left(14 a^{13} + 19 a^{12} + 3 a^{11} + 15 a^{10} + 9 a^{8} + 8 a^{7} + 7 a^{6} + 19 a^{5} + 20 a^{4} + 18 a^{3} + 8 a^{2} + 15\right)\cdot 23^{70} + \left(18 a^{14} + 18 a^{13} + 18 a^{12} + 3 a^{11} + 10 a^{10} + 21 a^{9} + 12 a^{8} + 3 a^{6} + 22 a^{5} + 21 a^{4} + 21 a^{3} + 3 a^{2} + 5 a + 13\right)\cdot 23^{71} + \left(19 a^{14} + 13 a^{13} + 11 a^{11} + 8 a^{10} + 17 a^{9} + 21 a^{8} + 12 a^{7} + 15 a^{6} + 5 a^{5} + 7 a^{4} + 4 a^{3} + 6 a^{2} + 19 a + 13\right)\cdot 23^{72} + \left(16 a^{14} + 9 a^{13} + 17 a^{12} + 11 a^{11} + 13 a^{10} + 4 a^{9} + 10 a^{8} + 20 a^{7} + a^{5} + 21 a^{4} + 14 a^{3} + 8 a^{2} + 7 a + 12\right)\cdot 23^{73} + \left(10 a^{14} + 5 a^{13} + 2 a^{12} + 11 a^{11} + 14 a^{10} + 22 a^{9} + 22 a^{8} + 6 a^{7} + 2 a^{6} + 9 a^{5} + 3 a^{4} + 20 a^{3} + 11 a^{2} + 4 a + 22\right)\cdot 23^{74} + \left(9 a^{14} + 14 a^{13} + 12 a^{12} + 8 a^{11} + 19 a^{10} + 22 a^{9} + 17 a^{8} + 8 a^{7} + 11 a^{6} + 17 a^{5} + 5 a^{4} + 18 a^{3} + 13 a^{2} + 18 a + 11\right)\cdot 23^{75} + \left(16 a^{14} + 7 a^{13} + 21 a^{12} + 3 a^{11} + 22 a^{10} + 21 a^{9} + 20 a^{8} + 17 a^{7} + 17 a^{6} + 12 a^{5} + 8 a^{4} + 17 a^{3} + 6 a^{2} + 21 a + 15\right)\cdot 23^{76} + \left(15 a^{14} + 20 a^{13} + 20 a^{11} + 3 a^{10} + 22 a^{9} + 5 a^{8} + 20 a^{7} + 16 a^{6} + 7 a^{5} + 7 a^{4} + 8 a^{3} + 18 a^{2} + 17 a + 11\right)\cdot 23^{77} + \left(17 a^{14} + 16 a^{13} + 14 a^{12} + 2 a^{11} + 18 a^{10} + 6 a^{9} + 16 a^{8} + 18 a^{7} + 2 a^{6} + 15 a^{5} + 17 a^{4} + 22 a^{3} + 3 a^{2} + a + 18\right)\cdot 23^{78} + \left(10 a^{14} + 20 a^{13} + 19 a^{12} + 21 a^{11} + 8 a^{10} + 2 a^{9} + 4 a^{8} + 8 a^{7} + 17 a^{6} + 5 a^{5} + 11 a^{4} + 22 a^{3} + 10 a^{2} + 2 a + 18\right)\cdot 23^{79} + \left(16 a^{14} + 7 a^{13} + 6 a^{12} + a^{11} + 13 a^{10} + 22 a^{9} + 3 a^{8} + 19 a^{7} + 19 a^{6} + 2 a^{5} + 4 a^{4} + 14 a^{3} + 9 a^{2} + 13 a + 3\right)\cdot 23^{80} + \left(10 a^{14} + 8 a^{13} + 20 a^{12} + 21 a^{11} + 21 a^{10} + 4 a^{9} + 21 a^{7} + 18 a^{6} + 13 a^{5} + 12 a^{4} + 20 a^{3} + 4 a^{2} + 15 a + 7\right)\cdot 23^{81} + \left(11 a^{14} + a^{13} + 12 a^{12} + 11 a^{11} + 6 a^{10} + 5 a^{9} + 19 a^{8} + a^{7} + 4 a^{6} + 11 a^{5} + 11 a^{4} + 22 a^{3} + 8 a^{2} + 7 a + 14\right)\cdot 23^{82} + \left(22 a^{14} + 16 a^{13} + 15 a^{12} + 17 a^{11} + 9 a^{10} + 2 a^{9} + 22 a^{8} + a^{7} + 15 a^{6} + 13 a^{5} + 15 a^{4} + 4 a^{3} + 20 a^{2} + 7 a + 14\right)\cdot 23^{83} + \left(3 a^{14} + 18 a^{13} + 5 a^{11} + 8 a^{10} + 4 a^{9} + 14 a^{8} + 6 a^{7} + 14 a^{6} + 14 a^{5} + 14 a^{4} + 21 a^{3} + 16 a^{2} + 14 a\right)\cdot 23^{84} + \left(13 a^{13} + 19 a^{12} + 13 a^{11} + 11 a^{10} + a^{9} + 4 a^{8} + 17 a^{7} + 14 a^{6} + 15 a^{5} + 11 a^{4} + 16 a^{3} + 20 a^{2} + 9 a + 11\right)\cdot 23^{85} + \left(3 a^{14} + 5 a^{13} + 13 a^{12} + 21 a^{11} + 7 a^{10} + 8 a^{9} + 14 a^{8} + 12 a^{6} + 10 a^{5} + 21 a^{4} + 20 a^{3} + 14 a^{2} + 20\right)\cdot 23^{86} + \left(9 a^{14} + 22 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13\right)\cdot 23^{92} + \left(4 a^{13} + 5 a^{12} + 10 a^{11} + 20 a^{10} + 3 a^{9} + 13 a^{8} + 12 a^{7} + 2 a^{6} + 6 a^{5} + 19 a^{4} + 10 a^{3} + 2 a + 7\right)\cdot 23^{93} + \left(12 a^{14} + 8 a^{13} + 19 a^{12} + 10 a^{11} + 12 a^{10} + 15 a^{9} + 13 a^{8} + 17 a^{7} + 8 a^{6} + 8 a^{5} + 16 a^{3} + 15 a^{2} + 8 a + 3\right)\cdot 23^{94} + \left(5 a^{14} + 13 a^{13} + 19 a^{12} + 3 a^{11} + 4 a^{10} + 21 a^{9} + 22 a^{8} + 10 a^{7} + 21 a^{6} + 3 a^{5} + 17 a^{4} + 17 a^{3} + 13 a^{2} + 13 a + 6\right)\cdot 23^{95} + \left(17 a^{14} + 14 a^{13} + 11 a^{12} + 9 a^{11} + 20 a^{10} + a^{9} + 3 a^{8} + 7 a^{7} + 10 a^{6} + 20 a^{5} + 10 a^{4} + 22 a^{3} + 8 a^{2} + 15 a + 14\right)\cdot 23^{96} + \left(11 a^{13} + 15 a^{12} + 8 a^{11} + 5 a^{10} + 12 a^{9} + 18 a^{8} + 3 a^{7} + 7 a^{6} + 15 a^{5} + 21 a^{4} + 12 a^{3} + 4 a^{2} + a + 14\right)\cdot 23^{97} + \left(6 a^{14} + 6 a^{13} + 12 a^{11} + 15 a^{9} + 4 a^{8} + a^{7} + 11 a^{6} + 18 a^{5} + 15 a^{4} + 17 a^{2} + 5 a + 17\right)\cdot 23^{98} + \left(13 a^{14} + 16 a^{13} + 7 a^{12} + 7 a^{11} + 21 a^{10} + 9 a^{8} + 22 a^{7} + 20 a^{6} + 12 a^{5} + 7 a^{4} + 14 a^{3} + 13 a^{2} + 4\right)\cdot 23^{99} + \left(10 a^{14} + 22 a^{13} + 9 a^{12} + 21 a^{11} + 6 a^{10} + 15 a^{9} + 10 a^{8} + 3 a^{7} + 20 a^{6} + 22 a^{5} + 2 a^{4} + 14 a^{3} + 11 a^{2} + 4 a + 5\right)\cdot 23^{100} + \left(3 a^{14} + 16 a^{13} + a^{12} + 8 a^{11} + a^{10} + 8 a^{9} + 22 a^{8} + 22 a^{7} + 15 a^{6} + 9 a^{5} + 22 a^{4} + 19 a^{3} + 18 a^{2} + 19 a + 21\right)\cdot 23^{101} + \left(4 a^{14} + 5 a^{13} + 15 a^{12} + 12 a^{11} + 4 a^{10} + 5 a^{9} + 6 a^{8} + 5 a^{7} + 22 a^{6} + 18 a^{5} + 17 a^{4} + 19 a^{3} + 8 a^{2} + 16 a + 16\right)\cdot 23^{102} + \left(6 a^{14} + 2 a^{13} + 2 a^{12} + 17 a^{11} + a^{10} + 17 a^{9} + 21 a^{7} + 17 a^{6} + 2 a^{5} + 15 a^{4} + 7 a^{3} + 2 a^{2} + 16 a + 4\right)\cdot 23^{103} + \left(21 a^{14} + 20 a^{12} + 15 a^{11} + 9 a^{10} + 11 a^{9} + 17 a^{8} + 18 a^{7} + a^{6} + 18 a^{5} + 13 a^{4} + a^{3} + 19 a^{2} + 21 a + 15\right)\cdot 23^{104} + \left(10 a^{14} + 15 a^{13} + 18 a^{12} + 5 a^{11} + 19 a^{10} + 18 a^{9} + 7 a^{8} + 12 a^{7} + 22 a^{6} + 20 a^{5} + 19 a^{4} + 13 a^{3} + 12 a^{2} + 4 a + 17\right)\cdot 23^{105} + \left(a^{14} + 16 a^{13} + 22 a^{12} + 12 a^{11} + 16 a^{10} + 3 a^{9} + 5 a^{8} + 18 a^{7} + 15 a^{6} + 4 a^{5} + 9 a^{4} + 9 a^{3} + 18 a^{2} + 4 a + 5\right)\cdot 23^{106} + \left(13 a^{14} + 16 a^{13} + 21 a^{12} + 19 a^{11} + 18 a^{10} + 16 a^{9} + 9 a^{8} + 11 a^{7} + 15 a^{6} + 14 a^{5} + 6 a^{4} + 17 a^{3} + 22 a^{2} + 11 a + 4\right)\cdot 23^{107} + \left(9 a^{14} + 2 a^{13} + 9 a^{12} + a^{11} + 22 a^{10} + 17 a^{9} + 11 a^{8} + 12 a^{7} + 8 a^{6} + 14 a^{5} + 13 a^{4} + 6 a^{2} + 18 a + 14\right)\cdot 23^{108} + \left(17 a^{14} + 8 a^{13} + 19 a^{12} + 11 a^{11} + 5 a^{10} + 19 a^{9} + 22 a^{8} + 11 a^{7} + 6 a^{6} + 3 a^{5} + 16 a^{4} + 2 a^{3} + 5 a^{2} + 16 a + 21\right)\cdot 23^{109} + \left(7 a^{14} + 16 a^{13} + 22 a^{12} + 13 a^{11} + 11 a^{10} + 3 a^{9} + 9 a^{7} + 3 a^{6} + a^{5} + 21 a^{3} + 10 a^{2} + 18 a + 9\right)\cdot 23^{110} + \left(7 a^{14} + 21 a^{13} + 7 a^{12} + 16 a^{11} + 8 a^{10} + 17 a^{9} + 10 a^{8} + 15 a^{7} + 15 a^{6} + 8 a^{5} + 14 a^{4} + 5 a^{3} + 10 a^{2} + 4 a + 9\right)\cdot 23^{111} + \left(14 a^{14} + 11 a^{13} + 14 a^{12} + 8 a^{11} + 2 a^{10} + 7 a^{9} + 22 a^{8} + 12 a^{7} + 16 a^{6} + 21 a^{5} + 10 a^{4} + 15 a^{3} + 15 a^{2} + 6 a + 6\right)\cdot 23^{112} + \left(10 a^{14} + 14 a^{13} + 5 a^{12} + 4 a^{11} + 11 a^{10} + 22 a^{9} + 16 a^{8} + 11 a^{7} + 14 a^{6} + 21 a^{4} + 6 a^{3} + 14 a^{2} + 21 a + 20\right)\cdot 23^{113} + \left(10 a^{14} + 19 a^{13} + 18 a^{12} + 4 a^{11} + 10 a^{10} + 15 a^{9} + 7 a^{8} + 3 a^{7} + 19 a^{6} + 3 a^{5} + 10 a^{3} + 14 a^{2} + 8 a + 2\right)\cdot 23^{114} + \left(11 a^{14} + 19 a^{12} + 13 a^{11} + 8 a^{10} + 11 a^{9} + 15 a^{8} + 11 a^{7} + 13 a^{6} + 6 a^{5} + 6 a^{4} + 21 a^{3} + 15 a^{2} + 19 a + 18\right)\cdot 23^{115} + \left(15 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9\right)\cdot 23^{121} + \left(3 a^{14} + 2 a^{13} + 20 a^{12} + 10 a^{11} + 19 a^{10} + 3 a^{8} + 22 a^{7} + 2 a^{6} + 16 a^{5} + 9 a^{4} + 2 a^{3} + 16 a^{2} + 14 a + 6\right)\cdot 23^{122} + \left(a^{14} + 13 a^{13} + 4 a^{12} + 6 a^{11} + 3 a^{10} + 20 a^{9} + 10 a^{8} + 10 a^{7} + 13 a^{6} + 19 a^{5} + 21 a^{4} + 10 a^{3} + 19 a^{2} + 7 a + 8\right)\cdot 23^{123} + \left(3 a^{14} + 12 a^{13} + 20 a^{12} + 14 a^{11} + 9 a^{10} + 5 a^{9} + 18 a^{8} + 20 a^{7} + 9 a^{6} + 6 a^{5} + a^{4} + 16 a^{3} + 4 a^{2} + 11 a + 4\right)\cdot 23^{124} + \left(13 a^{14} + 7 a^{13} + 13 a^{12} + a^{11} + 18 a^{10} + a^{9} + 9 a^{8} + 7 a^{6} + 10 a^{5} + 15 a^{4} + 16 a^{3} + 11 a^{2} + 14 a + 4\right)\cdot 23^{125} + \left(21 a^{13} + 21 a^{12} + 15 a^{11} + 15 a^{10} + 17 a^{9} + 12 a^{8} + 12 a^{7} + 5 a^{6} + 2 a^{5} + a^{4} + a^{3} + 19 a^{2} + 7 a + 15\right)\cdot 23^{126} + \left(3 a^{14} + 16 a^{13} + 17 a^{12} + 11 a^{11} + 3 a^{10} + 10 a^{9} + 10 a^{8} + 14 a^{7} + 19 a^{6} + 14 a^{5} + 11 a^{4} + 22 a^{3} + a^{2} + 21 a + 1\right)\cdot 23^{127} + \left(2 a^{14} + 7 a^{13} + 7 a^{12} + 9 a^{11} + 11 a^{10} + 16 a^{9} + 7 a^{8} + 18 a^{7} + a^{6} + 18 a^{5} + 21 a^{4} + 7 a^{3} + 16 a^{2} + 10 a + 14\right)\cdot 23^{128} + \left(2 a^{14} + 11 a^{13} + 16 a^{12} + 19 a^{11} + 21 a^{10} + 10 a^{9} + 7 a^{8} + 4 a^{7} + 12 a^{6} + 10 a^{5} + 6 a^{4} + 21 a^{3} + 2 a^{2} + 11 a + 21\right)\cdot 23^{129} + \left(12 a^{14} + 11 a^{13} + 9 a^{12} + 18 a^{11} + 12 a^{10} + 6 a^{9} + 11 a^{8} + 6 a^{7} + 16 a^{6} + 9 a^{5} + 13 a^{4} + 15 a^{3} + 9 a^{2} + 12 a + 20\right)\cdot 23^{130} + \left(17 a^{14} + 10 a^{13} + 18 a^{12} + 14 a^{11} + 13 a^{10} + 3 a^{9} + 9 a^{8} + 22 a^{7} + 16 a^{6} + 12 a^{5} + 9 a^{4} + 5 a^{3} + 4 a^{2} + 4\right)\cdot 23^{131} + \left(22 a^{14} + 17 a^{13} + 3 a^{12} + 11 a^{11} + a^{10} + 12 a^{9} + 16 a^{8} + 12 a^{7} + 5 a^{6} + 12 a^{5} + 15 a^{4} + 10 a^{3} + a^{2} + 5 a + 15\right)\cdot 23^{132} + \left(20 a^{14} + a^{13} + a^{12} + a^{11} + 6 a^{10} + 13 a^{9} + 10 a^{8} + 20 a^{7} + 10 a^{6} + 4 a^{5} + 18 a^{4} + a^{3} + 14 a^{2} + 4 a + 9\right)\cdot 23^{133} + \left(16 a^{14} + 11 a^{13} + 3 a^{12} + 18 a^{11} + a^{10} + 16 a^{9} + 14 a^{8} + 2 a^{7} + 22 a^{6} + 14 a^{5} + 10 a^{4} + 22 a^{3} + 21 a^{2} + 5 a + 9\right)\cdot 23^{134} + \left(3 a^{14} + 20 a^{13} + 20 a^{12} + 12 a^{11} + 17 a^{10} + 21 a^{9} + 20 a^{8} + 11 a^{7} + 13 a^{6} + 12 a^{5} + 7 a^{4} + 10 a^{3} + 7 a^{2} + 18 a + 19\right)\cdot 23^{135} + \left(5 a^{14} + a^{13} + 10 a^{12} + a^{11} + 4 a^{10} + 3 a^{9} + 11 a^{8} + 12 a^{7} + 14 a^{6} + 22 a^{5} + 14 a^{4} + 6 a^{3} + 3 a^{2} + 4 a + 15\right)\cdot 23^{136} + \left(18 a^{14} + 18 a^{13} + 22 a^{12} + 8 a^{11} + a^{10} + 6 a^{9} + a^{8} + 16 a^{7} + 12 a^{6} + 19 a^{5} + 20 a^{4} + 18 a^{2} + 18 a + 5\right)\cdot 23^{137} + \left(8 a^{14} + 10 a^{13} + 9 a^{12} + 22 a^{10} + 17 a^{9} + 22 a^{8} + 19 a^{7} + 18 a^{5} + 21 a^{4} + 20 a^{3} + 21 a + 8\right)\cdot 23^{138} + \left(20 a^{14} + 11 a^{13} + 8 a^{12} + 21 a^{11} + 22 a^{10} + 21 a^{9} + 19 a^{8} + 18 a^{7} + 8 a^{6} + 7 a^{5} + 8 a^{4} + 10 a^{3} + 9 a^{2} + a + 6\right)\cdot 23^{139} + \left(12 a^{14} + 6 a^{13} + 10 a^{12} + 12 a^{11} + 3 a^{10} + 16 a^{9} + 10 a^{8} + 5 a^{7} + 16 a^{6} + 10 a^{5} + 17 a^{4} + 13 a^{3} + 22 a^{2} + a + 15\right)\cdot 23^{140} + \left(15 a^{14} + 10 a^{13} + 2 a^{12} + 7 a^{11} + 7 a^{10} + 2 a^{9} + 2 a^{8} + 15 a^{7} + 22 a^{6} + 20 a^{5} + 2 a^{4} + 7 a^{3} + 17 a + 22\right)\cdot 23^{141} + \left(21 a^{14} + 10 a^{13} + 7 a^{12} + 20 a^{11} + 18 a^{10} + 13 a^{9} + 16 a^{8} + 20 a^{7} + 8 a^{6} + 22 a^{5} + 22 a^{4} + 20 a^{2} + 3 a + 10\right)\cdot 23^{142} + \left(20 a^{14} + 12 a^{13} + 6 a^{12} + 14 a^{11} + 17 a^{10} + a^{9} + 3 a^{8} + 16 a^{6} + 2 a^{5} + 3 a^{4} + 19 a^{3} + 22 a^{2} + 8\right)\cdot 23^{143} + \left(5 a^{14} + 4 a^{12} + 7 a^{11} + 7 a^{10} + 19 a^{9} + 22 a^{8} + 20 a^{7} + 12 a^{6} + 7 a^{5} + 18 a^{4} + 13 a^{3} + 7 a^{2} + a + 17\right)\cdot 23^{144} + \left(21 a^{14} + 10 a^{13} + 15 a^{12} + 3 a^{11} + 6 a^{10} + 11 a^{9} + 15 a^{8} + 19 a^{7} + 16 a^{6} + 2 a^{5} + 19 a^{4} + 5 a^{2} + 7 a + 1\right)\cdot 23^{145} + \left(13 a^{14} + 11 a^{13} + 10 a^{12} + 7 a^{11} + 10 a^{10} + 5 a^{9} + 18 a^{8} + a^{7} + 21 a^{6} + 6 a^{5} + 19 a^{4} + 11 a^{3} + 13 a^{2} + 3 a + 22\right)\cdot 23^{146} + \left(12 a^{14} + 12 a^{13} + 5 a^{12} + 17 a^{11} + 2 a^{10} + 18 a^{9} + 4 a^{8} + 19 a^{7} + 20 a^{6} + 10 a^{5} + 10 a^{4} + 7 a^{3} + 9 a^{2} + 16\right)\cdot 23^{147} + \left(16 a^{14} + a^{13} + 3 a^{12} + 9 a^{11} + 6 a^{10} + 22 a^{9} + 9 a^{8} + 22 a^{7} + 3 a^{6} + 2 a^{5} + 16 a^{4} + 17 a^{3} + 16 a^{2} + 5 a + 6\right)\cdot 23^{148} + \left(11 a^{14} + 13 a^{13} + 19 a^{12} + 14 a^{11} + 10 a^{10} + 10 a^{9} + 3 a^{8} + 8 a^{7} + 9 a^{6} + 5 a^{5} + 22 a^{4} + 20 a^{3} + 15 a^{2} + 10 a + 10\right)\cdot 23^{149} + \left(18 a^{14} + 22 a^{13} + 11 a^{12} + 17 a^{11} + 12 a^{10} + 20 a^{9} + 22 a^{8} + 2 a^{7} + 11 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21 a^{4} + 22 a^{3} + 2 a^{2} + 2 a + 20\right)\cdot 23^{173} + \left(14 a^{14} + 12 a^{13} + 22 a^{12} + 10 a^{11} + 12 a^{10} + 22 a^{9} + 21 a^{8} + 8 a^{7} + 2 a^{6} + 7 a^{5} + 2 a^{4} + 3 a^{3} + 22 a^{2} + 12 a + 16\right)\cdot 23^{174} + \left(6 a^{14} + 16 a^{13} + 13 a^{12} + 2 a^{11} + 6 a^{10} + 9 a^{9} + 15 a^{8} + 9 a^{7} + 17 a^{6} + 6 a^{5} + 7 a^{4} + 13 a^{3} + 4 a^{2} + 11 a + 18\right)\cdot 23^{175} + \left(9 a^{13} + 17 a^{12} + 4 a^{11} + 5 a^{10} + 17 a^{9} + 20 a^{8} + 3 a^{7} + 19 a^{6} + 17 a^{5} + 14 a^{4} + 8 a^{3} + 8 a^{2} + 20 a + 18\right)\cdot 23^{176} + \left(21 a^{14} + 16 a^{13} + 21 a^{12} + 14 a^{11} + 15 a^{10} + 3 a^{7} + 8 a^{6} + 22 a^{5} + 7 a^{4} + 17 a^{2} + 14 a + 5\right)\cdot 23^{177} + \left(21 a^{14} + 7 a^{13} + 20 a^{12} + 17 a^{11} + 19 a^{10} + 15 a^{9} + 2 a^{8} + 12 a^{7} + a^{6} + 15 a^{5} + 18 a^{4} + 12 a^{3} + a^{2} + 7 a + 16\right)\cdot 23^{178} + \left(5 a^{14} + 13 a^{13} + 18 a^{12} + 15 a^{10} + 20 a^{9} + 22 a^{8} + 20 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23^{190} + \left(a^{14} + 17 a^{12} + 17 a^{11} + 11 a^{10} + 10 a^{9} + 18 a^{8} + 18 a^{7} + 11 a^{6} + 3 a^{5} + 2 a^{4} + 9 a^{3} + 6 a^{2} + 11\right)\cdot 23^{191} + \left(17 a^{14} + 10 a^{13} + 4 a^{12} + 5 a^{11} + 12 a^{10} + 22 a^{9} + 16 a^{8} + 19 a^{7} + 7 a^{6} + 10 a^{5} + 3 a^{4} + 7 a^{3} + 22 a^{2} + 19 a + 1\right)\cdot 23^{192} + \left(22 a^{14} + 5 a^{13} + 6 a^{12} + 17 a^{11} + 5 a^{10} + 22 a^{9} + 2 a^{8} + 13 a^{7} + 9 a^{6} + 7 a^{5} + 16 a^{4} + 2 a^{3} + 10 a^{2} + 8 a + 8\right)\cdot 23^{193} + \left(15 a^{14} + 16 a^{13} + 8 a^{12} + 15 a^{11} + 10 a^{10} + 5 a^{9} + 21 a^{8} + 18 a^{7} + 10 a^{6} + 13 a^{5} + 17 a^{4} + 7 a^{3} + 9 a^{2} + a + 9\right)\cdot 23^{194} + \left(2 a^{14} + 19 a^{13} + 11 a^{12} + 2 a^{11} + 17 a^{10} + 6 a^{9} + 19 a^{8} + 2 a^{7} + 10 a^{6} + 14 a^{5} + 4 a^{4} + 3 a^{3} + 11 a^{2} + 20 a + 17\right)\cdot 23^{195} + \left(19 a^{14} + 22 a^{13} + 17 a^{12} + 19 a^{11} + 12 a^{10} + 7 a^{9} + 17 a^{8} + 15 a^{7} + 19 a^{6} + 16 a^{5} + 13 a^{3} + 11 a^{2} + 12 a + 9\right)\cdot 23^{196} + \left(5 a^{14} + 2 a^{13} + 20 a^{12} + 10 a^{11} + 12 a^{10} + 15 a^{9} + 5 a^{8} + 6 a^{7} + 9 a^{6} + 12 a^{5} + 10 a^{4} + 12 a^{3} + 16 a^{2} + 8 a + 16\right)\cdot 23^{197} + \left(4 a^{14} + 10 a^{13} + 5 a^{12} + 11 a^{11} + 9 a^{10} + 17 a^{9} + 21 a^{8} + 12 a^{7} + 12 a^{6} + 6 a^{5} + 13 a^{4} + a^{3} + 21 a^{2} + 16 a + 6\right)\cdot 23^{198} + \left(13 a^{14} + 22 a^{13} + 4 a^{12} + 11 a^{11} + 21 a^{10} + 4 a^{8} + 9 a^{7} + 9 a^{6} + 8 a^{5} + 22 a^{4} + 16 a^{3} + a^{2} + 13 a + 13\right)\cdot 23^{199} + \left(14 a^{14} + 10 a^{13} + 7 a^{12} + 17 a^{11} + 12 a^{10} + 11 a^{9} + 2 a^{8} + 16 a^{6} + 13 a^{5} + 15 a^{4} + 3 a^{3} + 18 a + 5\right)\cdot 23^{200} + \left(22 a^{14} + 12 a^{13} + 22 a^{12} + 7 a^{11} + 4 a^{10} + 13 a^{9} + 16 a^{8} + 13 a^{7} + 12 a^{6} + 7 a^{5} + a^{4} + 18 a^{3} + 3 a^{2} + 15 a + 12\right)\cdot 23^{201} + \left(10 a^{14} + 6 a^{13} + 5 a^{12} + 15 a^{11} + 19 a^{10} + 19 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19\right)\cdot 23^{213} + \left(3 a^{14} + 13 a^{13} + 5 a^{11} + 11 a^{10} + a^{9} + a^{8} + a^{7} + 4 a^{6} + 9 a^{5} + 11 a^{4} + 19 a^{3} + 19 a^{2} + 15 a\right)\cdot 23^{214} + \left(18 a^{14} + 4 a^{13} + 10 a^{12} + a^{11} + 7 a^{10} + 2 a^{9} + 12 a^{8} + 5 a^{7} + 16 a^{6} + 13 a^{5} + 22 a^{3} + 10 a^{2} + 21 a + 9\right)\cdot 23^{215} + \left(11 a^{14} + 6 a^{13} + 4 a^{12} + 4 a^{11} + 16 a^{10} + 19 a^{9} + 6 a^{8} + 19 a^{7} + 4 a^{6} + 19 a^{5} + 14 a^{4} + 12 a^{3} + 19 a^{2} + 18 a + 9\right)\cdot 23^{216} + \left(6 a^{14} + 13 a^{13} + 10 a^{12} + 10 a^{11} + 7 a^{10} + a^{9} + 6 a^{7} + 4 a^{6} + 3 a^{5} + 6 a^{4} + 6 a^{3} + 20 a^{2} + 4 a + 3\right)\cdot 23^{217} + \left(12 a^{14} + 11 a^{13} + 17 a^{12} + 16 a^{11} + 22 a^{10} + 14 a^{9} + 22 a^{8} + 10 a^{7} + 19 a^{6} + 10 a^{5} + 14 a^{4} + 21 a^{3} + 12 a^{2} + 20 a + 18\right)\cdot 23^{218} + \left(17 a^{14} + 9 a^{13} + 19 a^{12} + 19 a^{11} + 20 a^{10} + 12 a^{8} + 5 a^{7} + 5 a^{6} + 17 a^{5} + 16 a^{4} + 12 a^{3} + 9 a^{2} + 10 a + 16\right)\cdot 23^{219} + \left(2 a^{14} + 5 a^{13} + 10 a^{12} + 13 a^{11} + 13 a^{10} + 7 a^{9} + 8 a^{8} + 11 a^{7} + 2 a^{6} + 18 a^{5} + 11 a^{4} + 20 a^{3} + 16 a^{2} + 15 a + 12\right)\cdot 23^{220} + \left(13 a^{14} + 2 a^{13} + 9 a^{12} + 6 a^{11} + 20 a^{10} + 15 a^{9} + 22 a^{7} + 15 a^{6} + 19 a^{5} + 20 a^{3} + a^{2} + 2 a + 6\right)\cdot 23^{221} + \left(5 a^{14} + 18 a^{12} + 15 a^{11} + 2 a^{10} + 18 a^{8} + 11 a^{7} + 16 a^{6} + 22 a^{5} + 17 a^{4} + 22 a^{3} + 4 a^{2} + a + 16\right)\cdot 23^{222} + \left(12 a^{14} + 12 a^{13} + 10 a^{12} + 16 a^{11} + 15 a^{10} + 4 a^{9} + 7 a^{8} + 19 a^{7} + 17 a^{6} + 4 a^{5} + 10 a^{4} + 20 a^{3} + 5 a^{2} + 11 a + 6\right)\cdot 23^{223} + \left(11 a^{14} + 7 a^{13} + 14 a^{12} + 20 a^{11} + 8 a^{10} + 19 a^{8} + 3 a^{7} + 11 a^{6} + 4 a^{5} + 15 a^{4} + 2 a^{3} + 15 a^{2} + 8 a + 5\right)\cdot 23^{224} + \left(11 a^{14} + 10 a^{13} + 21 a^{12} + 5 a^{11} + 16 a^{10} + 13 a^{9} + 15 a^{8} + 19 a^{7} + a^{6} + 14 a^{5} + 20 a^{4} + 17 a^{3} + 7 a^{2} + 20 a + 6\right)\cdot 23^{225} + \left(20 a^{13} + 20 a^{12} + 6 a^{11} + 20 a^{10} + 21 a^{9} + 9 a^{8} + 5 a^{7} + 5 a^{6} + 10 a^{5} + 6 a^{4} + 12 a^{3} + 15 a^{2} + 20 a + 17\right)\cdot 23^{226} + \left(4 a^{14} + 17 a^{13} + 17 a^{12} + 2 a^{11} + 7 a^{10} + 19 a^{9} + 13 a^{8} + 14 a^{7} + 8 a^{6} + 5 a^{5} + 6 a^{4} + 18 a^{3} + 6 a^{2} + 21 a + 10\right)\cdot 23^{227} + \left(18 a^{14} + 18 a^{13} + a^{12} + 14 a^{11} + 8 a^{10} + 4 a^{9} + 21 a^{8} + 21 a^{7} + 4 a^{6} + 22 a^{5} + a^{4} + 5 a^{3} + 20 a^{2} + 5 a + 9\right)\cdot 23^{228} + \left(22 a^{14} + 11 a^{13} + 5 a^{12} + 20 a^{11} + 6 a^{10} + 12 a^{9} + 2 a^{8} + a^{7} + 3 a^{6} + 12 a^{5} + 15 a^{4} + 8 a^{3} + 18 a^{2} + 17 a + 19\right)\cdot 23^{229} + \left(10 a^{14} + 13 a^{13} + 3 a^{12} + 17 a^{11} + a^{10} + 13 a^{9} + 2 a^{8} + 6 a^{7} + 14 a^{6} + 3 a^{5} + 22 a^{4} + 5 a^{3} + 6 a + 1\right)\cdot 23^{230} + \left(13 a^{14} + 20 a^{13} + 3 a^{12} + 17 a^{11} + 14 a^{10} + 13 a^{9} + 12 a^{8} + 8 a^{7} + 18 a^{6} + 6 a^{5} + 4 a^{4} + 13 a^{3} + 22 a^{2} + 11 a + 21\right)\cdot 23^{231} + \left(13 a^{14} + 16 a^{13} + 17 a^{12} + 7 a^{11} + 20 a^{10} + 11 a^{9} + 8 a^{8} + 13 a^{7} + 13 a^{6} + 20 a^{5} + 4 a^{4} + 17 a^{3} + 19 a^{2} + 19 a + 19\right)\cdot 23^{232} + \left(16 a^{14} + 8 a^{13} + 17 a^{12} + 17 a^{11} + 10 a^{10} + 16 a^{9} + 5 a^{8} + 14 a^{7} + 20 a^{6} + 15 a^{5} + 7 a^{4} + 7 a^{3} + 6 a^{2} + 13 a + 5\right)\cdot 23^{233} + \left(13 a^{14} + 7 a^{13} + 21 a^{12} + 5 a^{10} + 21 a^{9} + 3 a^{8} + 2 a^{7} + 2 a^{6} + 17 a^{5} + 4 a^{4} + 14 a^{3} + 9 a^{2} + 3 a + 19\right)\cdot 23^{234} + \left(18 a^{14} + 8 a^{13} + 10 a^{12} + 3 a^{11} + 11 a^{10} + 9 a^{8} + 8 a^{7} + 14 a^{6} + 13 a^{5} + 20 a^{4} + 3 a^{3} + 17 a^{2} + 13 a + 20\right)\cdot 23^{235} + \left(4 a^{14} + 9 a^{13} + 18 a^{12} + 22 a^{11} + 22 a^{10} + 12 a^{9} + 9 a^{8} + 4 a^{7} + 21 a^{6} + 3 a^{5} + 20 a^{4} + 13 a^{3} + 12 a^{2} + 2 a + 22\right)\cdot 23^{236} + \left(2 a^{14} + 6 a^{13} + 4 a^{12} + 10 a^{11} + 3 a^{10} + 21 a^{9} + 18 a^{8} + 13 a^{7} + 6 a^{6} + 2 a^{5} + 2 a^{4} + 17 a^{3} + 8 a^{2} + 9 a + 2\right)\cdot 23^{237} + \left(6 a^{14} + 14 a^{13} + a^{12} + 17 a^{11} + 8 a^{10} + 9 a^{9} + 16 a^{8} + 4 a^{7} + a^{6} + 20 a^{5} + 11 a^{4} + 3 a^{2} + 2 a + 12\right)\cdot 23^{238} + \left(13 a^{14} + 11 a^{13} + 21 a^{12} + 18 a^{11} + a^{10} + 10 a^{9} + 11 a^{8} + 2 a^{7} + 3 a^{6} + 16 a^{5} + 19 a^{4} + 14 a^{3} + 2 a^{2} + 10 a + 3\right)\cdot 23^{239} + \left(13 a^{14} + 6 a^{13} + 7 a^{12} + 13 a^{11} + 18 a^{10} + 7 a^{9} + 12 a^{8} + 18 a^{7} + 19 a^{6} + 16 a^{5} + 6 a^{4} + 14 a^{3} + 14 a^{2} + 3 a + 16\right)\cdot 23^{240} + \left(2 a^{14} + 3 a^{13} + 22 a^{12} + 22 a^{11} + 7 a^{10} + 13 a^{9} + 13 a^{8} + 5 a^{7} + 20 a^{6} + 14 a^{5} + 10 a^{4} + 18 a^{3} + 13 a^{2} + 7 a + 10\right)\cdot 23^{241} + \left(21 a^{14} + 13 a^{13} + 22 a^{12} + 12 a^{11} + 21 a^{10} + 4 a^{9} + 10 a^{8} + 18 a^{7} + 22 a^{6} + 15 a^{5} + 5 a^{4} + 2 a^{3} + 17 a^{2} + 17 a + 17\right)\cdot 23^{242} + \left(3 a^{14} + 3 a^{12} + 10 a^{11} + 14 a^{10} + 17 a^{9} + 14 a^{8} + 15 a^{7} + 9 a^{6} + 6 a^{4} + 13 a^{3} + 5 a^{2} + 15 a + 22\right)\cdot 23^{243} + \left(19 a^{14} + 10 a^{13} + 21 a^{12} + 17 a^{11} + 17 a^{10} + 8 a^{8} + 7 a^{7} + 14 a^{6} + 4 a^{5} + 17 a^{4} + 9 a^{3} + 7 a^{2} + 17 a + 2\right)\cdot 23^{244} + \left(2 a^{14} + 2 a^{13} + 18 a^{12} + 4 a^{11} + 14 a^{10} + 2 a^{9} + 21 a^{8} + 5 a^{6} + 7 a^{5} + 20 a^{4} + 19 a^{3} + a^{2} + 8 a + 18\right)\cdot 23^{245} + \left(15 a^{14} + 22 a^{13} + 22 a^{12} + 15 a^{11} + 6 a^{10} + 5 a^{9} + 2 a^{8} + 14 a^{7} + 9 a^{6} + 13 a^{5} + 10 a^{4} + 4 a^{3} + 14 a^{2} + 11 a + 16\right)\cdot 23^{246} + \left(4 a^{14} + 21 a^{13} + 7 a^{12} + 7 a^{11} + 21 a^{10} + 17 a^{9} + 16 a^{8} + 12 a^{7} + 12 a^{6} + a^{5} + 22 a^{4} + 19 a^{3} + 21 a^{2} + 6 a + 10\right)\cdot 23^{247} + \left(11 a^{14} + 12 a^{13} + 17 a^{12} + 14 a^{11} + 4 a^{10} + 14 a^{9} + 17 a^{8} + 2 a^{7} + 17 a^{6} + 13 a^{5} + 5 a^{4} + 11 a^{3} + 15 a^{2} + 3 a + 4\right)\cdot 23^{248} + \left(2 a^{13} + a^{12} + 22 a^{11} + 9 a^{10} + 10 a^{9} + 5 a^{8} + a^{7} + 13 a^{6} + 5 a^{5} + 12 a^{4} + 21 a^{3} + 21 a^{2} + 8 a\right)\cdot 23^{249} + \left(a^{14} + 12 a^{13} + 19 a^{12} + 3 a^{11} + 17 a^{10} + 22 a^{9} + 18 a^{8} + 9 a^{7} + 4 a^{6} + a^{5} + 18 a^{4} + 5 a^{3} + 21 a^{2} + 21 a + 18\right)\cdot 23^{250} + \left(21 a^{13} + 15 a^{12} + 5 a^{11} + 4 a^{10} + 4 a^{9} + 19 a^{8} + 20 a^{7} + 11 a^{6} + 7 a^{5} + 11 a^{4} + 6 a^{3} + 10 a^{2} + 6 a + 8\right)\cdot 23^{251} + \left(2 a^{14} + 11 a^{13} + 18 a^{12} + 13 a^{11} + 14 a^{10} + 4 a^{9} + 7 a^{8} + 13 a^{7} + 8 a^{6} + 13 a^{5} + 12 a^{4} + 8 a^{3} + 14 a^{2} + 8 a + 1\right)\cdot 23^{252} + \left(8 a^{14} + 19 a^{12} + 13 a^{11} + 21 a^{10} + 12 a^{9} + 21 a^{8} + 3 a^{7} + 22 a^{6} + 19 a^{5} + 7 a^{3} + 19 a^{2} + 16 a + 3\right)\cdot 23^{253} + \left(15 a^{14} + 10 a^{13} + 12 a^{12} + 4 a^{11} + 6 a^{10} + 16 a^{9} + 15 a^{7} + 5 a^{6} + 19 a^{5} + 14 a^{4} + 7 a^{3} + 20 a^{2} + 2 a + 3\right)\cdot 23^{254} + \left(4 a^{14} + 22 a^{13} + 10 a^{12} + 21 a^{11} + 9 a^{10} + 20 a^{9} + 2 a^{8} + a^{7} + 5 a^{6} + 21 a^{5} + 17 a^{4} + 14 a^{3} + 11 a^{2} + 16 a + 1\right)\cdot 23^{255} + \left(13 a^{14} + 16 a^{13} + 14 a^{12} + 4 a^{11} + 13 a^{10} + 22 a^{9} + 9 a^{8} + a^{7} + 12 a^{6} + 19 a^{5} + 6 a^{4} + 8 a^{3} + 14 a^{2} + 21 a + 5\right)\cdot 23^{256} + \left(18 a^{14} + 8 a^{13} + 17 a^{12} + 6 a^{11} + 15 a^{10} + 13 a^{9} + 8 a^{8} + 13 a^{7} + 19 a^{6} + 18 a^{4} + 5 a^{3} + 17 a^{2} + 4 a + 18\right)\cdot 23^{257} + \left(18 a^{12} + 18 a^{11} + 21 a^{10} + 17 a^{9} + 13 a^{8} + 22 a^{7} + 6 a^{6} + 10 a^{5} + 7 a^{4} + 6 a^{3} + 22 a^{2} + 13 a + 22\right)\cdot 23^{258} + \left(21 a^{14} + 11 a^{13} + 6 a^{12} + 5 a^{11} + 10 a^{10} + 4 a^{9} + 15 a^{8} + 17 a^{7} + 8 a^{6} + 15 a^{5} + 22 a^{4} + 5 a^{3} + 17 a^{2} + 14\right)\cdot 23^{259} + \left(8 a^{13} + 3 a^{12} + 20 a^{11} + 11 a^{10} + 12 a^{9} + 4 a^{8} + 5 a^{7} + a^{6} + 17 a^{5} + 11 a^{4} + 3 a^{3} + 19 a^{2} + 15\right)\cdot 23^{260} + \left(12 a^{14} + 14 a^{13} + 20 a^{12} + 15 a^{11} + 2 a^{10} + 5 a^{9} + 18 a^{8} + 22 a^{7} + 14 a^{6} + 5 a^{5} + 7 a^{4} + a^{3} + 4 a^{2} + 12 a + 17\right)\cdot 23^{261} + \left(18 a^{14} + 10 a^{13} + 21 a^{12} + 15 a^{11} + 8 a^{10} + 16 a^{9} + 7 a^{8} + 14 a^{7} + 10 a^{6} + 6 a^{5} + 12 a^{4} + 12 a^{3} + 21 a^{2} + 19 a + 20\right)\cdot 23^{262} + \left(7 a^{14} + a^{13} + 22 a^{12} + 4 a^{11} + 11 a^{10} + 15 a^{9} + 2 a^{8} + 3 a^{7} + 21 a^{6} + 4 a^{5} + 13 a^{4} + 4 a^{3} + 18 a^{2} + 20 a + 9\right)\cdot 23^{263} + \left(14 a^{14} + 22 a^{13} + 17 a^{12} + 2 a^{11} + 14 a^{10} + 16 a^{9} + a^{7} + 15 a^{6} + 2 a^{5} + 17 a^{4} + 6 a^{3} + 5 a^{2} + 16 a + 8\right)\cdot 23^{264} + \left(5 a^{14} + 16 a^{13} + 6 a^{12} + 5 a^{11} + 13 a^{10} + 8 a^{9} + 22 a^{8} + 8 a^{7} + 21 a^{6} + 7 a^{4} + 14 a^{3} + 15 a^{2} + 19 a + 2\right)\cdot 23^{265} + \left(13 a^{14} + 20 a^{13} + 20 a^{11} + 14 a^{10} + 20 a^{9} + 5 a^{8} + 21 a^{7} + 15 a^{6} + 22 a^{5} + 11 a^{4} + 12 a^{3} + 22 a^{2} + 5 a\right)\cdot 23^{266} + \left(3 a^{14} + 13 a^{12} + 4 a^{11} + 18 a^{10} + 18 a^{9} + 8 a^{8} + 6 a^{7} + 4 a^{6} + 16 a^{4} + 5 a^{3} + 3 a^{2} + 21 a + 7\right)\cdot 23^{267} + \left(3 a^{14} + 15 a^{13} + 3 a^{12} + 8 a^{11} + a^{10} + 14 a^{9} + 17 a^{7} + 17 a^{6} + 21 a^{5} + 19 a^{4} + 20 a^{3} + 22 a^{2} + 15 a + 6\right)\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 5 }$ $=$ $15 a^{14} + 18 a^{13} + 13 a^{12} + 18 a^{11} + 19 a^{10} + 16 a^{9} + 6 a^{8} + 4 a^{7} + 10 a^{6} + 11 a^{5} + 7 a^{4} + 15 a^{3} + 15 a^{2} + 3 a + 14 + \left(14 a^{14} + 13 a^{13} + 6 a^{12} + 11 a^{11} + 13 a^{10} + 12 a^{9} + 16 a^{8} + 12 a^{7} + 9 a^{6} + 19 a^{5} + 10 a^{4} + 6 a^{3} + 11 a^{2} + 5\right)\cdot 23 + \left(11 a^{14} + 14 a^{13} + 17 a^{12} + 17 a^{10} + 7 a^{9} + 16 a^{8} + 17 a^{7} + 5 a^{6} + 12 a^{5} + 14 a^{4} + 15 a^{3} + 2 a^{2} + 15\right)\cdot 23^{2} + \left(15 a^{14} + 10 a^{13} + 6 a^{12} + 2 a^{11} + 4 a^{10} + 9 a^{9} + 2 a^{7} + 3 a^{6} + 12 a^{5} + 17 a^{4} + 2 a^{3} + 13 a^{2} + 2 a + 8\right)\cdot 23^{3} + \left(13 a^{14} + 17 a^{13} + 4 a^{12} + a^{11} + 12 a^{10} + 5 a^{9} + 14 a^{8} + a^{7} + 10 a^{6} + 15 a^{5} + 10 a^{4} + a^{3} + 4 a^{2} + 9 a + 7\right)\cdot 23^{4} + \left(6 a^{14} + 8 a^{13} + 5 a^{12} + 15 a^{11} + 8 a^{10} + 14 a^{9} + 16 a^{8} + 16 a^{7} + 9 a^{6} + 9 a^{5} + 19 a^{4} + 5 a^{3} + 9 a^{2} + 10 a + 12\right)\cdot 23^{5} + \left(19 a^{14} + 8 a^{13} + a^{12} + 4 a^{11} + 17 a^{10} + 7 a^{9} + 2 a^{8} + 9 a^{7} + 2 a^{6} + 22 a^{5} + 4 a^{4} + 5 a^{2} + 20 a + 22\right)\cdot 23^{6} + \left(17 a^{14} + 16 a^{13} + 8 a^{12} + 12 a^{11} + 20 a^{10} + 10 a^{9} + 5 a^{8} + 4 a^{7} + 16 a^{5} + 8 a^{4} + 15 a^{3} + 8 a^{2} + 19 a + 22\right)\cdot 23^{7} + \left(9 a^{14} + 10 a^{13} + 18 a^{12} + 4 a^{11} + 11 a^{10} + 12 a^{9} + 12 a^{8} + 5 a^{7} + 19 a^{6} + 16 a^{5} + 10 a^{4} + 17 a^{3} + 9 a^{2} + 10 a + 20\right)\cdot 23^{8} + \left(3 a^{14} + 13 a^{13} + 13 a^{12} + 7 a^{11} + 2 a^{10} + 22 a^{9} + 7 a^{8} + 15 a^{7} + 2 a^{6} + a^{5} + 2 a^{4} + 6 a^{3} + 19 a^{2} + 4 a + 9\right)\cdot 23^{9} + \left(18 a^{14} + 2 a^{13} + 6 a^{12} + 12 a^{11} + 21 a^{10} + 20 a^{9} + 13 a^{8} + 4 a^{7} + 11 a^{6} + 10 a^{5} + 21 a^{4} + a^{3} + 2 a^{2} + 7\right)\cdot 23^{10} + \left(11 a^{14} + 19 a^{13} + 12 a^{12} + 10 a^{11} + 2 a^{10} + 22 a^{9} + 6 a^{8} + 2 a^{7} + 3 a^{6} + 11 a^{5} + 17 a^{4} + 7 a^{3} + 17 a^{2} + 9 a + 20\right)\cdot 23^{11} + \left(21 a^{14} + 21 a^{13} + 19 a^{12} + 22 a^{11} + 15 a^{10} + 9 a^{9} + 16 a^{8} + 21 a^{7} + a^{6} + 16 a^{5} + 21 a^{4} + 13 a^{3} + 21 a^{2} + 13 a + 16\right)\cdot 23^{12} + \left(6 a^{14} + 20 a^{13} + 13 a^{12} + 8 a^{11} + 12 a^{10} + 2 a^{9} + 9 a^{8} + 9 a^{7} + 15 a^{6} + 9 a^{5} + 11 a^{4} + 11 a^{3} + 14 a^{2} + 21 a + 13\right)\cdot 23^{13} + \left(15 a^{14} + 7 a^{13} + 18 a^{12} + 10 a^{11} + 9 a^{10} + 15 a^{9} + 10 a^{8} + a^{7} + 4 a^{6} + 20 a^{5} + 19 a^{4} + 13 a^{3} + 17 a^{2} + 20 a + 2\right)\cdot 23^{14} + \left(8 a^{14} + 20 a^{13} + 4 a^{12} + 22 a^{11} + 14 a^{10} + 20 a^{9} + 22 a^{8} + 20 a^{7} + 19 a^{6} + 13 a^{5} + 4 a^{4} + 18 a^{3} + 14 a^{2} + a + 18\right)\cdot 23^{15} + \left(8 a^{14} + 8 a^{13} + a^{12} + 18 a^{11} + 11 a^{10} + 16 a^{9} + 15 a^{8} + 12 a^{7} + a^{6} + 3 a^{5} + 20 a^{4} + 8 a^{3} + 7 a^{2} + 2 a + 14\right)\cdot 23^{16} + \left(18 a^{14} + 15 a^{13} + 6 a^{12} + 3 a^{11} + 3 a^{10} + 5 a^{9} + 6 a^{8} + 19 a^{7} + 22 a^{5} + 20 a^{4} + 11 a^{3} + 7 a^{2} + 22 a + 21\right)\cdot 23^{17} + \left(8 a^{14} + 15 a^{13} + 10 a^{12} + 9 a^{11} + 22 a^{10} + 3 a^{9} + 10 a^{7} + 19 a^{6} + a^{5} + 11 a^{4} + 13 a^{3} + 2 a^{2} + 6 a + 14\right)\cdot 23^{18} + \left(11 a^{14} + 15 a^{13} + 5 a^{12} + 18 a^{11} + 20 a^{10} + 14 a^{9} + 14 a^{8} + 3 a^{7} + 3 a^{6} + 17 a^{5} + 12 a^{4} + 8 a^{3} + 18 a^{2} + 8 a + 19\right)\cdot 23^{19} + \left(22 a^{14} + 9 a^{13} + 17 a^{12} + 20 a^{11} + 19 a^{10} + 13 a^{9} + a^{8} + 4 a^{7} + 22 a^{6} + 5 a^{5} + 20 a^{4} + a^{3} + 11 a^{2} + 15 a + 12\right)\cdot 23^{20} + \left(3 a^{14} + 15 a^{13} + 21 a^{12} + 19 a^{11} + 17 a^{10} + 2 a^{9} + 9 a^{8} + 13 a^{7} + 17 a^{6} + 22 a^{5} + 7 a^{4} + 6 a^{3} + 7 a^{2} + 13 a + 22\right)\cdot 23^{21} + \left(16 a^{14} + 12 a^{13} + 3 a^{12} + 5 a^{11} + 21 a^{10} + 15 a^{9} + 10 a^{8} + 19 a^{7} + 18 a^{6} + 5 a^{5} + 8 a^{4} + 14 a^{3} + 14 a^{2} + 9 a + 3\right)\cdot 23^{22} + \left(5 a^{14} + 16 a^{13} + 7 a^{12} + 14 a^{10} + 12 a^{9} + 14 a^{8} + 4 a^{7} + 9 a^{6} + 21 a^{5} + 12 a^{3} + 12 a^{2} + 22 a + 22\right)\cdot 23^{23} + \left(8 a^{14} + 2 a^{13} + 20 a^{12} + 4 a^{11} + 7 a^{10} + 17 a^{8} + 21 a^{7} + 12 a^{6} + 6 a^{5} + 12 a^{4} + 15 a^{3} + 16 a^{2} + 14 a + 3\right)\cdot 23^{24} + \left(21 a^{14} + 13 a^{12} + 8 a^{11} + 19 a^{10} + a^{9} + 8 a^{8} + 13 a^{7} + 2 a^{6} + 16 a^{5} + 15 a^{4} + 20 a^{3} + 4 a^{2} + 11 a + 21\right)\cdot 23^{25} + \left(19 a^{14} + 3 a^{13} + 7 a^{12} + 8 a^{11} + 8 a^{10} + a^{9} + 4 a^{8} + 18 a^{7} + 19 a^{6} + 2 a^{5} + 16 a^{4} + 9 a^{3} + 6 a^{2} + 13 a + 9\right)\cdot 23^{26} + \left(a^{14} + 22 a^{13} + 8 a^{12} + 4 a^{11} + 15 a^{10} + a^{9} + 11 a^{8} + 19 a^{7} + 14 a^{6} + 19 a^{5} + 17 a^{4} + 17 a^{3} + 10 a + 19\right)\cdot 23^{27} + \left(6 a^{14} + 8 a^{13} + 9 a^{12} + 17 a^{11} + 19 a^{10} + 3 a^{9} + 4 a^{8} + 5 a^{7} + 17 a^{6} + 17 a^{5} + 15 a^{4} + 15 a^{3} + 17 a^{2} + 19\right)\cdot 23^{28} + \left(3 a^{14} + 6 a^{13} + 18 a^{12} + 2 a^{10} + 8 a^{9} + 21 a^{8} + 2 a^{7} + 12 a^{6} + a^{5} + 18 a^{4} + 14 a^{2} + 11 a + 15\right)\cdot 23^{29} + \left(11 a^{14} + 18 a^{13} + 11 a^{12} + 3 a^{11} + 22 a^{10} + 21 a^{9} + 22 a^{8} + 22 a^{7} + 18 a^{6} + 2 a^{5} + 12 a^{4} + 3 a^{3} + 22 a^{2} + 4 a + 22\right)\cdot 23^{30} + \left(19 a^{14} + 8 a^{13} + 18 a^{12} + 2 a^{11} + 6 a^{10} + 22 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18 a^{9} + 2 a^{7} + 10 a^{6} + 15 a^{5} + 5 a^{4} + 5 a^{3} + 14 a^{2} + 10 a + 18\right)\cdot 23^{37} + \left(9 a^{14} + 20 a^{13} + 7 a^{12} + 9 a^{11} + 20 a^{10} + 9 a^{9} + 17 a^{8} + 10 a^{7} + 4 a^{6} + 16 a^{5} + 7 a^{4} + 6 a^{3} + 3 a^{2} + 7 a + 16\right)\cdot 23^{38} + \left(16 a^{14} + 13 a^{13} + 19 a^{12} + 6 a^{11} + 4 a^{10} + 22 a^{9} + 15 a^{8} + 17 a^{7} + 4 a^{6} + 4 a^{5} + 12 a^{4} + 13 a^{3} + 9 a^{2} + 2 a + 2\right)\cdot 23^{39} + \left(6 a^{14} + 2 a^{13} + 10 a^{12} + 6 a^{11} + 5 a^{10} + a^{9} + 16 a^{8} + 8 a^{7} + a^{6} + a^{5} + 11 a^{4} + 6 a^{3} + 11 a^{2} + 8 a + 13\right)\cdot 23^{40} + \left(16 a^{14} + 4 a^{13} + 8 a^{12} + 6 a^{11} + 17 a^{10} + 22 a^{9} + 17 a^{7} + 21 a^{6} + 15 a^{4} + 6 a^{3} + 5 a^{2} + 1\right)\cdot 23^{41} + \left(14 a^{14} + 21 a^{13} + 20 a^{12} + a^{11} + 11 a^{10} + 20 a^{9} + 17 a^{8} + 22 a^{5} + a^{4} + 19 a^{3} + 3 a^{2} + 9 a + 1\right)\cdot 23^{42} + \left(9 a^{14} + 16 a^{13} + 14 a^{12} + 10 a^{11} + 9 a^{10} + 3 a^{9} + 2 a^{8} + 7 a^{7} + 2 a^{6} + 5 a^{5} + 10 a^{4} + 15 a^{3} + 16 a^{2} + 12 a + 11\right)\cdot 23^{43} + \left(9 a^{14} + 10 a^{13} + 5 a^{12} + 22 a^{11} + 5 a^{10} + 17 a^{9} + 20 a^{8} + 13 a^{7} + 18 a^{6} + 17 a^{5} + 21 a^{4} + 14 a^{3} + 19 a^{2} + 10 a + 4\right)\cdot 23^{44} + \left(17 a^{14} + 22 a^{13} + 12 a^{12} + a^{11} + 6 a^{9} + 14 a^{8} + 11 a^{7} + 21 a^{6} + 12 a^{5} + 19 a^{4} + 3 a^{3} + 21 a^{2} + 8 a + 14\right)\cdot 23^{45} + \left(9 a^{14} + 13 a^{13} + 10 a^{12} + 17 a^{11} + 8 a^{10} + 2 a^{9} + 3 a^{8} + 10 a^{7} + 7 a^{6} + 18 a^{5} + 19 a^{4} + 9 a^{3} + 22 a^{2} + 21 a + 13\right)\cdot 23^{46} + \left(18 a^{14} + 12 a^{13} + 21 a^{12} + 15 a^{11} + 9 a^{10} + 15 a^{9} + 10 a^{8} + a^{7} + 7 a^{6} + 13 a^{5} + 10 a^{4} + 15 a^{3} + 15 a^{2} + 13 a + 2\right)\cdot 23^{47} + \left(22 a^{14} + 12 a^{13} + 11 a^{12} + 3 a^{11} + 20 a^{10} + 5 a^{9} + 8 a^{8} + 10 a^{7} + 19 a^{6} + 19 a^{5} + 21 a^{4} + 15 a^{3} + 5 a^{2} + 13 a + 10\right)\cdot 23^{48} + \left(21 a^{14} + 19 a^{13} + 2 a^{12} + 9 a^{11} + 15 a^{10} + 3 a^{9} + a^{8} + 4 a^{7} + 13 a^{6} + 4 a^{5} + 11 a^{4} + a^{3} + 18 a^{2} + 15 a + 17\right)\cdot 23^{49} + \left(a^{14} + 2 a^{13} + 21 a^{12} + 3 a^{11} + 12 a^{8} + 3 a^{7} + 9 a^{6} + 19 a^{5} + 10 a^{4} + 17 a^{3} + 4 a^{2} + 18 a + 18\right)\cdot 23^{50} + \left(22 a^{13} + 2 a^{12} + 3 a^{10} + 17 a^{9} + 8 a^{8} + 20 a^{7} + 14 a^{6} + 16 a^{5} + 19 a^{4} + 7 a^{3} + 8 a^{2} + 9 a + 8\right)\cdot 23^{51} + \left(17 a^{14} + 13 a^{13} + 21 a^{12} + 14 a^{11} + 12 a^{9} + 16 a^{8} + 19 a^{7} + 8 a^{6} + 9 a^{5} + 5 a^{4} + 3 a^{3} + 7 a^{2} + 3 a + 11\right)\cdot 23^{52} + \left(a^{14} + 10 a^{13} + 13 a^{12} + 8 a^{11} + 8 a^{10} + 15 a^{9} + 3 a^{8} + 10 a^{7} + 22 a^{5} + 18 a^{4} + 2 a^{3} + a^{2} + 6 a + 9\right)\cdot 23^{53} + \left(7 a^{14} + 9 a^{13} + 11 a^{12} + 2 a^{11} + 13 a^{9} + 11 a^{8} + 16 a^{7} + 9 a^{6} + 7 a^{5} + 19 a^{4} + 13 a^{3} + 15 a^{2} + 16 a + 6\right)\cdot 23^{54} + \left(7 a^{13} + 12 a^{12} + 16 a^{11} + 2 a^{10} + 18 a^{9} + 11 a^{8} + 14 a^{7} + 20 a^{6} + 3 a^{5} + 17 a^{4} + 17 a^{3} + 18 a^{2} + 1\right)\cdot 23^{55} + \left(5 a^{14} + 11 a^{13} + 18 a^{12} + 7 a^{11} + 16 a^{10} + 15 a^{9} + 22 a^{8} + 3 a^{7} + 18 a^{6} + 9 a^{5} + 17 a^{4} + 10 a^{3} + 15 a^{2} + 10 a + 12\right)\cdot 23^{56} + \left(12 a^{14} + 12 a^{13} + 5 a^{12} + 21 a^{11} + 18 a^{10} + 6 a^{9} + 10 a^{7} + 2 a^{6} + a^{5} + 19 a^{4} + 13 a^{3} + 16 a^{2} + 10 a + 11\right)\cdot 23^{57} + \left(18 a^{14} + 8 a^{13} + 3 a^{12} + 4 a^{11} + 19 a^{10} + 7 a^{9} + a^{8} + a^{7} + 14 a^{6} + 2 a^{5} + 2 a^{4} + 4 a^{3} + 15 a^{2} + 6 a + 14\right)\cdot 23^{58} + \left(22 a^{14} + 19 a^{13} + 22 a^{11} + 20 a^{10} + 4 a^{9} + 14 a^{8} + 19 a^{7} + 12 a^{6} + 18 a^{5} + 3 a^{4} + 2 a^{3} + 9 a^{2} + 22 a + 22\right)\cdot 23^{59} + \left(10 a^{14} + 16 a^{13} + 11 a^{12} + 22 a^{11} + 15 a^{10} + 9 a^{9} + 20 a^{8} + 17 a^{7} + 14 a^{6} + 9 a^{5} + 19 a^{4} + 13 a^{3} + 9 a^{2} + 4 a + 11\right)\cdot 23^{60} + \left(6 a^{14} + 2 a^{13} + 14 a^{12} + 4 a^{11} + 2 a^{10} + 7 a^{9} + 2 a^{8} + 15 a^{7} + 13 a^{6} + 2 a^{5} + 2 a^{4} + 2 a^{3} + 18 a^{2} + 11 a + 6\right)\cdot 23^{61} + \left(8 a^{14} + 22 a^{13} + 17 a^{12} + 21 a^{11} + 16 a^{10} + a^{9} + 10 a^{8} + 15 a^{7} + 17 a^{6} + 20 a^{5} + 15 a^{4} + 9 a^{3} + 17 a^{2} + 18 a + 3\right)\cdot 23^{62} + \left(2 a^{14} + 3 a^{13} + 10 a^{12} + 16 a^{11} + 12 a^{10} + 14 a^{9} + 20 a^{8} + 15 a^{7} + 4 a^{6} + 21 a^{5} + 11 a^{4} + 5 a^{3} + 16 a^{2} + 15 a + 10\right)\cdot 23^{63} + \left(9 a^{14} + a^{13} + 7 a^{12} + 4 a^{11} + 18 a^{10} + 13 a^{9} + 7 a^{7} + 5 a^{6} + 2 a^{4} + 18 a^{3} + 19 a^{2} + a + 21\right)\cdot 23^{64} + \left(4 a^{14} + 9 a^{13} + 11 a^{12} + 12 a^{11} + 9 a^{10} + 10 a^{9} + 21 a^{8} + 13 a^{7} + 19 a^{6} + 17 a^{5} + 3 a^{4} + 8 a^{3} + 15\right)\cdot 23^{65} + \left(19 a^{14} + 14 a^{13} + a^{12} + 17 a^{11} + 7 a^{10} + 10 a^{8} + 7 a^{7} + 15 a^{6} + 3 a^{5} + 22 a^{4} + 6 a^{3} + 2 a^{2} + 14 a + 19\right)\cdot 23^{66} + \left(13 a^{14} + 20 a^{13} + 9 a^{12} + 2 a^{11} + 15 a^{9} + 20 a^{8} + 9 a^{7} + 14 a^{6} + 7 a^{5} + 11 a^{4} + 3 a^{3} + 17 a^{2} + 14 a + 21\right)\cdot 23^{67} + \left(10 a^{14} + 19 a^{13} + 21 a^{12} + 6 a^{11} + 7 a^{10} + 7 a^{9} + 5 a^{8} + 7 a^{6} + 10 a^{5} + 2 a^{4} + 12 a^{3} + 19 a^{2} + 9 a + 12\right)\cdot 23^{68} + \left(18 a^{14} + 7 a^{13} + 16 a^{12} + 4 a^{11} + 20 a^{10} + 19 a^{9} + 13 a^{8} + 7 a^{7} + 20 a^{5} + 10 a^{4} + 20 a^{3} + 21 a^{2} + 14 a + 21\right)\cdot 23^{69} + \left(5 a^{14} + 16 a^{13} + 18 a^{11} + 22 a^{10} + 12 a^{9} + 17 a^{8} + 6 a^{7} + 3 a^{6} + 15 a^{4} + 2 a^{3} + 18 a^{2} + 22 a + 15\right)\cdot 23^{70} + \left(8 a^{14} + 2 a^{12} + 19 a^{11} + 2 a^{10} + 9 a^{9} + 7 a^{8} + 20 a^{7} + a^{6} + a^{5} + 19 a^{4} + 22 a^{3} + 18 a^{2} + 14 a + 19\right)\cdot 23^{71} + \left(4 a^{14} + 5 a^{13} + 16 a^{12} + 2 a^{11} + 11 a^{9} + 22 a^{8} + 19 a^{7} + 10 a^{6} + 18 a^{5} + 14 a^{4} + 22 a^{3} + 7 a^{2} + 18 a + 12\right)\cdot 23^{72} + \left(2 a^{14} + 21 a^{13} + 13 a^{12} + 16 a^{11} + 16 a^{10} + 16 a^{9} + 10 a^{8} + 10 a^{7} + 9 a^{6} + 6 a^{5} + 2 a^{4} + 6 a^{3} + 6 a^{2} + 8 a + 20\right)\cdot 23^{73} + \left(10 a^{14} + 3 a^{13} + 21 a^{12} + 8 a^{11} + 18 a^{10} + 5 a^{9} + 8 a^{8} + 8 a^{7} + 12 a^{6} + 17 a^{5} + 16 a^{4} + 8 a^{3} + 14 a^{2} + 21 a + 17\right)\cdot 23^{74} + \left(22 a^{14} + 12 a^{13} + 16 a^{12} + 6 a^{11} + 18 a^{10} + 5 a^{9} + 14 a^{8} + 18 a^{7} + 18 a^{6} + 17 a^{5} + 3 a^{4} + 10 a^{3} + 5 a^{2} + 12 a + 3\right)\cdot 23^{75} + \left(8 a^{14} + 19 a^{13} + 14 a^{12} + 3 a^{11} + 19 a^{10} + 20 a^{9} + 12 a^{8} + 4 a^{7} + 13 a^{6} + 17 a^{5} + 3 a^{4} + 16 a^{2} + 7 a + 11\right)\cdot 23^{76} + \left(21 a^{14} + 8 a^{13} + 5 a^{12} + 21 a^{11} + 15 a^{10} + 11 a^{9} + 5 a^{8} + 19 a^{7} + 20 a^{6} + 15 a^{5} + 2 a^{4} + 5 a^{3} + 9 a^{2} + 8 a + 6\right)\cdot 23^{77} + \left(10 a^{14} + 16 a^{13} + 5 a^{12} + 5 a^{11} + 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+ \left(18 a^{14} + a^{13} + 20 a^{12} + 21 a^{11} + 16 a^{10} + 5 a^{9} + 6 a^{8} + 6 a^{7} + 16 a^{6} + 14 a^{5} + 20 a^{4} + 17 a^{3} + 4 a^{2} + 3 a + 16\right)\cdot 23^{84} + \left(7 a^{14} + 3 a^{13} + 14 a^{12} + 19 a^{11} + 14 a^{10} + 15 a^{9} + 10 a^{8} + a^{7} + 20 a^{6} + 13 a^{5} + 2 a^{4} + 8 a^{3} + 4 a^{2} + 17 a + 18\right)\cdot 23^{85} + \left(12 a^{14} + 21 a^{13} + 4 a^{12} + 10 a^{10} + 2 a^{9} + 21 a^{8} + 8 a^{7} + 18 a^{6} + 7 a^{5} + 12 a^{4} + 9 a^{3} + 14 a^{2} + 5 a + 7\right)\cdot 23^{86} + \left(15 a^{14} + 16 a^{13} + 8 a^{11} + 10 a^{10} + 10 a^{9} + 2 a^{8} + 9 a^{7} + 13 a^{6} + 6 a^{4} + 9 a^{3} + 17 a^{2} + 21 a + 1\right)\cdot 23^{87} + \left(10 a^{14} + 5 a^{13} + 19 a^{12} + 5 a^{11} + 14 a^{10} + 8 a^{9} + 7 a^{8} + 12 a^{7} + 13 a^{6} + 3 a^{5} + 21 a^{4} + 11 a^{3} + 4 a^{2} + 19 a + 7\right)\cdot 23^{88} + \left(12 a^{14} + 16 a^{13} + 22 a^{12} + 22 a^{11} + 2 a^{10} + 3 a^{9} + 4 a^{8} + 20 a^{6} + 20 a^{5} + 14 a^{4} + 7 a^{3} + 9 a^{2} + 16 a + 22\right)\cdot 23^{89} + \left(3 a^{14} + 5 a^{13} + 18 a^{12} + 4 a^{11} + 16 a^{10} + 10 a^{9} + 13 a^{8} + 16 a^{7} + 8 a^{6} + 10 a^{5} + 11 a^{4} + 7 a^{3} + 3 a^{2} + 9 a + 11\right)\cdot 23^{90} + \left(20 a^{13} + 19 a^{12} + 6 a^{11} + 18 a^{10} + 8 a^{9} + 15 a^{8} + 7 a^{7} + 7 a^{6} + 3 a^{5} + 22 a^{4} + 18 a^{3} + 18 a^{2} + 11 a + 7\right)\cdot 23^{91} + \left(2 a^{14} + 14 a^{13} + 22 a^{12} + 9 a^{11} + 22 a^{10} + 10 a^{8} + 6 a^{7} + 6 a^{6} + 22 a^{4} + 17 a^{3} + 20 a^{2} + 3 a + 16\right)\cdot 23^{92} + \left(6 a^{14} + 6 a^{13} + 19 a^{12} + 13 a^{11} + 3 a^{10} + 21 a^{9} + 5 a^{8} + 4 a^{7} + 13 a^{5} + 20 a^{4} + 10 a^{3} + 14 a^{2} + 12 a + 12\right)\cdot 23^{93} + \left(4 a^{14} + 15 a^{13} + 19 a^{12} + 6 a^{11} + 15 a^{10} + 10 a^{9} + 17 a^{8} + 22 a^{7} + 19 a^{6} + 10 a^{5} + 9 a^{4} + 6 a^{3} + 6 a^{2} + 2 a + 16\right)\cdot 23^{94} + \left(a^{14} + 12 a^{13} + a^{12} + 6 a^{11} + 16 a^{10} + 4 a^{9} + 20 a^{8} + 22 a^{7} + 17 a^{6} + 4 a^{5} + 17 a^{4} + 14 a^{3} + 20 a^{2} + 4 a + 12\right)\cdot 23^{95} + \left(14 a^{14} + 8 a^{13} + 17 a^{12} + 22 a^{11} + 5 a^{10} + 3 a^{9} + 21 a^{8} + a^{7} + 16 a^{6} + 2 a^{5} + 2 a^{4} + 18 a^{3} + 22 a^{2} + 13 a + 16\right)\cdot 23^{96} + \left(20 a^{14} + 3 a^{13} + 16 a^{12} + 21 a^{11} + 22 a^{10} + 8 a^{9} + 2 a^{8} + 20 a^{7} + 6 a^{6} + 12 a^{5} + 7 a^{4} + 3 a^{3} + 2 a^{2} + 3 a + 3\right)\cdot 23^{97} + \left(a^{14} + 17 a^{12} + 15 a^{11} + 2 a^{10} + 19 a^{9} + 9 a^{8} + 13 a^{7} + 4 a^{6} + 4 a^{5} + 4 a^{4} + 12 a^{3} + 16 a^{2} + 11 a + 11\right)\cdot 23^{98} + \left(13 a^{14} + 20 a^{13} + a^{12} + 15 a^{11} + 10 a^{10} + 2 a^{9} + 14 a^{8} + 13 a^{7} + 9 a^{6} + 5 a^{5} + 20 a^{4} + 2 a^{3} + 20 a^{2} + 19 a + 8\right)\cdot 23^{99} + \left(7 a^{14} + 13 a^{13} + 8 a^{12} + 13 a^{11} + 4 a^{10} + a^{9} + 14 a^{8} + 19 a^{7} + 15 a^{6} + 19 a^{5} + 16 a^{4} + 6 a^{3} + 10 a^{2} + 11 a + 14\right)\cdot 23^{100} + \left(11 a^{14} + 18 a^{13} + 5 a^{12} + 9 a^{11} + a^{10} + 22 a^{9} + 9 a^{8} + 17 a^{7} + 4 a^{6} + 19 a^{5} + 5 a^{4} + 6 a^{3} + 15 a^{2} + 14 a + 13\right)\cdot 23^{101} + \left(10 a^{14} + 4 a^{13} + 7 a^{12} + 10 a^{11} + 12 a^{10} + 15 a^{9} + 22 a^{8} + 15 a^{7} + 20 a^{6} + 3 a^{5} + 16 a^{4} + 17 a^{3} + 7 a^{2} + 14 a + 1\right)\cdot 23^{102} + \left(4 a^{14} + 5 a^{13} + 5 a^{12} + 16 a^{11} + 22 a^{10} + 8 a^{9} + 15 a^{8} + 11 a^{7} + 14 a^{6} + 17 a^{5} + 15 a^{4} + 10 a^{3} + 12 a^{2} + 20 a + 14\right)\cdot 23^{103} + \left(5 a^{14} + 21 a^{13} + 13 a^{12} + 21 a^{11} + 9 a^{10} + 9 a^{9} + 9 a^{8} + 20 a^{7} + 4 a^{6} + 22 a^{4} + 15 a^{3} + 7 a^{2} + 5 a + 18\right)\cdot 23^{104} + \left(3 a^{13} + 7 a^{12} + 21 a^{11} + 4 a^{10} + 3 a^{9} + 22 a^{8} + 8 a^{7} + 22 a^{6} + 11 a^{5} + 16 a^{4} + 10 a^{3} + 15 a^{2} + 12 a + 1\right)\cdot 23^{105} + \left(5 a^{13} + 2 a^{12} + 6 a^{11} + 3 a^{10} + 3 a^{9} + 10 a^{8} + 16 a^{7} + a^{6} + 8 a^{5} + 11 a^{4} + 19 a^{3} + 11 a^{2} + 16 a + 20\right)\cdot 23^{106} + \left(18 a^{14} + 3 a^{13} + 3 a^{12} + 7 a^{11} + 18 a^{10} + 17 a^{9} + 7 a^{8} + 8 a^{7} + 8 a^{6} + 18 a^{5} + 21 a^{4} + 10 a^{3} + 21 a^{2} + 3 a + 14\right)\cdot 23^{107} + \left(4 a^{14} + 10 a^{13} + 11 a^{12} + 6 a^{11} + 3 a^{10} + 21 a^{9} + 14 a^{8} + 20 a^{7} + 18 a^{6} + 7 a^{5} + 6 a^{4} + 14 a^{3} + a^{2} + 21 a + 2\right)\cdot 23^{108} + \left(10 a^{14} + 3 a^{13} + 17 a^{12} + 12 a^{11} + 21 a^{10} + 12 a^{9} + 8 a^{8} + 6 a^{7} + 14 a^{6} + 9 a^{5} + 14 a^{4} + 13 a^{3} + 5 a^{2} + 9 a + 19\right)\cdot 23^{109} + \left(22 a^{14} + 12 a^{13} + 3 a^{11} + 14 a^{10} + 17 a^{9} + 2 a^{8} + 8 a^{7} + 8 a^{6} + 16 a^{5} + 16 a^{4} + 21 a^{3} + a^{2} + 4 a + 4\right)\cdot 23^{110} + \left(21 a^{14} + 2 a^{13} + 11 a^{12} + 8 a^{11} + 20 a^{10} + 2 a^{9} + 2 a^{8} + 6 a^{7} + 2 a^{6} + 11 a^{5} + 17 a^{4} + 10 a^{3} + 4 a^{2} + 16 a + 2\right)\cdot 23^{111} + \left(21 a^{14} + 4 a^{13} + 19 a^{12} + 17 a^{10} + a^{9} + 8 a^{8} + 6 a^{6} + 17 a^{5} + 9 a^{4} + 21 a^{3} + 22 a^{2} + 8 a + 21\right)\cdot 23^{112} + \left(9 a^{14} + 21 a^{13} + 8 a^{12} + 16 a^{11} + 3 a^{10} + 2 a^{9} + 18 a^{8} + 13 a^{6} + 6 a^{5} + 22 a^{4} + 3 a^{3} + 5 a^{2} + 21 a + 6\right)\cdot 23^{113} + \left(13 a^{14} + 5 a^{13} + 13 a^{12} + 9 a^{11} + 3 a^{10} + 17 a^{8} + 7 a^{7} + 14 a^{6} + 18 a^{5} + 15 a^{4} + 9 a^{3} + 9 a^{2} + 2 a + 15\right)\cdot 23^{114} + \left(2 a^{14} + 19 a^{13} + 14 a^{11} + 21 a^{10} + 6 a^{9} + 19 a^{8} + 22 a^{7} + 7 a^{6} + 8 a^{4} + 16 a^{3} + 21 a^{2} + 12 a + 22\right)\cdot 23^{115} + \left(14 a^{14} + 8 a^{13} + 11 a^{12} + 15 a^{11} + 10 a^{10} + 18 a^{9} + 8 a^{8} + 6 a^{7} + 17 a^{6} + 13 a^{5} + 7 a^{4} + 9 a^{3} + 19 a^{2} + 21 a + 13\right)\cdot 23^{116} + \left(13 a^{14} + 4 a^{13} + 20 a^{12} + 10 a^{10} + 8 a^{9} + 19 a^{8} + 22 a^{7} + 9 a^{6} + 15 a^{5} + 14 a^{4} + 11 a^{3} + 7 a^{2} + 17\right)\cdot 23^{117} + \left(17 a^{14} + 20 a^{13} + 7 a^{12} + 16 a^{11} + 12 a^{10} + 20 a^{9} + 16 a^{8} + 5 a^{7} + 12 a^{6} + 7 a^{5} + 21 a^{4} + 18 a^{3} + 21 a^{2} + a + 13\right)\cdot 23^{118} + \left(2 a^{14} + 20 a^{13} + 10 a^{12} + 17 a^{11} + 19 a^{10} + 17 a^{9} + 14 a^{8} + 18 a^{7} + 10 a^{6} + 13 a^{3} + 21 a^{2} + 18 a + 9\right)\cdot 23^{119} + \left(14 a^{14} + 17 a^{13} + 8 a^{12} + a^{11} + 2 a^{9} + 11 a^{8} + 12 a^{7} + 5 a^{6} + 6 a^{5} + 22 a^{4} + 19 a^{3} + 20 a^{2} + 10 a + 15\right)\cdot 23^{120} + \left(22 a^{14} + 21 a^{13} + 3 a^{12} + 13 a^{11} + 22 a^{10} + 5 a^{9} + 18 a^{8} + 3 a^{7} + a^{6} + 3 a^{5} + 21 a^{4} + 8 a^{3} + 11 a^{2} + 3 a + 17\right)\cdot 23^{121} + \left(13 a^{14} + 17 a^{13} + 19 a^{11} + 22 a^{10} + 20 a^{9} + 5 a^{8} + 10 a^{7} + 7 a^{6} + 3 a^{5} + 10 a^{4} + 19 a^{3} + 8 a^{2} + 3 a + 3\right)\cdot 23^{122} + \left(18 a^{14} + 9 a^{13} + 10 a^{12} + 2 a^{11} + 15 a^{10} + 14 a^{9} + 17 a^{8} + 5 a^{7} + 14 a^{5} + 8 a^{4} + a^{3} + 21 a^{2} + 10 a + 6\right)\cdot 23^{123} + \left(7 a^{14} + 8 a^{13} + 14 a^{12} + 7 a^{11} + 4 a^{10} + 15 a^{9} + 20 a^{8} + 3 a^{7} + 21 a^{6} + 13 a^{5} + 10 a^{4} + 3 a^{3} + 13 a^{2} + 18 a + 20\right)\cdot 23^{124} + \left(7 a^{14} + 2 a^{13} + 5 a^{12} + 11 a^{11} + 15 a^{10} + a^{9} + 19 a^{8} + 12 a^{5} + 20 a^{4} + 6 a^{3} + 10 a^{2} + 11\right)\cdot 23^{125} + \left(14 a^{14} + 20 a^{13} + 22 a^{12} + 4 a^{11} + 22 a^{10} + 5 a^{9} + 12 a^{8} + 21 a^{7} + 21 a^{6} + 22 a^{5} + 6 a^{4} + 19 a^{3} + 17 a^{2} + 3 a + 3\right)\cdot 23^{126} + \left(10 a^{14} + 9 a^{13} + 12 a^{12} + 21 a^{11} + 8 a^{10} + 21 a^{9} + 3 a^{8} + 8 a^{7} + 16 a^{6} + 7 a^{5} + 11 a^{4} + 6 a^{3} + 5 a^{2} + 11 a + 20\right)\cdot 23^{127} + \left(6 a^{14} + 22 a^{13} + 13 a^{12} + 19 a^{11} + 14 a^{10} + 20 a^{9} + 11 a^{8} + 11 a^{7} + 3 a^{6} + 19 a^{5} + 20 a^{4} + a^{3} + 9 a^{2} + 6 a + 15\right)\cdot 23^{128} + \left(3 a^{14} + a^{13} + 22 a^{12} + 12 a^{11} + 2 a^{10} + 15 a^{9} + 20 a^{8} + 2 a^{7} + 7 a^{6} + 10 a^{5} + 6 a^{4} + 6 a^{2} + 2 a + 13\right)\cdot 23^{129} + \left(13 a^{14} + 4 a^{13} + 10 a^{12} + 19 a^{11} + 2 a^{10} + 3 a^{9} + 20 a^{8} + 4 a^{7} + 8 a^{6} + 8 a^{5} + 2 a^{4} + 8 a^{3} + 15 a^{2} + 15\right)\cdot 23^{130} + \left(15 a^{14} + 15 a^{13} + 19 a^{12} + 16 a^{11} + 12 a^{10} + 5 a^{9} + 13 a^{8} + 4 a^{7} + 3 a^{6} + 8 a^{5} + a^{4} + 5 a^{3} + 20 a^{2} + 16 a + 18\right)\cdot 23^{131} + \left(18 a^{14} + 19 a^{13} + 6 a^{12} + 6 a^{11} + 2 a^{10} + 16 a^{9} + 20 a^{8} + a^{7} + 11 a^{6} + 5 a^{5} + 14 a^{4} + 19 a^{3} + 18 a^{2} + 10 a + 9\right)\cdot 23^{132} + \left(8 a^{14} + 17 a^{13} + 11 a^{12} + 19 a^{11} + 9 a^{10} + 4 a^{9} + 3 a^{8} + 4 a^{7} + 7 a^{5} + 13 a^{4} + 15 a^{3} + 9 a^{2} + 9 a + 14\right)\cdot 23^{133} + \left(5 a^{14} + 18 a^{13} + 8 a^{12} + 3 a^{11} + 15 a^{10} + 4 a^{9} + 13 a^{8} + 12 a^{7} + 9 a^{6} + 7 a^{5} + 22 a^{4} + 19 a^{3} + 11 a^{2} + 12 a + 12\right)\cdot 23^{134} + \left(18 a^{14} + 14 a^{13} + 17 a^{12} + 11 a^{11} + 9 a^{10} + 2 a^{9} + 12 a^{8} + 20 a^{7} + 20 a^{6} + 8 a^{5} + 18 a^{4} + 19 a^{3} + 13 a^{2} + 11 a + 11\right)\cdot 23^{135} + \left(16 a^{14} + 20 a^{13} + 2 a^{12} + 5 a^{11} + 18 a^{10} + 8 a^{9} + 5 a^{8} + 18 a^{7} + 7 a^{6} + 12 a^{5} + 15 a^{3} + 21 a^{2} + 18 a + 10\right)\cdot 23^{136} + \left(12 a^{14} + 13 a^{13} + 19 a^{12} + 14 a^{11} + 8 a^{10} + 21 a^{9} + 8 a^{8} + 7 a^{7} + 3 a^{6} + 15 a^{4} + 16 a^{3} + 6 a^{2} + 13 a + 13\right)\cdot 23^{137} + \left(10 a^{14} + 2 a^{13} + 8 a^{12} + 9 a^{11} + 16 a^{10} + a^{9} + 14 a^{8} + 3 a^{7} + 16 a^{6} + 22 a^{5} + 19 a^{4} + 7 a^{3} + 22 a^{2} + 5 a + 2\right)\cdot 23^{138} + \left(4 a^{14} + 21 a^{13} + 4 a^{12} + 18 a^{11} + a^{10} + 9 a^{9} + 17 a^{8} + 6 a^{7} + 9 a^{6} + 17 a^{5} + 9 a^{4} + 17 a^{3} + 20 a^{2} + 12 a + 2\right)\cdot 23^{139} + \left(9 a^{14} + 16 a^{13} + 19 a^{12} + 21 a^{11} + 7 a^{10} + 15 a^{9} + 19 a^{8} + 13 a^{7} + 13 a^{6} + 13 a^{5} + 16 a^{4} + 21 a^{3} + 21 a^{2} + 19 a + 22\right)\cdot 23^{140} + \left(16 a^{13} + 7 a^{12} + 5 a^{11} + 5 a^{10} + 18 a^{9} + 9 a^{8} + 3 a^{7} + 17 a^{6} + 19 a^{5} + 12 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+ 14 a^{3} + 16 a^{2} + 12 a + 18\right)\cdot 23^{147} + \left(7 a^{14} + 2 a^{13} + 9 a^{12} + 4 a^{11} + 14 a^{10} + 8 a^{9} + 17 a^{8} + 20 a^{7} + 5 a^{6} + 17 a^{5} + 15 a^{4} + 14 a^{3} + 2 a^{2} + 16 a + 12\right)\cdot 23^{148} + \left(17 a^{14} + 19 a^{13} + 12 a^{12} + 17 a^{11} + 6 a^{10} + a^{9} + 20 a^{8} + 7 a^{7} + 12 a^{6} + 12 a^{5} + 11 a^{4} + 14 a^{3} + a^{2} + 14 a + 7\right)\cdot 23^{149} + \left(3 a^{14} + 2 a^{13} + 18 a^{12} + 10 a^{11} + 10 a^{10} + 13 a^{9} + 14 a^{8} + 11 a^{7} + 4 a^{6} + 20 a^{5} + 14 a^{4} + 16 a^{3} + 15 a^{2} + 17 a + 4\right)\cdot 23^{150} + \left(2 a^{14} + 2 a^{13} + 6 a^{12} + 5 a^{11} + 21 a^{10} + 7 a^{9} + 3 a^{8} + 21 a^{7} + 11 a^{6} + 2 a^{5} + 11 a^{4} + 12 a^{3} + 6 a^{2} + 20 a + 12\right)\cdot 23^{151} + \left(2 a^{12} + 10 a^{11} + 16 a^{10} + 2 a^{9} + 10 a^{7} + a^{6} + 10 a^{5} + 3 a^{4} + 14 a^{3} + 14 a^{2} + 20 a + 19\right)\cdot 23^{152} + \left(3 a^{14} + a^{13} + 11 a^{12} + 22 a^{11} + 11 a^{10} + 4 a^{9} + 14 a^{8} + 7 a^{7} + a^{6} + 11 a^{5} + 4 a^{4} + 18 a^{3} + 22 a^{2} + 22 a + 11\right)\cdot 23^{153} + \left(5 a^{14} + 6 a^{13} + 3 a^{11} + 3 a^{10} + 12 a^{9} + 11 a^{8} + 20 a^{7} + 11 a^{6} + 11 a^{5} + 8 a^{4} + 3 a^{3} + 18 a^{2} + 20 a + 20\right)\cdot 23^{154} + \left(5 a^{14} + 4 a^{12} + 16 a^{11} + 22 a^{10} + 11 a^{9} + 5 a^{8} + 9 a^{7} + 11 a^{6} + 20 a^{5} + 13 a^{4} + 9 a^{3} + 22 a^{2} + 6 a + 10\right)\cdot 23^{155} + \left(a^{14} + 21 a^{13} + 9 a^{12} + 11 a^{11} + 11 a^{10} + 8 a^{9} + 19 a^{8} + 7 a^{7} + 15 a^{6} + 14 a^{5} + 14 a^{4} + 16 a^{3} + 7 a^{2} + 4 a + 4\right)\cdot 23^{156} + \left(19 a^{14} + 7 a^{13} + 13 a^{12} + 8 a^{11} + 12 a^{10} + 9 a^{9} + 15 a^{8} + 7 a^{7} + 5 a^{6} + 11 a^{5} + 11 a^{4} + 19 a^{3} + 18 a^{2} + 21 a + 21\right)\cdot 23^{157} + \left(2 a^{14} + 12 a^{13} + 10 a^{12} + 12 a^{11} + 17 a^{10} + 5 a^{9} + 21 a^{8} + 20 a^{7} + 4 a^{6} + 7 a^{5} + 5 a^{4} + a^{3} + 19 a^{2} + 21 a + 18\right)\cdot 23^{158} + \left(2 a^{14} + 13 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\left(13 a^{14} + 7 a^{13} + 19 a^{12} + a^{11} + 21 a^{10} + 6 a^{9} + 3 a^{8} + 11 a^{7} + 20 a^{6} + 14 a^{5} + 22 a^{4} + 20 a^{3} + 13 a^{2} + 20 a + 19\right)\cdot 23^{171} + \left(22 a^{14} + 14 a^{13} + 12 a^{12} + 17 a^{11} + 19 a^{10} + 21 a^{9} + a^{8} + 15 a^{7} + 12 a^{6} + 20 a^{5} + 20 a^{4} + 12 a^{3} + 20 a^{2} + 14 a + 14\right)\cdot 23^{172} + \left(10 a^{14} + 9 a^{13} + a^{12} + 16 a^{11} + 17 a^{10} + 9 a^{9} + 3 a^{8} + 19 a^{7} + 9 a^{6} + 14 a^{4} + 18 a^{3} + 22 a^{2} + 20 a + 8\right)\cdot 23^{173} + \left(20 a^{14} + 9 a^{13} + 13 a^{12} + 15 a^{10} + 14 a^{9} + 11 a^{8} + 8 a^{6} + 18 a^{5} + 22 a^{4} + 22 a^{3} + 3 a^{2} + 8 a + 12\right)\cdot 23^{174} + \left(20 a^{14} + 16 a^{13} + 19 a^{12} + 3 a^{11} + 11 a^{10} + 13 a^{9} + 6 a^{8} + 19 a^{7} + 10 a^{6} + 3 a^{5} + a^{4} + 20 a^{3} + 10 a^{2} + 14 a + 12\right)\cdot 23^{175} + \left(16 a^{14} + 2 a^{12} + 6 a^{11} + 7 a^{10} + 2 a^{9} + 14 a^{8} + 11 a^{6} + 5 a^{5} + 6 a^{4} + 15 a^{3} + 5 a^{2} + 11 a + 18\right)\cdot 23^{176} + \left(12 a^{14} + 3 a^{13} + 5 a^{12} + 17 a^{11} + a^{10} + 12 a^{9} + 12 a^{8} + 2 a^{7} + 21 a^{6} + a^{5} + 4 a^{4} + 14 a^{3} + 10 a^{2} + 5 a + 14\right)\cdot 23^{177} + \left(a^{14} + 11 a^{13} + 16 a^{12} + 16 a^{11} + 12 a^{10} + 5 a^{8} + 16 a^{7} + a^{6} + 10 a^{5} + 10 a^{4} + 19 a^{3} + 12 a^{2} + 9 a + 16\right)\cdot 23^{178} + \left(6 a^{13} + 5 a^{12} + a^{11} + 14 a^{9} + 5 a^{8} + 20 a^{7} + 2 a^{6} + 11 a^{5} + 11 a^{4} + 21 a^{2} + 11 a + 16\right)\cdot 23^{179} + \left(22 a^{14} + 20 a^{13} + 14 a^{12} + 11 a^{11} + 22 a^{10} + 9 a^{9} + 19 a^{8} + 17 a^{7} + 9 a^{6} + 13 a^{5} + 21 a^{4} + 8 a^{3} + 15 a^{2} + 14 a + 7\right)\cdot 23^{180} + \left(10 a^{14} + 12 a^{13} + 22 a^{12} + 7 a^{11} + 22 a^{9} + 21 a^{8} + 4 a^{7} + 8 a^{6} + 4 a^{5} + 22 a^{4} + 21 a^{3} + 7 a^{2} + 12 a + 15\right)\cdot 23^{181} + \left(5 a^{14} + 15 a^{12} + 16 a^{10} + 22 a^{9} + 18 a^{8} + 22 a^{7} + 16 a^{6} + 17 a^{5} + 10 a^{4} + 19 a^{3} + 17 a^{2} + 7 a + 4\right)\cdot 23^{182} + \left(5 a^{14} + 6 a^{13} + 6 a^{12} + 12 a^{11} + 3 a^{10} + 21 a^{9} + 7 a^{8} + 8 a^{7} + 16 a^{6} + 5 a^{5} + 7 a^{4} + 13 a^{3} + a^{2} + 11 a + 18\right)\cdot 23^{183} + \left(a^{14} + 9 a^{13} + 10 a^{12} + 16 a^{11} + 12 a^{10} + 2 a^{9} + 12 a^{8} + 11 a^{7} + 2 a^{6} + 2 a^{5} + 9 a^{4} + 10 a^{3} + 8 a^{2} + 18 a + 13\right)\cdot 23^{184} + \left(9 a^{14} + 18 a^{13} + 16 a^{12} + 3 a^{11} + 10 a^{10} + 14 a^{9} + 4 a^{8} + 16 a^{7} + a^{6} + 19 a^{5} + 6 a^{4} + 7 a^{3} + 9 a^{2} + 21 a + 20\right)\cdot 23^{185} + \left(17 a^{14} + 22 a^{13} + 12 a^{12} + 6 a^{11} + 19 a^{10} + 22 a^{9} + 5 a^{8} + 2 a^{7} + 15 a^{6} + 19 a^{5} + 20 a^{4} + 12 a^{3} + 5 a^{2} + 10 a + 3\right)\cdot 23^{186} + \left(18 a^{14} + 5 a^{13} + 6 a^{12} + 4 a^{11} + 2 a^{10} + 10 a^{9} + 4 a^{8} + 3 a^{7} + 6 a^{6} + 10 a^{5} + 12 a^{4} + 15 a^{3} + 12 a + 10\right)\cdot 23^{187} + \left(14 a^{14} + 4 a^{13} + 9 a^{12} + 17 a^{11} + 20 a^{10} + 12 a^{9} + 22 a^{8} + 11 a^{7} + 11 a^{6} + 7 a^{5} + 8 a^{4} + 13 a^{3} + 9 a^{2} + 17\right)\cdot 23^{188} + \left(8 a^{14} + 18 a^{13} + 11 a^{12} + 3 a^{11} + 19 a^{10} + 4 a^{9} + 22 a^{8} + 13 a^{7} + 12 a^{6} + 15 a^{5} + 8 a^{4} + 7 a^{3} + 21 a^{2} + 18 a + 18\right)\cdot 23^{189} + \left(10 a^{14} + 11 a^{13} + 18 a^{12} + 22 a^{11} + 11 a^{10} + 14 a^{9} + 21 a^{8} + 17 a^{7} + 14 a^{6} + 18 a^{5} + 10 a^{4} + a^{3} + 2 a^{2} + 8 a + 19\right)\cdot 23^{190} + \left(22 a^{14} + 3 a^{13} + 17 a^{12} + 21 a^{11} + 17 a^{9} + 8 a^{8} + 4 a^{7} + 20 a^{6} + 20 a^{5} + 8 a^{4} + 7 a^{3} + 3 a^{2} + 7 a + 3\right)\cdot 23^{191} + \left(12 a^{14} + 19 a^{12} + 4 a^{11} + a^{10} + 5 a^{9} + 21 a^{8} + 18 a^{7} + 18 a^{6} + 5 a^{5} + 7 a^{4} + 19 a^{3} + 3 a^{2} + 14 a + 22\right)\cdot 23^{192} + \left(16 a^{14} + 18 a^{13} + 5 a^{12} + 20 a^{11} + 8 a^{10} + 13 a^{9} + 13 a^{8} + 12 a^{7} + 19 a^{6} + 6 a^{5} + 5 a^{4} + 15 a^{3} + 11 a^{2} + 13 a + 19\right)\cdot 23^{193} + \left(2 a^{14} + 14 a^{13} + 10 a^{12} + 18 a^{11} + a^{10} + 8 a^{9} + 16 a^{8} + 3 a^{7} + 12 a^{6} + 6 a^{5} + 4 a^{4} + 4 a^{3} + 17 a^{2} + 21 a + 5\right)\cdot 23^{194} + \left(9 a^{14} + 12 a^{13} + 16 a^{12} + 17 a^{10} + 20 a^{8} + 12 a^{7} + 16 a^{6} + 13 a^{5} + 16 a^{4} + 15 a^{3} + 6 a^{2} + a + 11\right)\cdot 23^{195} + \left(11 a^{14} + 9 a^{13} + 13 a^{12} + 10 a^{11} + 20 a^{10} + 12 a^{9} + 17 a^{8} + 2 a^{7} + 15 a^{6} + 2 a^{5} + a^{4} + 19 a^{3} + 14 a^{2} + 22 a + 19\right)\cdot 23^{196} + \left(9 a^{14} + 4 a^{13} + a^{12} + 2 a^{11} + 22 a^{10} + 14 a^{9} + a^{8} + 10 a^{7} + 15 a^{5} + 6 a^{4} + 18 a^{3} + 5 a^{2} + 13 a + 17\right)\cdot 23^{197} + \left(3 a^{14} + 4 a^{13} + 7 a^{12} + 13 a^{11} + 21 a^{10} + 11 a^{9} + 19 a^{8} + a^{7} + 14 a^{6} + 14 a^{5} + 19 a^{4} + 22 a^{3} + 21 a^{2} + 4 a + 3\right)\cdot 23^{198} + \left(17 a^{14} + 9 a^{13} + 12 a^{12} + 15 a^{11} + 12 a^{10} + 19 a^{9} + 17 a^{8} + 11 a^{7} + 21 a^{6} + 6 a^{5} + 15 a^{4} + 5 a^{3} + 5 a + 15\right)\cdot 23^{199} + \left(12 a^{14} + 20 a^{13} + 11 a^{12} + 15 a^{11} + 8 a^{10} + 5 a^{9} + 5 a^{8} + 10 a^{7} + 4 a^{6} + 13 a^{5} + 10 a^{4} + 5 a^{3} + 13 a^{2} + a + 7\right)\cdot 23^{200} + \left(15 a^{14} + 16 a^{13} + 13 a^{12} + 12 a^{11} + 18 a^{10} + 6 a^{9} + 12 a^{8} + 18 a^{7} + 4 a^{6} + 22 a^{5} + a^{4} + 13 a^{3} + 9 a^{2} + 7 a + 3\right)\cdot 23^{201} + \left(13 a^{14} + 10 a^{13} + 8 a^{12} + 16 a^{11} + 5 a^{10} + 16 a^{9} + 6 a^{8} + 16 a^{7} + 16 a^{6} + 2 a^{5} + 10 a^{4} + 16 a^{3} + 2 a^{2} + 19 a + 14\right)\cdot 23^{202} + \left(9 a^{14} + 17 a^{13} + 21 a^{12} + a^{11} + 16 a^{10} + 5 a^{9} + 9 a^{8} + 7 a^{7} + 14 a^{6} + 2 a^{5} + 11 a^{4} + 9 a^{3} + 22 a^{2} + 19 a + 16\right)\cdot 23^{203} + \left(a^{14} + 17 a^{13} + 7 a^{12} + 15 a^{11} + 11 a^{10} + 20 a^{9} + 11 a^{8} + 20 a^{7} + 16 a^{6} + 2 a^{5} + 22 a^{4} + 22 a^{3} + 22 a^{2} + 10 a + 6\right)\cdot 23^{204} + \left(4 a^{14} + 16 a^{13} + 8 a^{11} + 9 a^{10} + 18 a^{9} + 15 a^{8} + 14 a^{7} + 14 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\left(20 a^{14} + 13 a^{13} + 3 a^{12} + 9 a^{11} + 3 a^{10} + 21 a^{9} + 6 a^{8} + 17 a^{7} + 7 a^{6} + 16 a^{5} + 5 a^{4} + 20 a^{3} + 17 a^{2} + 7 a + 12\right)\cdot 23^{217} + \left(11 a^{14} + 13 a^{13} + 8 a^{12} + 22 a^{11} + 12 a^{10} + 8 a^{9} + 14 a^{8} + a^{7} + 15 a^{6} + 11 a^{5} + 13 a^{4} + 4 a^{2} + 1\right)\cdot 23^{218} + \left(8 a^{13} + 17 a^{12} + 18 a^{11} + 4 a^{10} + a^{9} + 13 a^{7} + 8 a^{6} + 22 a^{5} + 6 a^{4} + 4 a^{3} + 14 a^{2} + 3 a + 1\right)\cdot 23^{219} + \left(16 a^{14} + 21 a^{13} + 6 a^{12} + 14 a^{11} + 21 a^{10} + 22 a^{9} + a^{8} + 22 a^{7} + 8 a^{6} + 8 a^{5} + 18 a^{4} + 14 a^{3} + 10 a^{2} + 21 a + 13\right)\cdot 23^{220} + \left(2 a^{14} + 9 a^{13} + 7 a^{12} + 2 a^{11} + 21 a^{10} + 4 a^{9} + 17 a^{8} + 15 a^{7} + 20 a^{6} + 9 a^{5} + 8 a^{4} + 7 a^{3} + 18 a^{2} + 3 a\right)\cdot 23^{221} + \left(9 a^{14} + 15 a^{13} + 10 a^{12} + 22 a^{11} + 7 a^{10} + 6 a^{9} + 14 a^{8} + 19 a^{7} + 14 a^{6} + 16 a^{5} + 5 a^{4} + 3 a^{3} + 19 a^{2} + a + 20\right)\cdot 23^{222} + \left(8 a^{13} + a^{12} + 9 a^{10} + 4 a^{9} + 20 a^{8} + 8 a^{7} + 7 a^{6} + 17 a^{5} + 7 a^{4} + 16 a^{3} + 14 a^{2} + 15 a + 22\right)\cdot 23^{223} + \left(14 a^{14} + 20 a^{13} + a^{12} + 3 a^{11} + 19 a^{10} + 13 a^{9} + 5 a^{8} + 12 a^{7} + 7 a^{6} + 19 a^{5} + 8 a^{4} + 20 a^{3} + 14 a^{2} + 3 a + 1\right)\cdot 23^{224} + \left(12 a^{14} + 12 a^{13} + 19 a^{12} + 15 a^{11} + 18 a^{10} + 11 a^{9} + 2 a^{8} + 8 a^{7} + 2 a^{6} + 9 a^{5} + 17 a^{4} + 21 a^{3} + 5 a^{2} + 3 a + 12\right)\cdot 23^{225} + \left(7 a^{14} + 3 a^{13} + 21 a^{12} + 7 a^{11} + 16 a^{10} + 9 a^{8} + 9 a^{7} + 19 a^{6} + 16 a^{5} + 22 a^{4} + a^{3} + 17 a^{2} + 15 a + 10\right)\cdot 23^{226} + \left(5 a^{14} + 4 a^{13} + 11 a^{12} + 3 a^{10} + 21 a^{9} + 8 a^{8} + 8 a^{7} + 22 a^{6} + 6 a^{5} + 2 a^{4} + 20 a^{3} + a^{2} + 11 a + 14\right)\cdot 23^{227} + \left(3 a^{14} + 19 a^{13} + 17 a^{12} + 20 a^{11} + 15 a^{10} + 9 a^{9} + 17 a^{8} + 19 a^{7} + 17 a^{6} + 12 a^{5} + 3 a^{4} + 14 a^{3} + 13 a^{2} + 6 a + 1\right)\cdot 23^{228} + \left(2 a^{14} + 16 a^{13} + 18 a^{12} + 2 a^{11} + 4 a^{10} + 13 a^{9} + 10 a^{8} + 4 a^{7} + 13 a^{6} + 3 a^{5} + 6 a^{4} + 4 a^{3} + 11 a^{2} + 19 a + 8\right)\cdot 23^{229} + \left(15 a^{14} + 18 a^{13} + 18 a^{12} + 19 a^{11} + 17 a^{10} + 2 a^{9} + 20 a^{8} + 20 a^{7} + 9 a^{6} + 14 a^{5} + 21 a^{4} + 14 a^{3} + 7 a^{2} + a + 16\right)\cdot 23^{230} + \left(7 a^{14} + 19 a^{13} + 7 a^{12} + 17 a^{11} + 12 a^{10} + 11 a^{9} + 13 a^{8} + 5 a^{7} + 17 a^{6} + 18 a^{5} + 20 a^{4} + 14 a^{3} + 17 a^{2} + 19 a + 2\right)\cdot 23^{231} + \left(7 a^{14} + 17 a^{13} + a^{12} + 19 a^{11} + 2 a^{10} + 22 a^{9} + 20 a^{8} + 21 a^{7} + 18 a^{6} + 22 a^{4} + 16 a^{3} + 14 a^{2} + 9 a + 9\right)\cdot 23^{232} + \left(5 a^{14} + 8 a^{13} + 13 a^{12} + 19 a^{11} + 17 a^{10} + 11 a^{9} + 13 a^{8} + 10 a^{7} + 13 a^{6} + 14 a^{5} + 20 a^{4} + 13 a^{3} + 15 a^{2} + 19\right)\cdot 23^{233} + \left(5 a^{14} + 5 a^{13} + 9 a^{12} + 7 a^{11} + 3 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a^{13} + 8 a^{12} + 14 a^{11} + 10 a^{10} + 12 a^{9} + 19 a^{8} + 9 a^{7} + 15 a^{6} + 4 a^{5} + a^{4} + 8 a^{3} + 5 a^{2} + 7 a + 6\right)\cdot 23^{240} + \left(5 a^{14} + 11 a^{13} + 9 a^{12} + 2 a^{11} + 5 a^{10} + 9 a^{9} + 15 a^{8} + 10 a^{7} + 4 a^{6} + 22 a^{5} + 2 a^{4} + a^{3} + 14 a^{2} + 17 a + 5\right)\cdot 23^{241} + \left(5 a^{14} + 21 a^{13} + 3 a^{12} + 8 a^{11} + 12 a^{10} + 8 a^{9} + 14 a^{8} + 21 a^{7} + 6 a^{6} + 13 a^{5} + 17 a^{4} + 13 a^{3} + 8 a^{2} + 21 a + 15\right)\cdot 23^{242} + \left(14 a^{14} + 18 a^{13} + 15 a^{12} + 15 a^{11} + 10 a^{9} + 22 a^{8} + 21 a^{7} + 19 a^{6} + 18 a^{5} + 4 a^{4} + 19 a^{3} + 15 a^{2} + 9 a + 2\right)\cdot 23^{243} + \left(8 a^{14} + 18 a^{13} + 11 a^{12} + 7 a^{11} + 2 a^{10} + 20 a^{9} + 7 a^{8} + 17 a^{7} + 2 a^{6} + 10 a^{5} + 17 a^{4} + 12 a^{3} + 18 a^{2} + 15 a + 8\right)\cdot 23^{244} + \left(13 a^{14} + 10 a^{13} + 13 a^{12} + 14 a^{11} + 14 a^{10} + 11 a^{9} + 19 a^{8} + 15 a^{5} + 17 a^{4} + a^{3} + 15 a^{2} + 2 a + 9\right)\cdot 23^{245} + \left(14 a^{14} + 16 a^{13} + 3 a^{12} + 4 a^{11} + 15 a^{10} + 15 a^{9} + 7 a^{8} + 5 a^{7} + 22 a^{6} + 22 a^{5} + 9 a^{4} + 3 a^{3} + 3 a^{2} + 7 a + 7\right)\cdot 23^{246} + \left(6 a^{14} + 15 a^{13} + 5 a^{12} + 11 a^{11} + 14 a^{10} + 22 a^{9} + 6 a^{8} + a^{7} + 7 a^{6} + 6 a^{5} + 12 a^{4} + 11 a^{3} + 12 a^{2} + 15 a + 17\right)\cdot 23^{247} + \left(13 a^{14} + 8 a^{13} + 19 a^{12} + 22 a^{11} + 13 a^{10} + 10 a^{9} + 8 a^{8} + 14 a^{7} + 4 a^{6} + 21 a^{5} + 2 a^{4} + 11 a^{3} + 15 a^{2} + 4 a + 20\right)\cdot 23^{248} + \left(14 a^{14} + 6 a^{13} + 13 a^{12} + 7 a^{11} + 9 a^{10} + 8 a^{9} + 13 a^{8} + 14 a^{7} + 20 a^{6} + 8 a^{5} + 21 a^{4} + 4 a^{3} + 3 a + 2\right)\cdot 23^{249} + \left(6 a^{14} + 12 a^{13} + 3 a^{12} + 3 a^{11} + 3 a^{10} + 12 a^{9} + 4 a^{8} + 14 a^{7} + 2 a^{6} + 6 a^{5} + 10 a^{4} + 18 a^{3} + 21 a^{2} + 2 a + 20\right)\cdot 23^{250} + \left(17 a^{14} + 19 a^{13} + 15 a^{11} + 13 a^{10} + 6 a^{9} + 19 a^{8} + 12 a^{7} + 18 a^{6} + 2 a^{5} + 14 a^{4} + 15 a^{3} + 20 a^{2} + 19 a + 7\right)\cdot 23^{251} + \left(15 a^{14} + a^{13} + 4 a^{12} + 8 a^{11} + 14 a^{10} + 11 a^{9} + 14 a^{8} + 17 a^{7} + 11 a^{6} + 11 a^{5} + 11 a^{4} + 10 a^{3} + 17 a^{2} + 12 a + 10\right)\cdot 23^{252} + \left(2 a^{14} + 8 a^{13} + 21 a^{12} + 16 a^{11} + 7 a^{9} + a^{8} + 7 a^{7} + 16 a^{6} + 6 a^{5} + 3 a^{4} + 17 a^{3} + a^{2} + 9 a + 5\right)\cdot 23^{253} + \left(20 a^{14} + 12 a^{13} + 21 a^{12} + 17 a^{11} + 12 a^{10} + 15 a^{9} + 12 a^{8} + 3 a^{7} + 2 a^{6} + 20 a^{5} + 14 a^{4} + 19 a^{3} + 19 a^{2} + 9 a + 18\right)\cdot 23^{254} + \left(2 a^{14} + 13 a^{13} + 21 a^{12} + 3 a^{11} + a^{10} + 7 a^{9} + 13 a^{8} + 14 a^{7} + 10 a^{6} + 19 a^{5} + 14 a^{4} + 17 a^{3} + a^{2} + 13 a + 7\right)\cdot 23^{255} + \left(5 a^{14} + 11 a^{13} + 2 a^{12} + 11 a^{11} + 10 a^{10} + 3 a^{9} + 8 a^{8} + 11 a^{7} + 6 a^{6} + 21 a^{5} + a^{4} + 5 a^{3} + 22 a^{2} + 17 a + 17\right)\cdot 23^{256} + \left(22 a^{14} + 11 a^{13} + 16 a^{12} + 11 a^{11} + 11 a^{10} + 16 a^{9} + 14 a^{8} + 2 a^{7} + 15 a^{6} + a^{5} + 11 a^{4} + 9 a^{3} + 9 a^{2} + 12 a + 22\right)\cdot 23^{257} + \left(a^{13} + 17 a^{12} + 15 a^{11} + a^{10} + 8 a^{8} + 14 a^{7} + 16 a^{6} + 18 a^{5} + 9 a^{4} + 17 a^{3} + 9 a^{2} + 21 a + 17\right)\cdot 23^{258} + \left(a^{14} + 4 a^{13} + 17 a^{12} + a^{11} + 19 a^{10} + 22 a^{9} + 20 a^{8} + 13 a^{6} + 16 a^{5} + 8 a^{4} + 18 a^{3} + 7 a^{2} + 14 a + 22\right)\cdot 23^{259} + \left(6 a^{14} + 16 a^{13} + 2 a^{12} + 3 a^{11} + 20 a^{10} + 17 a^{9} + 13 a^{8} + 14 a^{7} + 7 a^{6} + 16 a^{4} + 6 a^{3} + 5 a^{2} + 11 a + 7\right)\cdot 23^{260} + \left(12 a^{14} + 5 a^{13} + 5 a^{12} + 8 a^{11} + 17 a^{10} + 20 a^{8} + 8 a^{7} + 7 a^{6} + 22 a^{5} + 11 a^{4} + 10 a^{3} + 9 a^{2} + 18 a + 8\right)\cdot 23^{261} + \left(6 a^{14} + 21 a^{13} + 18 a^{12} + 7 a^{11} + 7 a^{10} + 6 a^{9} + 10 a^{8} + 14 a^{7} + 15 a^{6} + 15 a^{5} + 18 a^{4} + 7 a^{3} + 13 a^{2} + 8 a + 9\right)\cdot 23^{262} + \left(5 a^{14} + 18 a^{13} + 11 a^{12} + 21 a^{11} + 12 a^{10} + 19 a^{9} + 18 a^{8} + 19 a^{7} + 2 a^{6} + 13 a^{5} + 2 a^{4} + 17 a^{3} + 12 a^{2} + 6 a + 12\right)\cdot 23^{263} + \left(20 a^{14} + 22 a^{13} + 18 a^{12} + 3 a^{11} + 4 a^{10} + 11 a^{9} + 14 a^{8} + 5 a^{7} + 8 a^{6} + 7 a^{5} + 14 a^{4} + 17 a^{3} + 21 a^{2} + 6 a + 20\right)\cdot 23^{264} + \left(12 a^{13} + 3 a^{12} + 4 a^{11} + 5 a^{10} + 19 a^{9} + 3 a^{8} + 7 a^{7} + 21 a^{6} + 12 a^{5} + 22 a^{4} + 11 a^{3} + 4 a^{2} + 17 a + 15\right)\cdot 23^{265} + \left(22 a^{14} + 4 a^{13} + 17 a^{12} + 6 a^{11} + 3 a^{10} + 8 a^{9} + 8 a^{8} + 3 a^{7} + 8 a^{6} + 20 a^{5} + 2 a^{4} + 5 a^{3} + 15 a^{2} + 7 a + 6\right)\cdot 23^{266} + \left(16 a^{14} + 14 a^{13} + 14 a^{12} + 4 a^{11} + 11 a^{10} + 14 a^{9} + 11 a^{8} + 8 a^{7} + 2 a^{6} + 19 a^{5} + 17 a^{4} + 7 a^{3} + 16 a^{2} + 15\right)\cdot 23^{267} + \left(8 a^{14} + 17 a^{12} + 12 a^{11} + 22 a^{10} + 16 a^{9} + 12 a^{8} + 20 a^{7} + 6 a^{6} + 11 a^{5} + 17 a^{4} + 6 a^{3} + 4 a^{2} + 14 a + 21\right)\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 6 }$ $=$ $20 a^{14} + 16 a^{13} + 13 a^{12} + 13 a^{11} + 12 a^{10} + 2 a^{9} + 7 a^{8} + 9 a^{7} + 20 a^{6} + 6 a^{5} + 3 a^{4} + 6 a^{3} + 3 a^{2} + 18 a + 12 + \left(20 a^{14} + a^{13} + 19 a^{12} + 7 a^{11} + 6 a^{10} + 15 a^{9} + 20 a^{8} + 12 a^{7} + 5 a^{6} + 20 a^{5} + 8 a^{4} + 16 a^{3} + 12 a^{2} + 10 a + 9\right)\cdot 23 + \left(12 a^{14} + 21 a^{13} + 3 a^{12} + 5 a^{10} + 2 a^{9} + 10 a^{8} + 6 a^{7} + 9 a^{6} + 16 a^{5} + 2 a^{4} + 12 a^{3} + 20 a^{2} + 22 a + 14\right)\cdot 23^{2} + \left(18 a^{14} + 20 a^{13} + 14 a^{12} + 15 a^{11} + a^{10} + 11 a^{9} + 14 a^{8} + 8 a^{7} + 6 a^{6} + 5 a^{5} + 15 a^{3} + 17 a^{2} + 16 a + 10\right)\cdot 23^{3} + \left(5 a^{13} + 12 a^{12} + 11 a^{11} + 3 a^{10} + 13 a^{9} + 14 a^{8} + 2 a^{7} + 11 a^{6} + 19 a^{5} + 6 a^{4} + 18 a^{3} + 21 a^{2} + 21 a + 11\right)\cdot 23^{4} + \left(20 a^{14} + 11 a^{13} + 11 a^{12} + 21 a^{11} + 5 a^{10} + 19 a^{9} + a^{8} + 11 a^{7} + 7 a^{6} + 12 a^{5} + 11 a^{4} + 19 a^{3} + 16 a^{2} + 7 a + 16\right)\cdot 23^{5} + \left(5 a^{14} + 11 a^{13} + 20 a^{12} + 5 a^{11} + 21 a^{10} + 21 a^{9} + 20 a^{8} + 8 a^{7} + 20 a^{6} + 10 a^{5} + 3 a^{4} + 12 a^{3} + 4 a^{2} + 8 a + 15\right)\cdot 23^{6} + \left(17 a^{14} + 12 a^{13} + 8 a^{12} + 9 a^{11} + 17 a^{10} + 19 a^{9} + 11 a^{8} + 6 a^{7} + 11 a^{6} + 18 a^{5} + 5 a^{4} + 12 a^{3} + 2 a^{2} + 16 a + 22\right)\cdot 23^{7} + \left(2 a^{14} + 22 a^{13} + 6 a^{12} + 7 a^{11} + 14 a^{10} + 5 a^{9} + 4 a^{8} + 19 a^{7} + 16 a^{6} + 2 a^{5} + 12 a^{4} + 20 a^{3} + 14 a^{2} + 14 a + 11\right)\cdot 23^{8} + \left(18 a^{14} + 20 a^{13} + 11 a^{12} + 14 a^{11} + 16 a^{10} + 20 a^{9} + 17 a^{8} + 7 a^{7} + 19 a^{6} + 6 a^{5} + 18 a^{4} + 4 a^{3} + 2 a^{2} + 10 a + 13\right)\cdot 23^{9} + \left(19 a^{12} + 8 a^{11} + 2 a^{10} + a^{9} + 16 a^{8} + 18 a^{7} + 12 a^{6} + 3 a^{5} + 19 a^{3} + 4 a^{2} + 18 a + 15\right)\cdot 23^{10} + \left(12 a^{14} + a^{13} + 2 a^{11} + 18 a^{10} + 13 a^{9} + 13 a^{8} + 17 a^{7} + 6 a^{6} + 4 a^{4} + 8 a^{3} + a^{2} + a + 6\right)\cdot 23^{11} + \left(9 a^{14} + 4 a^{13} + 21 a^{12} + 9 a^{11} + 12 a^{10} + 17 a^{9} + 18 a^{8} + 7 a^{7} + 12 a^{6} + 5 a^{5} + 19 a^{4} + 11 a^{3} + 9 a + 14\right)\cdot 23^{12} + \left(12 a^{14} + 8 a^{13} + 21 a^{12} + 14 a^{11} + 18 a^{10} + 7 a^{9} + 22 a^{8} + 15 a^{7} + 19 a^{6} + 4 a^{5} + 12 a^{4} + 13 a^{3} + 18 a^{2} + 10 a + 13\right)\cdot 23^{13} + \left(6 a^{14} + 2 a^{13} + 7 a^{12} + 2 a^{11} + 8 a^{10} + 13 a^{9} + 20 a^{8} + 11 a^{7} + 18 a^{6} + 7 a^{5} + 3 a^{4} + 13 a^{3} + 19 a^{2} + 9 a + 1\right)\cdot 23^{14} + \left(11 a^{14} + 13 a^{13} + 11 a^{11} + 16 a^{10} + 2 a^{9} + 6 a^{8} + 10 a^{7} + 15 a^{6} + 10 a^{5} + 11 a^{4} + 19 a^{3} + 22 a^{2} + 11 a + 8\right)\cdot 23^{15} + \left(10 a^{14} + 11 a^{13} + 5 a^{12} + 3 a^{11} + 10 a^{10} + 11 a^{9} + 19 a^{8} + 4 a^{7} + 12 a^{6} + 18 a^{5} + 8 a^{4} + 14 a^{3} + 9 a^{2} + 6 a + 8\right)\cdot 23^{16} + \left(15 a^{14} + 3 a^{13} + 15 a^{12} + 15 a^{11} + 19 a^{10} + 2 a^{9} + 16 a^{8} + 5 a^{7} + a^{6} + 18 a^{5} + 16 a^{4} + 4 a^{3} + 3 a^{2} + 16 a + 5\right)\cdot 23^{17} + \left(21 a^{14} + 2 a^{12} + 20 a^{11} + 10 a^{10} + 15 a^{9} + 8 a^{8} + 17 a^{7} + 15 a^{6} + a^{5} + 13 a^{4} + a^{3} + 17 a^{2} + 19 a + 3\right)\cdot 23^{18} + \left(20 a^{14} + 10 a^{13} + 15 a^{12} + 3 a^{11} + 14 a^{10} + 19 a^{9} + 22 a^{8} + 9 a^{7} + 7 a^{6} + 15 a^{5} + 4 a^{4} + 9 a^{3} + 2 a^{2} + 20 a + 14\right)\cdot 23^{19} + \left(9 a^{14} + 13 a^{13} + 4 a^{12} + 9 a^{11} + 2 a^{10} + 2 a^{9} + 18 a^{8} + 6 a^{7} + 13 a^{6} + 13 a^{5} + 10 a^{3} + 16 a^{2} + 9 a + 6\right)\cdot 23^{20} + \left(15 a^{14} + 14 a^{13} + 15 a^{12} + 2 a^{11} + 14 a^{10} + 17 a^{8} + 20 a^{7} + 5 a^{6} + 10 a^{5} + a^{4} + 7 a^{3} + a^{2} + 16 a + 19\right)\cdot 23^{21} + \left(4 a^{14} + a^{13} + 10 a^{12} + 21 a^{11} + 3 a^{10} + 17 a^{9} + 8 a^{8} + 5 a^{7} + 2 a^{6} + 22 a^{5} + 20 a^{4} + 13 a^{2} + 6 a + 10\right)\cdot 23^{22} + \left(11 a^{14} + 14 a^{13} + 10 a^{12} + 5 a^{10} + 17 a^{9} + 10 a^{8} + 21 a^{7} + 12 a^{6} + 14 a^{5} + 19 a^{4} + 19 a^{3} + 12 a^{2} + 6 a + 18\right)\cdot 23^{23} + \left(11 a^{14} + 3 a^{13} + a^{12} + 11 a^{11} + 14 a^{10} + 2 a^{9} + 9 a^{8} + 5 a^{7} + 16 a^{6} + a^{5} + 11 a^{4} + 13 a^{3} + 10 a^{2} + 18 a + 7\right)\cdot 23^{24} + \left(19 a^{14} + 11 a^{13} + 13 a^{12} + 8 a^{11} + 10 a^{10} + 12 a^{9} + 13 a^{8} + 10 a^{7} + 16 a^{6} + 14 a^{5} + 4 a^{4} + 9 a^{3} + 8 a^{2} + 3 a + 4\right)\cdot 23^{25} + \left(5 a^{14} + 4 a^{13} + 17 a^{12} + 19 a^{11} + 15 a^{10} + 12 a^{9} + 18 a^{8} + 14 a^{7} + 16 a^{6} + 7 a^{5} + 16 a^{4} + 22 a^{3} + 13 a + 18\right)\cdot 23^{26} + \left(8 a^{14} + 9 a^{13} + 19 a^{12} + 7 a^{11} + 5 a^{9} + 3 a^{8} + 14 a^{7} + 15 a^{6} + 5 a^{5} + 20 a^{4} + 12 a^{3} + 19 a^{2} + 2 a + 3\right)\cdot 23^{27} + \left(a^{14} + 19 a^{13} + 14 a^{12} + a^{11} + 18 a^{10} + 6 a^{9} + 9 a^{8} + 21 a^{7} + 18 a^{6} + 7 a^{5} + 11 a^{4} + a^{3} + 3 a^{2} + 3 a + 2\right)\cdot 23^{28} + \left(4 a^{14} + 13 a^{13} + 14 a^{11} + 10 a^{10} + 22 a^{9} + 4 a^{8} + 14 a^{7} + 12 a^{6} + 12 a^{5} + 16 a^{4} + 19 a^{3} + 4 a^{2} + 20 a + 8\right)\cdot 23^{29} + \left(21 a^{14} + 22 a^{13} + 20 a^{12} + 14 a^{11} + 9 a^{10} + 10 a^{9} + 4 a^{8} + 19 a^{7} + 12 a^{6} + 22 a^{5} + 8 a^{4} + 10 a^{3} + 7 a^{2} + a + 11\right)\cdot 23^{30} + \left(a^{14} + a^{13} + 18 a^{12} + 12 a^{11} + 18 a^{10} + a^{9} + a^{8} + 2 a^{7} + 3 a^{6} + 5 a^{5} + 7 a^{4} + 15 a^{3} + 3 a^{2} + a + 12\right)\cdot 23^{31} + \left(17 a^{14} + 5 a^{13} + 15 a^{12} + 15 a^{11} + 10 a^{10} + 5 a^{9} + 16 a^{8} + 14 a^{7} + 16 a^{6} + 3 a^{5} + 11 a^{3} + 20 a + 16\right)\cdot 23^{32} + \left(17 a^{14} + a^{13} + 22 a^{12} + 3 a^{11} + 8 a^{10} + 10 a^{9} + 15 a^{8} + 7 a^{7} + a^{6} + 8 a^{5} + 15 a^{4} + 17 a^{3} + 6 a^{2} + 18 a + 3\right)\cdot 23^{33} + \left(11 a^{14} + 16 a^{13} + 16 a^{12} + 3 a^{11} + 17 a^{10} + 15 a^{9} + 17 a^{8} + 5 a^{7} + 3 a^{6} + 8 a^{5} + 22 a^{4} + 7 a^{3} + 19 a^{2} + 17 a + 4\right)\cdot 23^{34} + \left(a^{14} + 15 a^{13} + 13 a^{12} + 13 a^{11} + 10 a^{10} + 13 a^{9} + 21 a^{8} + 21 a^{6} + 6 a^{5} + 6 a^{4} + 9 a^{3} + 15 a^{2} + 20 a + 19\right)\cdot 23^{35} + \left(17 a^{14} + 10 a^{13} + 18 a^{12} + 12 a^{11} + 14 a^{10} + 15 a^{9} + 6 a^{8} + 22 a^{7} + 19 a^{6} + 5 a^{5} + 5 a^{4} + 3 a^{3} + 9 a^{2} + 19 a + 22\right)\cdot 23^{36} + \left(16 a^{14} + a^{13} + 4 a^{12} + a^{11} + 11 a^{10} + 2 a^{9} + 4 a^{8} + 20 a^{7} + 14 a^{6} + 12 a^{5} + 7 a^{4} + 20 a^{3} + 3 a^{2} + 17 a + 3\right)\cdot 23^{37} + \left(11 a^{14} + 8 a^{13} + 5 a^{12} + 2 a^{11} + 6 a^{10} + 16 a^{9} + 17 a^{8} + 4 a^{7} + 2 a^{5} + 13 a^{4} + 21 a^{3} + 10 a^{2} + 22 a + 14\right)\cdot 23^{38} + \left(15 a^{14} + 12 a^{13} + 15 a^{12} + 3 a^{11} + 10 a^{10} + 10 a^{9} + 11 a^{8} + 20 a^{6} + 13 a^{5} + 16 a^{4} + 19 a^{2} + 14 a + 15\right)\cdot 23^{39} + \left(10 a^{14} + 12 a^{13} + 21 a^{12} + 9 a^{11} + 2 a^{10} + 19 a^{9} + 22 a^{8} + 18 a^{7} + 6 a^{6} + 15 a^{5} + a^{4} + 16 a^{3} + 22 a^{2} + 4 a + 11\right)\cdot 23^{40} + \left(8 a^{13} + 18 a^{12} + 15 a^{11} + 18 a^{9} + 13 a^{8} + 15 a^{7} + 13 a^{6} + 18 a^{5} + 22 a^{4} + 5 a^{3} + 3 a^{2} + 9 a + 22\right)\cdot 23^{41} + \left(a^{14} + 14 a^{13} + 6 a^{12} + a^{11} + 11 a^{10} + 16 a^{9} + 10 a^{8} + 13 a^{7} + 22 a^{6} + 14 a^{5} + 16 a^{4} + 17 a^{3} + 21 a^{2} + 8 a + 7\right)\cdot 23^{42} + \left(15 a^{14} + 14 a^{13} + 14 a^{12} + 4 a^{11} + 3 a^{10} + 9 a^{9} + a^{8} + 12 a^{7} + 12 a^{6} + 8 a^{5} + 8 a^{4} + 4 a^{3} + 17 a^{2} + 7 a\right)\cdot 23^{43} + \left(19 a^{14} + 10 a^{13} + a^{12} + 2 a^{11} + 8 a^{10} + 10 a^{9} + 10 a^{8} + 17 a^{7} + 20 a^{6} + 14 a^{5} + 18 a^{4} + 17 a^{3} + 13 a^{2} + 20 a + 1\right)\cdot 23^{44} + \left(19 a^{14} + 12 a^{13} + 12 a^{12} + 18 a^{11} + 9 a^{10} + 15 a^{9} + 13 a^{8} + 7 a^{7} + 2 a^{6} + 4 a^{5} + 6 a^{4} + 4 a^{3} + 20 a^{2} + 21 a + 15\right)\cdot 23^{45} + \left(17 a^{14} + 20 a^{13} + 5 a^{12} + 18 a^{11} + 3 a^{10} + 7 a^{9} + 13 a^{8} + 8 a^{7} + 4 a^{6} + 7 a^{5} + 3 a^{4} + 15 a^{3} + 21 a^{2} + 5 a + 14\right)\cdot 23^{46} + \left(17 a^{14} + 15 a^{13} + 17 a^{12} + 16 a^{11} + 13 a^{10} + 19 a^{9} + 12 a^{8} + 11 a^{7} + 5 a^{6} + 14 a^{5} + 3 a^{4} + 7 a^{3} + 9 a^{2} + 4 a + 4\right)\cdot 23^{47} + \left(9 a^{14} + 11 a^{13} + 8 a^{12} + 19 a^{10} + 19 a^{9} + 11 a^{8} + 14 a^{7} + 22 a^{6} + 19 a^{5} + 15 a^{4} + 10 a^{3} + 22 a^{2} + 20 a + 19\right)\cdot 23^{48} + \left(5 a^{14} + 11 a^{13} + 14 a^{12} + 12 a^{11} + 21 a^{10} + 15 a^{9} + 12 a^{8} + 3 a^{7} + 15 a^{6} + 6 a^{5} + 15 a^{4} + 22 a^{2} + 21 a + 16\right)\cdot 23^{49} + \left(19 a^{14} + 11 a^{13} + 3 a^{12} + 4 a^{11} + 5 a^{10} + 16 a^{9} + 21 a^{8} + 11 a^{7} + 16 a^{6} + 18 a^{5} + 21 a^{4} + 13 a^{3} + 11 a^{2} + 11 a + 10\right)\cdot 23^{50} + \left(7 a^{14} + 13 a^{13} + 13 a^{12} + 21 a^{10} + 4 a^{9} + 14 a^{8} + 22 a^{6} + 22 a^{4} + 12 a^{3} + 2 a^{2} + 21 a + 13\right)\cdot 23^{51} + \left(22 a^{14} + 12 a^{13} + 5 a^{12} + 11 a^{11} + 10 a^{10} + 10 a^{9} + 12 a^{8} + 22 a^{7} + 7 a^{6} + 17 a^{5} + 15 a^{4} + 14 a^{3} + 3 a + 17\right)\cdot 23^{52} + \left(18 a^{14} + a^{13} + 7 a^{12} + 14 a^{11} + 2 a^{10} + 10 a^{9} + 12 a^{7} + 9 a^{6} + 3 a^{5} + 3 a^{4} + 18 a^{3} + 13 a^{2} + 12 a + 11\right)\cdot 23^{53} + \left(7 a^{14} + 14 a^{13} + 4 a^{12} + 20 a^{11} + 22 a^{10} + 14 a^{9} + 3 a^{8} + 18 a^{7} + 3 a^{5} + 12 a^{4} + 14 a^{3} + 7 a^{2} + 8 a + 2\right)\cdot 23^{54} + \left(9 a^{14} + 18 a^{13} + 7 a^{12} + 13 a^{11} + 5 a^{10} + 4 a^{9} + 21 a^{8} + 5 a^{7} + 5 a^{6} + 22 a^{5} + 16 a^{4} + 4 a^{3} + 15 a^{2} + 9 a + 20\right)\cdot 23^{55} + \left(19 a^{14} + 11 a^{13} + 16 a^{12} + 8 a^{11} + 19 a^{10} + a^{9} + 5 a^{8} + 19 a^{6} + 7 a^{5} + 6 a^{4} + 22 a^{3} + 6 a^{2} + 12 a + 10\right)\cdot 23^{56} + \left(22 a^{14} + 6 a^{13} + 16 a^{12} + 14 a^{11} + 15 a^{10} + 20 a^{9} + 5 a^{8} + 3 a^{7} + 14 a^{6} + 10 a^{5} + 16 a^{4} + 12 a^{3} + 8 a^{2} + 14 a + 18\right)\cdot 23^{57} + \left(5 a^{14} + 15 a^{13} + 19 a^{12} + 8 a^{11} + a^{10} + 17 a^{9} + 19 a^{8} + 12 a^{7} + 7 a^{6} + 16 a^{5} + a^{4} + 9 a^{3} + 10 a^{2} + 3 a + 22\right)\cdot 23^{58} + \left(15 a^{14} + 19 a^{13} + 16 a^{12} + 14 a^{11} + 22 a^{10} + 10 a^{9} + 15 a^{8} + 4 a^{7} + 15 a^{6} + 3 a^{5} + 18 a^{4} + 22 a^{3} + 11 a^{2} + 14 a + 14\right)\cdot 23^{59} + \left(4 a^{14} + 8 a^{13} + 18 a^{12} + a^{11} + 22 a^{10} + 21 a^{9} + 10 a^{8} + 17 a^{7} + 10 a^{6} + 4 a^{5} + 2 a^{4} + 2 a^{3} + 13 a^{2} + 19 a + 15\right)\cdot 23^{60} + \left(12 a^{14} + 21 a^{13} + 18 a^{12} + 6 a^{11} + 3 a^{10} + a^{9} + 18 a^{7} + 21 a^{6} + 4 a^{5} + 18 a^{4} + 19 a^{3} + 3 a^{2} + 15 a + 1\right)\cdot 23^{61} + \left(6 a^{14} + 4 a^{13} + 9 a^{12} + 10 a^{11} + 6 a^{10} + 21 a^{9} + 12 a^{8} + 19 a^{7} + 6 a^{6} + a^{5} + 6 a^{4} + 11 a^{3} + 11 a^{2} + 22 a + 10\right)\cdot 23^{62} + \left(16 a^{14} + 5 a^{13} + 20 a^{12} + 5 a^{11} + 6 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4 a^{9} + 16 a^{8} + 18 a^{7} + 5 a^{6} + 6 a^{5} + 3 a^{4} + 2 a^{3} + 4 a^{2} + 6 a + 21\right)\cdot 23^{69} + \left(8 a^{14} + 19 a^{13} + 9 a^{12} + 16 a^{11} + 6 a^{10} + 15 a^{9} + 16 a^{8} + 13 a^{7} + 3 a^{6} + 13 a^{5} + 14 a^{3} + 10 a^{2} + 13 a + 11\right)\cdot 23^{70} + \left(15 a^{14} + 13 a^{13} + 9 a^{12} + 6 a^{11} + 20 a^{10} + 11 a^{9} + 22 a^{8} + 8 a^{7} + 14 a^{6} + 8 a^{5} + 4 a^{4} + 12 a^{3} + 5 a^{2} + 14 a + 17\right)\cdot 23^{71} + \left(6 a^{14} + 8 a^{13} + 3 a^{12} + 9 a^{11} + 9 a^{10} + 20 a^{9} + 21 a^{8} + 19 a^{7} + 17 a^{6} + 21 a^{5} + 10 a^{4} + 21 a^{3} + 17 a^{2} + 9 a + 3\right)\cdot 23^{72} + \left(16 a^{14} + 18 a^{13} + 9 a^{12} + 9 a^{11} + 10 a^{10} + 13 a^{9} + 16 a^{8} + 14 a^{7} + 13 a^{6} + 21 a^{4} + 2 a^{3} + 13 a^{2} + 20 a + 10\right)\cdot 23^{73} + \left(9 a^{14} + 2 a^{13} + 4 a^{12} + 6 a^{10} + 12 a^{9} + 17 a^{8} + 7 a^{7} + 3 a^{6} + 20 a^{5} + 22 a^{4} + 7 a^{3} + 10 a^{2} + 16 a + 5\right)\cdot 23^{74} + \left(6 a^{14} + 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a^{2} + 15 a + 10\right)\cdot 23^{86} + \left(5 a^{14} + 16 a^{13} + 15 a^{12} + 11 a^{11} + a^{10} + 19 a^{9} + 17 a^{8} + 14 a^{7} + 8 a^{6} + 6 a^{5} + 12 a^{4} + 12 a^{3} + 21 a^{2} + 5 a + 4\right)\cdot 23^{87} + \left(5 a^{14} + 5 a^{13} + 21 a^{12} + 16 a^{11} + 2 a^{10} + 18 a^{9} + 10 a^{8} + 17 a^{7} + 20 a^{6} + 7 a^{5} + 3 a^{4} + 19 a^{3} + 2 a^{2} + 10 a + 13\right)\cdot 23^{88} + \left(7 a^{14} + 12 a^{13} + 10 a^{12} + 13 a^{11} + 18 a^{10} + 19 a^{9} + 21 a^{8} + 8 a^{7} + 7 a^{6} + 5 a^{5} + 4 a^{4} + 18 a^{3} + 8 a^{2} + 22 a + 18\right)\cdot 23^{89} + \left(21 a^{14} + 17 a^{13} + 7 a^{12} + 2 a^{11} + 5 a^{10} + 11 a^{9} + 20 a^{8} + 7 a^{7} + 5 a^{6} + 3 a^{5} + 2 a^{4} + 21 a^{3} + 21 a^{2} + 10 a + 14\right)\cdot 23^{90} + \left(21 a^{14} + 17 a^{13} + 10 a^{12} + 16 a^{11} + 5 a^{10} + 22 a^{9} + 16 a^{8} + 5 a^{7} + 17 a^{6} + 19 a^{5} + 9 a^{4} + 6 a^{3} + 6 a^{2} + 22 a + 12\right)\cdot 23^{91} + \left(22 a^{14} + 11 a^{13} + 11 a^{12} + 16 a^{11} + 6 a^{10} + 12 a^{8} + 8 a^{7} + 7 a^{6} + 14 a^{5} + 18 a^{4} + 22 a^{3} + 3 a^{2} + 22 a\right)\cdot 23^{92} + \left(2 a^{14} + 13 a^{12} + 3 a^{11} + 18 a^{10} + 21 a^{9} + 9 a^{7} + 19 a^{6} + 2 a^{5} + 21 a^{4} + 9 a^{3} + 12 a^{2} + 9 a + 6\right)\cdot 23^{93} + \left(2 a^{14} + 16 a^{13} + 8 a^{12} + 11 a^{11} + 12 a^{10} + 6 a^{9} + 5 a^{8} + 16 a^{7} + 12 a^{6} + 22 a^{4} + 4 a^{3} + 14 a^{2} + 10 a + 4\right)\cdot 23^{94} + \left(2 a^{14} + 14 a^{13} + 3 a^{12} + 4 a^{11} + 11 a^{10} + 18 a^{9} + 2 a^{8} + 6 a^{7} + 5 a^{6} + 3 a^{5} + 21 a^{4} + 3 a^{3} + 10 a^{2} + 6 a + 19\right)\cdot 23^{95} + \left(5 a^{14} + 14 a^{13} + 18 a^{12} + 13 a^{11} + 17 a^{10} + 18 a^{9} + 6 a^{8} + 16 a^{7} + 11 a^{6} + 10 a^{5} + 6 a^{4} + 11 a^{3} + 17 a^{2} + 21 a + 8\right)\cdot 23^{96} + \left(7 a^{14} + 22 a^{13} + 22 a^{12} + 16 a^{11} + 10 a^{10} + 17 a^{9} + 21 a^{8} + 20 a^{7} + 6 a^{6} + 18 a^{5} + 5 a^{4} + 2 a^{3} + 18 a^{2} + 4 a\right)\cdot 23^{97} + \left(4 a^{14} + 8 a^{13} + 10 a^{12} 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15 a + 17\right)\cdot 23^{103} + \left(11 a^{14} + 3 a^{13} + 17 a^{12} + 21 a^{11} + 5 a^{10} + 3 a^{9} + 13 a^{8} + 13 a^{7} + 7 a^{6} + 10 a^{5} + 18 a^{4} + 14 a^{3} + 6 a^{2} + 18 a + 4\right)\cdot 23^{104} + \left(8 a^{14} + 12 a^{13} + 12 a^{12} + 4 a^{11} + 4 a^{10} + 22 a^{9} + 14 a^{8} + 14 a^{7} + 16 a^{6} + 10 a^{5} + 19 a^{4} + 22 a^{3} + 4 a^{2} + 6 a + 7\right)\cdot 23^{105} + \left(3 a^{14} + 21 a^{13} + 9 a^{12} + 3 a^{11} + 2 a^{10} + a^{9} + 14 a^{8} + 14 a^{7} + 14 a^{6} + 7 a^{5} + 20 a^{4} + 12 a^{3} + 10 a^{2} + a + 17\right)\cdot 23^{106} + \left(6 a^{14} + 3 a^{13} + 2 a^{11} + 9 a^{10} + 8 a^{9} + 15 a^{8} + 7 a^{7} + 11 a^{6} + 22 a^{5} + 17 a^{4} + 4 a^{3} + 9 a^{2} + 13 a + 3\right)\cdot 23^{107} + \left(16 a^{14} + 9 a^{13} + 7 a^{12} + 4 a^{11} + 10 a^{10} + a^{9} + 9 a^{8} + 20 a^{7} + 9 a^{6} + 7 a^{5} + 17 a^{4} + 15 a^{3} + 6 a + 22\right)\cdot 23^{108} + \left(14 a^{14} + 5 a^{13} + 10 a^{12} + 8 a^{11} + 16 a^{10} + 15 a^{9} + 17 a^{8} + 17 a^{7} + 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a^{3} + 21 a + 13\right)\cdot 23^{126} + \left(20 a^{14} + 11 a^{13} + 5 a^{12} + a^{11} + 4 a^{10} + 2 a^{9} + 19 a^{8} + 5 a^{7} + 4 a^{6} + 14 a^{5} + 16 a^{4} + 15 a^{3} + 16 a^{2} + 16 a + 2\right)\cdot 23^{127} + \left(9 a^{14} + 12 a^{13} + 18 a^{12} + 14 a^{11} + 12 a^{10} + 7 a^{9} + 18 a^{8} + 9 a^{7} + 10 a^{6} + 8 a^{5} + 6 a^{4} + a^{3} + 16 a^{2} + 16 a + 17\right)\cdot 23^{128} + \left(8 a^{14} + 18 a^{13} + 22 a^{12} + 5 a^{11} + 19 a^{10} + 13 a^{9} + 14 a^{8} + 16 a^{7} + 4 a^{6} + 11 a^{5} + 4 a^{4} + 22 a^{3} + 7 a^{2} + 14 a + 17\right)\cdot 23^{129} + \left(13 a^{14} + 21 a^{13} + 6 a^{12} + 11 a^{11} + 2 a^{10} + 22 a^{9} + 19 a^{8} + 2 a^{7} + 9 a^{6} + 8 a^{5} + 13 a^{4} + 3 a^{3} + 16 a^{2} + 21 a + 11\right)\cdot 23^{130} + \left(18 a^{14} + 4 a^{13} + 9 a^{12} + 12 a^{11} + 17 a^{10} + 13 a^{9} + 21 a^{8} + 10 a^{7} + 6 a^{6} + 9 a^{5} + 8 a^{4} + 21 a^{3} + 2 a^{2} + 13 a + 21\right)\cdot 23^{131} + \left(14 a^{14} + 5 a^{13} + 22 a^{12} + 21 a^{11} + 8 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9 a^{2} + 6 a\right)\cdot 23^{143} + \left(21 a^{14} + a^{13} + 12 a^{12} + 2 a^{11} + 6 a^{10} + 13 a^{9} + 10 a^{8} + 6 a^{7} + 13 a^{6} + 10 a^{5} + 4 a^{4} + 12 a^{3} + 5 a^{2} + 18 a + 8\right)\cdot 23^{144} + \left(18 a^{14} + 10 a^{13} + 12 a^{12} + 22 a^{11} + 18 a^{10} + 15 a^{9} + 2 a^{8} + 3 a^{7} + 6 a^{6} + 7 a^{5} + 7 a^{4} + 17 a^{3} + 7 a^{2} + 4 a + 13\right)\cdot 23^{145} + \left(11 a^{14} + 2 a^{13} + 2 a^{12} + 9 a^{11} + 2 a^{10} + 18 a^{9} + 11 a^{8} + 9 a^{7} + 16 a^{6} + 10 a^{4} + 22 a^{3} + 17 a^{2} + 7 a + 15\right)\cdot 23^{146} + \left(2 a^{14} + 12 a^{13} + 11 a^{12} + 19 a^{11} + 5 a^{10} + 16 a^{9} + 12 a^{8} + 14 a^{6} + 2 a^{5} + 20 a^{4} + 16 a^{3} + 14 a^{2} + 5 a + 16\right)\cdot 23^{147} + \left(13 a^{14} + 22 a^{13} + 16 a^{12} + 14 a^{11} + 6 a^{10} + 8 a^{9} + 7 a^{8} + a^{7} + 4 a^{6} + 6 a^{5} + 14 a^{4} + 7 a^{2} + 9 a + 19\right)\cdot 23^{148} + \left(20 a^{14} + 9 a^{12} + 16 a^{11} + 22 a^{10} + 5 a^{9} + 10 a^{7} + 11 a^{5} + 9 a^{4} + 15 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5 a^{7} + 21 a^{6} + 4 a^{4} + a^{3} + 11 a^{2} + 4 a + 13\right)\cdot 23^{155} + \left(9 a^{14} + 4 a^{11} + 7 a^{10} + 16 a^{8} + 3 a^{7} + 2 a^{6} + 13 a^{5} + 15 a^{4} + 5 a^{3} + 13 a^{2} + 18 a + 18\right)\cdot 23^{156} + \left(20 a^{14} + 12 a^{13} + 13 a^{12} + 11 a^{11} + 16 a^{10} + 4 a^{9} + 2 a^{8} + 10 a^{7} + 22 a^{6} + 20 a^{5} + 2 a^{4} + 11 a^{3} + 8 a^{2} + 12 a + 11\right)\cdot 23^{157} + \left(2 a^{14} + 9 a^{13} + 14 a^{12} + 2 a^{11} + 2 a^{10} + 11 a^{9} + 12 a^{8} + 14 a^{7} + 11 a^{6} + 18 a^{5} + 4 a^{4} + 3 a^{3} + 7 a^{2} + a + 8\right)\cdot 23^{158} + \left(17 a^{14} + 5 a^{13} + 13 a^{12} + 11 a^{11} + 20 a^{10} + 14 a^{9} + 12 a^{8} + 22 a^{7} + 19 a^{6} + 14 a^{4} + 18 a^{3} + 21 a^{2} + 14 a + 22\right)\cdot 23^{159} + \left(a^{14} + 22 a^{13} + 13 a^{12} + 19 a^{11} + 13 a^{10} + 4 a^{9} + 22 a^{8} + 22 a^{7} + 10 a^{6} + 9 a^{5} + 19 a^{4} + 20 a^{3} + 9 a^{2} + 10 a + 4\right)\cdot 23^{160} + \left(22 a^{14} + 22 a^{13} + a^{11} + a^{10} + 11 a^{9} + 3 a^{8} + 20 a^{7} + 12 a^{6} + 11 a^{5} + 20 a^{4} + 14 a^{3} + 12 a^{2} + 17 a + 19\right)\cdot 23^{161} + \left(14 a^{14} + 7 a^{13} + 15 a^{11} + 18 a^{9} + 15 a^{8} + 2 a^{7} + 12 a^{6} + 12 a^{5} + 11 a^{4} + 12 a^{3} + 9 a^{2} + 6 a + 6\right)\cdot 23^{162} + \left(18 a^{14} + 14 a^{13} + 19 a^{12} + 12 a^{11} + 12 a^{10} + 5 a^{9} + 19 a^{8} + 2 a^{7} + 9 a^{6} + 5 a^{4} + 22 a^{3} + a^{2} + 18 a + 16\right)\cdot 23^{163} + \left(8 a^{14} + 14 a^{13} + 20 a^{12} + 2 a^{11} + 18 a^{10} + a^{9} + 15 a^{8} + 6 a^{7} + 8 a^{6} + 16 a^{5} + 3 a^{4} + 20 a^{3} + 6 a^{2} + 3 a + 9\right)\cdot 23^{164} + \left(14 a^{14} + 13 a^{13} + 12 a^{12} + 20 a^{11} + 9 a^{9} + a^{7} + 6 a^{6} + 14 a^{5} + 3 a^{4} + 11 a^{3} + 10 a^{2} + 2 a + 10\right)\cdot 23^{165} + \left(6 a^{13} + a^{12} + 15 a^{11} + 13 a^{10} + 12 a^{9} + 2 a^{8} + 5 a^{7} + 20 a^{6} + 5 a^{5} + 2 a^{4} + 15 a^{3} + 4 a^{2} + 15 a + 5\right)\cdot 23^{166} + \left(12 a^{14} + 8 a^{13} + 10 a^{12} + 13 a^{11} + 3 a^{10} + 19 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+ 7 a^{3} + 17 a^{2} + 6 a + 19\right)\cdot 23^{178} + \left(5 a^{14} + 16 a^{13} + a^{12} + 18 a^{11} + 12 a^{10} + 8 a^{9} + 6 a^{8} + 19 a^{7} + 22 a^{6} + 13 a^{5} + 10 a^{4} + 13 a^{3} + 11 a^{2} + 17 a + 10\right)\cdot 23^{179} + \left(10 a^{14} + 8 a^{13} + 8 a^{12} + 3 a^{11} + 7 a^{10} + 7 a^{9} + 14 a^{8} + 13 a^{7} + 6 a^{6} + 20 a^{5} + 2 a^{4} + 21 a^{3} + 21 a^{2} + 15 a + 12\right)\cdot 23^{180} + \left(9 a^{14} + 4 a^{13} + 19 a^{12} + a^{11} + 16 a^{10} + 12 a^{9} + 8 a^{8} + 12 a^{7} + 3 a^{6} + 2 a^{5} + 19 a^{4} + 9 a^{3} + 2 a^{2} + 22\right)\cdot 23^{181} + \left(6 a^{14} + 17 a^{13} + 21 a^{12} + 9 a^{11} + 2 a^{10} + 21 a^{9} + 12 a^{8} + 12 a^{7} + 5 a^{6} + 7 a^{5} + 21 a^{4} + 22 a^{3} + 6 a^{2} + 10 a + 20\right)\cdot 23^{182} + \left(7 a^{14} + 20 a^{13} + 8 a^{12} + 9 a^{11} + 21 a^{10} + 10 a^{9} + 5 a^{8} + 13 a^{7} + 22 a^{6} + 10 a^{5} + 3 a^{4} + 19 a^{3} + 14 a^{2} + 14 a + 4\right)\cdot 23^{183} + \left(16 a^{14} + 14 a^{13} + 4 a^{12} + 9 a^{11} + 20 a^{10} + 18 a^{9} + 10 a^{8} + 19 a^{7} + 22 a^{6} + 20 a^{5} + 16 a^{3} + 16 a^{2} + 22\right)\cdot 23^{184} + \left(5 a^{14} + 9 a^{13} + 12 a^{12} + 2 a^{11} + 15 a^{9} + 5 a^{8} + 18 a^{7} + a^{5} + 17 a^{4} + 15 a^{3} + 5 a^{2} + 11 a\right)\cdot 23^{185} + \left(11 a^{14} + 16 a^{13} + 13 a^{12} + 20 a^{11} + 6 a^{10} + 19 a^{9} + 22 a^{8} + 11 a^{7} + 13 a^{6} + 6 a^{5} + 17 a^{4} + 20 a^{3} + 4 a^{2} + 10 a + 12\right)\cdot 23^{186} + \left(a^{14} + a^{13} + 20 a^{12} + a^{11} + 10 a^{10} + 12 a^{9} + 2 a^{8} + 19 a^{7} + 21 a^{6} + 19 a^{5} + a^{3} + 4 a^{2} + 12 a + 11\right)\cdot 23^{187} + \left(17 a^{14} + 3 a^{13} + 4 a^{12} + 15 a^{11} + 10 a^{10} + 16 a^{9} + 21 a^{8} + 15 a^{7} + 18 a^{6} + 8 a^{5} + 14 a^{4} + 9 a^{3} + 16 a^{2} + 2 a + 12\right)\cdot 23^{188} + \left(12 a^{14} + 14 a^{13} + 19 a^{12} + 4 a^{11} + 20 a^{10} + 21 a^{9} + 19 a^{8} + 15 a^{7} + 12 a^{6} + 15 a^{5} + 11 a^{4} + 5 a^{3} + 6 a^{2} + 9 a + 19\right)\cdot 23^{189} + \left(10 a^{14} + 15 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+ 14 a + 6\right)\cdot 23^{195} + \left(13 a^{14} + 6 a^{13} + 10 a^{12} + 6 a^{11} + 17 a^{10} + 9 a^{9} + 21 a^{8} + 20 a^{7} + 22 a^{6} + 4 a^{5} + 7 a^{4} + 2 a^{3} + 5 a^{2} + 11 a + 2\right)\cdot 23^{196} + \left(2 a^{14} + 11 a^{13} + 11 a^{12} + 16 a^{11} + 16 a^{10} + 17 a^{9} + 2 a^{8} + 19 a^{7} + a^{6} + a^{5} + 20 a^{4} + 11 a^{3} + 3 a^{2} + a + 7\right)\cdot 23^{197} + \left(3 a^{14} + 7 a^{13} + 19 a^{12} + a^{11} + 19 a^{10} + 18 a^{9} + 17 a^{8} + 16 a^{7} + 5 a^{6} + 5 a^{5} + 3 a^{4} + 20 a^{3} + 16 a^{2} + 3 a + 2\right)\cdot 23^{198} + \left(7 a^{14} + 3 a^{12} + 7 a^{11} + 14 a^{10} + 6 a^{9} + 20 a^{8} + 6 a^{7} + 7 a^{6} + 6 a^{5} + 19 a^{4} + 19 a^{3} + 12 a^{2} + 8 a + 17\right)\cdot 23^{199} + \left(11 a^{14} + 12 a^{13} + 12 a^{11} + 8 a^{10} + 14 a^{9} + 8 a^{8} + 4 a^{7} + 14 a^{6} + 18 a^{5} + 4 a^{4} + 12 a^{3} + 12 a^{2} + 6 a + 3\right)\cdot 23^{200} + \left(12 a^{14} + 8 a^{13} + 18 a^{12} + 6 a^{11} + 15 a^{10} + 13 a^{9} + 17 a^{8} + 12 a^{7} + 11 a^{6} + 10 a^{5} + 5 a^{4} + 14 a^{3} + 3 a^{2} + 6 a + 12\right)\cdot 23^{201} + \left(2 a^{14} + 19 a^{13} + 3 a^{11} + 20 a^{10} + 19 a^{9} + 12 a^{8} + 15 a^{7} + 13 a^{6} + 15 a^{5} + 11 a^{4} + 14 a^{3} + 17 a^{2} + 22 a + 7\right)\cdot 23^{202} + \left(18 a^{14} + 22 a^{13} + 15 a^{12} + 9 a^{11} + 2 a^{10} + 3 a^{9} + 10 a^{8} + 6 a^{7} + 22 a^{6} + 9 a^{5} + 18 a^{4} + 8 a^{3} + 7 a^{2} + 3 a + 15\right)\cdot 23^{203} + \left(5 a^{13} + 21 a^{12} + a^{11} + 16 a^{10} + 17 a^{9} + 10 a^{8} + 18 a^{7} + 21 a^{6} + 17 a^{5} + 11 a^{4} + 20 a^{3} + 21 a^{2} + 19 a + 15\right)\cdot 23^{204} + \left(3 a^{14} + 9 a^{13} + 11 a^{12} + 8 a^{11} + 22 a^{10} + 6 a^{9} + 4 a^{8} + 12 a^{7} + 13 a^{6} + 15 a^{5} + 8 a^{4} + 4 a^{3} + 9 a^{2} + 18 a + 15\right)\cdot 23^{205} + \left(21 a^{14} + 22 a^{12} + 5 a^{11} + 6 a^{10} + 2 a^{9} + 16 a^{8} + 4 a^{7} + 8 a^{5} + 7 a^{4} + 4 a^{2} + 4 a + 8\right)\cdot 23^{206} + \left(22 a^{14} + a^{13} + 16 a^{10} + 4 a^{9} + 22 a^{8} + 21 a^{7} + 21 a^{6} + 8 a^{5} + 8 a^{4} + 21 a^{3} + 4 a^{2} + 19 a + 1\right)\cdot 23^{207} + \left(5 a^{14} + 18 a^{13} + 12 a^{12} + 3 a^{11} + 15 a^{10} + 10 a^{9} + 17 a^{8} + 15 a^{7} + 17 a^{6} + 20 a^{5} + 20 a^{4} + 16 a^{3} + 9 a^{2} + 16 a + 20\right)\cdot 23^{208} + \left(9 a^{14} + 14 a^{13} + a^{12} + 14 a^{11} + 10 a^{10} + a^{9} + 5 a^{8} + 20 a^{7} + 18 a^{6} + 5 a^{5} + 15 a^{4} + 4 a^{3} + 7 a^{2} + 3 a + 14\right)\cdot 23^{209} + \left(17 a^{14} + 8 a^{13} + 7 a^{12} + 16 a^{11} + 20 a^{10} + a^{9} + 7 a^{8} + 8 a^{7} + 18 a^{4} + 10 a^{3} + 21 a^{2} + a + 12\right)\cdot 23^{210} + \left(19 a^{14} + a^{13} + 10 a^{12} + 4 a^{11} + 10 a^{10} + 6 a^{9} + 3 a^{8} + 9 a^{7} + 22 a^{6} + 5 a^{5} + 18 a^{4} + 4 a^{3} + 16 a^{2} + 12 a + 11\right)\cdot 23^{211} + \left(8 a^{14} + 4 a^{13} + 12 a^{12} + 5 a^{11} + 21 a^{10} + a^{9} + 13 a^{8} + 22 a^{7} + 12 a^{6} + 17 a^{5} + 4 a^{4} + 12 a^{3} + 21 a^{2} + 21 a\right)\cdot 23^{212} + \left(7 a^{14} + 20 a^{13} + 15 a^{12} + 18 a^{11} + 14 a^{10} + 16 a^{9} + 8 a^{8} + 20 a^{7} + 18 a^{6} + 15 a^{5} + 17 a^{4} + a^{3} + 4 a^{2} + 15 a + 11\right)\cdot 23^{213} + \left(12 a^{13} + 12 a^{12} + 16 a^{11} + 13 a^{10} + 19 a^{9} + 18 a^{8} + 21 a^{7} + 8 a^{6} + 15 a^{5} + 20 a^{4} + 13 a^{3} + a^{2} + 16 a + 14\right)\cdot 23^{214} + \left(16 a^{14} + 2 a^{13} + 18 a^{12} + 12 a^{11} + 20 a^{10} + a^{9} + 13 a^{8} + 21 a^{7} + 18 a^{6} + 10 a^{4} + 10 a^{3} + 8 a^{2} + 15 a + 8\right)\cdot 23^{215} + \left(5 a^{14} + 15 a^{13} + 4 a^{11} + 18 a^{10} + 11 a^{9} + a^{8} + 20 a^{7} + 5 a^{6} + 7 a^{5} + 16 a^{4} + 18 a^{3} + 18 a^{2} + 14 a + 4\right)\cdot 23^{216} + \left(5 a^{14} + 3 a^{13} + 20 a^{11} + 9 a^{10} + 7 a^{9} + 2 a^{8} + 5 a^{7} + 16 a^{6} + 22 a^{5} + 7 a^{4} + 22 a^{3} + 3 a^{2} + a + 11\right)\cdot 23^{217} + \left(2 a^{14} + 12 a^{13} + 8 a^{12} + 5 a^{11} + 8 a^{10} + 9 a^{8} + 9 a^{7} + 6 a^{6} + 7 a^{5} + 12 a^{4} + 12 a^{3} + 11 a^{2} + 19 a + 19\right)\cdot 23^{218} + \left(9 a^{14} + 5 a^{13} + 7 a^{12} + a^{11} + 2 a^{10} + 12 a^{9} + 15 a^{8} + 14 a^{7} + 6 a^{6} + 12 a^{5} + 16 a^{4} + 10 a^{3} + 20 a^{2} + 10 a + 18\right)\cdot 23^{219} + \left(12 a^{14} + 18 a^{13} + 3 a^{12} + 19 a^{11} + 11 a^{10} + 4 a^{8} + 15 a^{7} + 8 a^{6} + 12 a^{5} + 3 a^{4} + 5 a^{3} + 17 a^{2} + 14 a + 17\right)\cdot 23^{220} + \left(20 a^{14} + 2 a^{13} + 8 a^{12} + 10 a^{11} + 11 a^{10} + 12 a^{9} + 9 a^{8} + 16 a^{7} + 9 a^{6} + 8 a^{5} + 11 a^{4} + 22 a^{3} + 14 a + 20\right)\cdot 23^{221} + \left(6 a^{14} + 5 a^{13} + 21 a^{12} + 15 a^{11} + 8 a^{10} + 2 a^{9} + 4 a^{8} + a^{7} + 21 a^{6} + 17 a^{5} + 9 a^{4} + 8 a^{3} + 5 a^{2} + 6 a + 13\right)\cdot 23^{222} + \left(16 a^{14} + 10 a^{13} + 9 a^{12} + 3 a^{10} + 5 a^{9} + 13 a^{8} + 16 a^{7} + 16 a^{6} + 13 a^{5} + 6 a^{4} + 15 a^{3} + 14 a^{2} + 10\right)\cdot 23^{223} + \left(19 a^{14} + 7 a^{13} + 5 a^{12} + 2 a^{11} + 9 a^{10} + 15 a^{9} + 2 a^{8} + 17 a^{7} + 6 a^{6} + 13 a^{5} + 17 a^{4} + 20 a^{3} + 18 a^{2} + 20 a + 22\right)\cdot 23^{224} + \left(5 a^{14} + 15 a^{13} + 21 a^{12} + 14 a^{11} + 7 a^{10} + 4 a^{9} + 9 a^{8} + 5 a^{7} + 15 a^{6} + 18 a^{5} + 5 a^{4} + 5 a^{3} + 14 a^{2} + 20 a + 13\right)\cdot 23^{225} + \left(13 a^{14} + 4 a^{13} + 14 a^{12} + 22 a^{11} + 17 a^{10} + 6 a^{9} + 8 a^{8} + 3 a^{7} + 15 a^{6} + a^{5} + 13 a^{4} + 7 a^{3} + 19 a^{2} + 6 a + 16\right)\cdot 23^{226} + \left(14 a^{14} + 20 a^{13} + 12 a^{12} + 9 a^{11} + 11 a^{10} + 18 a^{9} + 6 a^{8} + a^{7} + 5 a^{6} + 9 a^{5} + 17 a^{4} + a^{3} + 21 a^{2} + 2 a + 20\right)\cdot 23^{227} + \left(4 a^{13} + 13 a^{12} + 7 a^{11} + 14 a^{10} + 4 a^{9} + 22 a^{8} + 22 a^{7} + 3 a^{6} + 10 a^{4} + 18 a^{3} + 19 a + 5\right)\cdot 23^{228} + \left(a^{14} + 6 a^{12} + 17 a^{11} + 5 a^{10} + 19 a^{9} + 2 a^{8} + 2 a^{7} + 14 a^{6} + 22 a^{5} + 5 a^{3} + 14 a^{2} + 19 a + 7\right)\cdot 23^{229} + \left(17 a^{14} + 9 a^{13} + 19 a^{12} + 13 a^{11} + 11 a^{10} + 6 a^{9} + 16 a^{8} + 20 a^{7} + 11 a^{6} + 12 a^{5} + 4 a^{4} + 19 a^{3} + 19 a^{2} + 13 a + 1\right)\cdot 23^{230} + \left(22 a^{14} + 16 a^{13} + 13 a^{12} + 7 a^{11} + 8 a^{10} + 19 a^{9} + 4 a^{8} + 14 a^{7} + 9 a^{6} + 14 a^{5} + 19 a^{4} + 6 a^{3} + 6\right)\cdot 23^{231} + \left(5 a^{14} + a^{13} + 5 a^{11} + 10 a^{10} + 4 a^{9} + 12 a^{8} + 4 a^{7} + 21 a^{6} + 4 a^{5} + 18 a^{4} + 11 a^{3} + 18 a^{2} + 21 a + 16\right)\cdot 23^{232} + \left(21 a^{14} + 5 a^{13} + 9 a^{12} + 19 a^{11} + 5 a^{10} + 18 a^{9} + 16 a^{8} + a^{7} + 4 a^{6} + 2 a^{4} + 20 a^{3} + 21 a^{2} + 15 a + 17\right)\cdot 23^{233} + \left(17 a^{14} + 8 a^{13} + 9 a^{12} + 13 a^{11} + 20 a^{10} + 18 a^{8} + 9 a^{7} + 21 a^{6} + 4 a^{5} + 4 a^{4} + 10 a^{3} + 11 a^{2} + 22 a + 14\right)\cdot 23^{234} + \left(7 a^{14} + 22 a^{13} + 3 a^{12} + 2 a^{11} + 19 a^{10} + 11 a^{9} + 8 a^{8} + 2 a^{7} + 15 a^{6} + 16 a^{5} + 4 a^{3} + 5 a^{2} + 14 a + 1\right)\cdot 23^{235} + \left(16 a^{14} + 5 a^{13} + 20 a^{12} + 10 a^{11} + 21 a^{10} + 11 a^{9} + 15 a^{8} + 6 a^{7} + 13 a^{6} + 6 a^{5} + 18 a^{4} + 14 a^{3} + 17 a^{2} + 9 a + 12\right)\cdot 23^{236} + \left(5 a^{14} + 8 a^{13} + a^{12} + 6 a^{11} + 18 a^{10} + 8 a^{9} + 19 a^{7} + 6 a^{6} + 2 a^{5} + 17 a^{4} + 14 a^{3} + 17 a^{2} + 18 a\right)\cdot 23^{237} + \left(19 a^{13} + 17 a^{12} + 9 a^{11} + 4 a^{10} + 9 a^{9} + 18 a^{8} + 14 a^{7} + 13 a^{6} + 7 a^{5} + 11 a^{4} + 17 a^{3} + 4 a^{2} + 16 a + 7\right)\cdot 23^{238} + \left(11 a^{13} + 11 a^{12} + 19 a^{11} + 21 a^{10} + 3 a^{9} + 17 a^{8} + 5 a^{7} + 6 a^{6} + 4 a^{5} + 15 a^{4} + 9 a^{3} + 7 a^{2} + 9 a + 8\right)\cdot 23^{239} + \left(16 a^{14} + 11 a^{13} + a^{12} + 10 a^{10} + 2 a^{9} + 20 a^{8} + 9 a^{7} + a^{6} + 10 a^{5} + 7 a^{4} + 6 a^{3} + 10 a^{2} + 11 a + 7\right)\cdot 23^{240} + \left(21 a^{14} + 21 a^{13} + 19 a^{12} + 5 a^{11} + 3 a^{10} + 11 a^{9} + 3 a^{8} + 3 a^{7} + 8 a^{6} + 9 a^{5} + 19 a^{4} + 13 a^{3} + 21 a^{2} + 16 a + 20\right)\cdot 23^{241} + \left(3 a^{14} + 22 a^{13} + 16 a^{12} + 3 a^{11} + 19 a^{10} + 13 a^{9} + 10 a^{8} + 3 a^{7} + 5 a^{6} + 2 a^{5} + 15 a^{4} + 6 a^{2} + 13 a + 6\right)\cdot 23^{242} + \left(15 a^{14} + 5 a^{13} + 5 a^{12} + 14 a^{11} + 19 a^{10} + 11 a^{9} + 4 a^{8} + 6 a^{7} + 19 a^{6} + 11 a^{5} + 7 a^{4} + 18 a^{3} + 16 a^{2} + 12 a + 9\right)\cdot 23^{243} + \left(15 a^{14} + 4 a^{13} + 6 a^{12} + 13 a^{11} + 15 a^{10} + 6 a^{9} + a^{8} + 2 a^{7} + 15 a^{6} + 21 a^{5} + 15 a^{4} + 9 a^{3} + 11 a^{2} + 12 a + 20\right)\cdot 23^{244} + \left(10 a^{14} + a^{13} + 14 a^{12} + 11 a^{11} + 11 a^{10} + 14 a^{9} + 8 a^{6} + 17 a^{5} + 8 a^{3} + 2 a^{2} + a + 22\right)\cdot 23^{245} + \left(2 a^{14} + 9 a^{12} + 3 a^{11} + 22 a^{9} + 10 a^{8} + 8 a^{7} + a^{6} + 4 a^{5} + 16 a^{4} + a^{3} + 10 a^{2} + 6 a + 15\right)\cdot 23^{246} + \left(17 a^{14} + 3 a^{13} + 17 a^{12} + 3 a^{11} + 5 a^{10} + 21 a^{9} + 10 a^{8} + 10 a^{7} + 2 a^{6} + 8 a^{5} + 22 a^{4} + 7 a^{3} + 20 a^{2} + 16 a\right)\cdot 23^{247} + \left(3 a^{14} + 2 a^{13} + 3 a^{12} + 2 a^{11} + 18 a^{10} + 11 a^{9} + 9 a^{8} + a^{7} + 3 a^{6} + 18 a^{5} + 16 a^{4} + 3 a^{3} + 5 a^{2} + 8 a + 17\right)\cdot 23^{248} + \left(5 a^{14} + 2 a^{13} + 22 a^{12} + 18 a^{11} + 6 a^{10} + 5 a^{9} + 10 a^{8} + 13 a^{6} + a^{5} + 8 a^{4} + 5 a^{3} + 3 a^{2} + 13 a + 6\right)\cdot 23^{249} + \left(8 a^{14} + 4 a^{13} + 11 a^{12} + 20 a^{11} + 14 a^{10} + 21 a^{9} + 9 a^{8} + 4 a^{7} + a^{6} + 16 a^{5} + 14 a^{4} + 3 a^{3} + 13 a^{2} + 15 a + 14\right)\cdot 23^{250} + \left(8 a^{14} + 3 a^{13} + 5 a^{12} + 5 a^{11} + 13 a^{10} + 15 a^{9} + 9 a^{8} + 22 a^{7} + 13 a^{6} + 15 a^{5} + 16 a^{4} + 15 a^{3} + 8 a^{2} + 6 a + 20\right)\cdot 23^{251} + \left(12 a^{14} + 13 a^{13} + 20 a^{12} + 21 a^{11} + a^{10} + 18 a^{9} + 13 a^{8} + 15 a^{7} + 2 a^{6} + 12 a^{5} + 7 a^{4} + 4 a^{3} + 14 a^{2} + a + 3\right)\cdot 23^{252} + \left(13 a^{14} + 18 a^{13} + 10 a^{12} + 5 a^{11} + 4 a^{10} + 20 a^{9} + 5 a^{8} + 16 a^{7} + 5 a^{6} + a^{5} + 21 a^{4} + 10 a^{3} + 5 a^{2} + 3 a + 6\right)\cdot 23^{253} + \left(16 a^{14} + 16 a^{13} + 22 a^{12} + 12 a^{11} + 17 a^{10} + 4 a^{9} + 2 a^{8} + 4 a^{7} + 9 a^{6} + 6 a^{5} + 12 a^{4} + 19 a^{3} + 6\right)\cdot 23^{254} + \left(9 a^{14} + 11 a^{13} + 18 a^{11} + 18 a^{10} + 10 a^{9} + 14 a^{7} + 6 a^{6} + 18 a^{5} + 4 a^{4} + 17 a^{3} + a^{2} + 17 a + 17\right)\cdot 23^{255} + \left(18 a^{14} + 8 a^{13} + 19 a^{12} + 8 a^{11} + 12 a^{10} + 7 a^{9} + 6 a^{8} + 13 a^{7} + 17 a^{6} + 2 a^{5} + 2 a^{4} + 11 a^{3} + 4 a^{2} + 3 a + 18\right)\cdot 23^{256} + \left(18 a^{13} + 6 a^{12} + a^{11} + 9 a^{10} + 11 a^{9} + a^{8} + 19 a^{7} + 22 a^{6} + 6 a^{5} + 22 a^{4} + 10 a^{3} + 22 a^{2} + 10 a + 2\right)\cdot 23^{257} + \left(a^{14} + 8 a^{13} + 16 a^{12} + 4 a^{11} + 2 a^{10} + 14 a^{9} + 19 a^{8} + 10 a^{7} + 7 a^{6} + 7 a^{5} + 12 a^{4} + 18 a^{3} + a^{2} + 17 a + 4\right)\cdot 23^{258} + \left(16 a^{14} + 6 a^{13} + 15 a^{12} + 17 a^{11} + 13 a^{10} + 20 a^{9} + 20 a^{8} + 5 a^{7} + 4 a^{6} + 2 a^{5} + 18 a^{4} + 20 a^{3} + 20 a^{2} + 10 a\right)\cdot 23^{259} + \left(6 a^{14} + 4 a^{13} + 6 a^{12} + 5 a^{11} + 22 a^{10} + 13 a^{9} + 15 a^{8} + 15 a^{6} + 16 a^{5} + 14 a^{4} + 3 a^{3} + 19 a^{2} + 2 a + 21\right)\cdot 23^{260} + \left(17 a^{14} + 6 a^{13} + 13 a^{12} + 7 a^{11} + 21 a^{10} + 11 a^{9} + 2 a^{8} + 12 a^{7} + 22 a^{6} + 19 a^{5} + 10 a^{4} + 19 a^{3} + 3 a^{2} + 10\right)\cdot 23^{261} + \left(5 a^{14} + 20 a^{13} + 22 a^{12} + 4 a^{11} + 3 a^{10} + a^{9} + 22 a^{8} + 21 a^{7} + 6 a^{6} + 9 a^{5} + 4 a^{4} + 5 a^{2} + 13 a + 11\right)\cdot 23^{262} + \left(9 a^{14} + 14 a^{13} + 13 a^{11} + 19 a^{10} + 7 a^{9} + 22 a^{8} + 19 a^{7} + 16 a^{6} + 18 a^{5} + 11 a^{4} + 6 a^{3} + 14 a^{2} + 21 a + 22\right)\cdot 23^{263} + \left(3 a^{14} + a^{12} + 9 a^{11} + 15 a^{10} + 11 a^{9} + 21 a^{8} + 17 a^{6} + 15 a^{5} + 6 a^{3} + 4 a^{2} + 13 a\right)\cdot 23^{264} + \left(5 a^{14} + 16 a^{13} + 5 a^{12} + 22 a^{11} + 14 a^{10} + 14 a^{9} + 16 a^{8} + 14 a^{7} + 7 a^{6} + 8 a^{5} + 15 a^{4} + 5 a^{3} + 5 a^{2} + 13 a + 6\right)\cdot 23^{265} + \left(21 a^{14} + 4 a^{13} + 20 a^{12} + 20 a^{11} + 6 a^{10} + 18 a^{9} + 17 a^{8} + 14 a^{7} + 2 a^{6} + 18 a^{5} + 4 a^{4} + 20 a^{3} + 18 a^{2} + 9 a + 15\right)\cdot 23^{266} + \left(a^{14} + 14 a^{13} + 4 a^{12} + 17 a^{11} + 17 a^{10} + a^{9} + 14 a^{8} + 2 a^{7} + 4 a^{6} + 6 a^{5} + 3 a^{2} + 13 a + 6\right)\cdot 23^{267} + \left(14 a^{14} + 16 a^{13} + 20 a^{12} + 2 a^{11} + 19 a^{10} + 5 a^{9} + 22 a^{8} + 22 a^{7} + 22 a^{6} + 20 a^{5} + 15 a^{4} + 14 a^{3} + 12 a^{2} + 14 a + 9\right)\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 7 }$ $=$ $17 a^{14} + a^{13} + 17 a^{12} + 8 a^{11} + 15 a^{10} + 18 a^{9} + 4 a^{8} + 21 a^{7} + 22 a^{6} + 19 a^{5} + 21 a^{4} + 5 a^{3} + 8 a^{2} + 17 a + 13 + \left(17 a^{14} + 18 a^{13} + 2 a^{12} + 16 a^{11} + 17 a^{10} + 10 a^{9} + 12 a^{8} + a^{7} + 18 a^{6} + 18 a^{4} + 13 a^{3} + 5 a^{2} + 15 a + 7\right)\cdot 23 + \left(12 a^{14} + 18 a^{13} + 19 a^{12} + 19 a^{11} + 2 a^{10} + 3 a^{9} + 13 a^{8} + 14 a^{7} + 7 a^{6} + a^{5} + 14 a^{4} + 16 a^{3} + 10 a^{2} + 10 a + 5\right)\cdot 23^{2} + \left(14 a^{14} + 14 a^{13} + 2 a^{12} + 19 a^{11} + 14 a^{10} + 21 a^{9} + 11 a^{8} + 6 a^{7} + 20 a^{6} + 14 a^{5} + 6 a^{4} + 2 a^{3} + 14 a^{2} + a + 3\right)\cdot 23^{3} + \left(3 a^{14} + 16 a^{13} + 11 a^{12} + 8 a^{9} + 19 a^{8} + 16 a^{7} + 6 a^{6} + 11 a^{5} + 3 a^{4} + 6 a^{3} + 14 a + 22\right)\cdot 23^{4} + \left(6 a^{14} + 5 a^{13} + 20 a^{12} + 6 a^{11} + 14 a^{9} + 6 a^{8} + 18 a^{7} + 2 a^{6} + 16 a^{5} + 15 a^{4} + 2 a^{3} + 2 a^{2} + 12 a + 22\right)\cdot 23^{5} + \left(15 a^{13} + 15 a^{11} + 7 a^{10} + 19 a^{9} + 18 a^{8} + 18 a^{6} + 17 a^{5} + 19 a^{4} + 21 a^{3} + 8 a^{2} + 3 a + 21\right)\cdot 23^{6} + \left(17 a^{14} + 16 a^{13} + 14 a^{12} + 16 a^{11} + 20 a^{10} + 15 a^{9} + 9 a^{8} + 16 a^{7} + 21 a^{6} + 9 a^{5} + 16 a^{4} + 21 a^{3} + 22 a^{2} + a + 22\right)\cdot 23^{7} + \left(14 a^{14} + 19 a^{13} + 17 a^{12} + a^{11} + 16 a^{10} + 8 a^{9} + 8 a^{8} + 19 a^{7} + 9 a^{6} + 14 a^{5} + 4 a^{4} + 6 a^{3} + 16 a^{2} + 4 a + 9\right)\cdot 23^{8} + \left(a^{14} + 12 a^{13} + 20 a^{12} + 12 a^{11} + 22 a^{10} + 14 a^{9} + 9 a^{8} + 18 a^{7} + 18 a^{6} + 16 a^{5} + 10 a^{4} + 6 a^{3} + 19 a^{2} + 3 a + 9\right)\cdot 23^{9} + \left(4 a^{13} + 10 a^{12} + 6 a^{11} + 14 a^{10} + 8 a^{9} + 14 a^{8} + 16 a^{7} + 15 a^{6} + 17 a^{5} + 8 a^{3} + 7 a^{2} + 4 a + 20\right)\cdot 23^{10} + \left(12 a^{14} + 14 a^{13} + 17 a^{12} + 5 a^{11} + 9 a^{10} + 10 a^{9} + 14 a^{8} + 22 a^{7} + 14 a^{6} + 20 a^{5} + 13 a^{4} + 21 a^{3} + 14 a^{2} + 7 a + 9\right)\cdot 23^{11} + \left(17 a^{14} + 8 a^{13} + 18 a^{12} + 17 a^{11} + 7 a^{10} + 3 a^{9} + 20 a^{8} + 20 a^{7} + 4 a^{6} + 14 a^{5} + a^{4} + 9 a^{3} + 4 a^{2} + 9 a + 1\right)\cdot 23^{12} + \left(a^{14} + a^{13} + 13 a^{12} + 5 a^{11} + 10 a^{10} + 20 a^{9} + 2 a^{7} + 15 a^{6} + 12 a^{5} + 10 a^{4} + 9 a^{3} + 16 a^{2} + 4 a + 8\right)\cdot 23^{13} + \left(4 a^{14} + 20 a^{13} + 19 a^{12} + 3 a^{11} + 6 a^{9} + 16 a^{8} + a^{7} + 20 a^{5} + 5 a^{3} + 8 a^{2} + 13 a 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a^{4} + 12 a^{3} + 8 a + 8\right)\cdot 23^{20} + \left(6 a^{14} + 19 a^{13} + 7 a^{12} + 9 a^{11} + 2 a^{10} + 9 a^{9} + 6 a^{8} + 20 a^{7} + 14 a^{6} + 14 a^{5} + 16 a^{4} + a^{3} + 11 a^{2} + 11 a + 4\right)\cdot 23^{21} + \left(a^{14} + 20 a^{13} + 11 a^{12} + a^{11} + 12 a^{10} + 19 a^{9} + 21 a^{8} + 3 a^{7} + a^{5} + 22 a^{4} + 13 a^{3} + a^{2} + 8 a + 8\right)\cdot 23^{22} + \left(6 a^{14} + 14 a^{13} + a^{12} + 17 a^{11} + 13 a^{10} + 21 a^{9} + 10 a^{8} + 16 a^{7} + 6 a^{6} + 11 a^{5} + 22 a^{4} + 22 a^{3} + 14 a^{2} + 9 a + 18\right)\cdot 23^{23} + \left(11 a^{14} + 8 a^{13} + 4 a^{12} + 21 a^{11} + 5 a^{10} + 7 a^{9} + 2 a^{8} + a^{7} + 4 a^{6} + 22 a^{5} + 11 a^{4} + 15 a^{3} + a^{2} + 17 a + 17\right)\cdot 23^{24} + \left(13 a^{14} + 22 a^{13} + 17 a^{12} + 8 a^{11} + 2 a^{10} + 7 a^{9} + 16 a^{8} + 2 a^{7} + 9 a^{6} + 4 a^{5} + 7 a^{4} + a^{3} + 8 a^{2} + 8 a + 1\right)\cdot 23^{25} + \left(5 a^{14} + 22 a^{13} + 20 a^{12} + 19 a^{11} + 14 a^{10} + 7 a^{9} + 12 a^{8} + a^{7} + 21 a^{6} + 3 a^{5} + 12 a^{4} + 20 a^{3} + 6 a^{2} + 15 a + 22\right)\cdot 23^{26} + \left(11 a^{14} + a^{13} + 19 a^{12} + 9 a^{11} + 15 a^{10} + 8 a^{9} + 7 a^{8} + 17 a^{7} + 11 a^{6} + 10 a^{5} + 13 a^{4} + 7 a^{3} + 10 a^{2} + 8 a + 5\right)\cdot 23^{27} + \left(20 a^{14} + 4 a^{13} + 19 a^{12} + 8 a^{11} + 8 a^{10} + 3 a^{9} + 22 a^{8} + 9 a^{7} + a^{6} + 10 a^{5} + 10 a^{4} + 20 a^{3} + 22 a^{2} + 20 a + 12\right)\cdot 23^{28} + \left(10 a^{14} + 8 a^{13} + 21 a^{12} + 17 a^{11} + 11 a^{10} + 9 a^{9} + 6 a^{8} + 9 a^{7} + 21 a^{6} + 15 a^{5} + 12 a^{4} + 7 a^{3} + 15 a^{2} + 4 a + 10\right)\cdot 23^{29} + \left(9 a^{14} + 12 a^{13} + 16 a^{12} + 7 a^{11} + 20 a^{10} + 3 a^{9} + 18 a^{8} + 13 a^{7} + 12 a^{6} + 5 a^{5} + 13 a^{4} + 20 a^{3} + 14 a^{2} + 8 a + 14\right)\cdot 23^{30} + \left(19 a^{14} + 19 a^{13} + 6 a^{12} + 21 a^{11} + 16 a^{10} + 15 a^{9} + 10 a^{8} + 18 a^{7} + 21 a^{6} + 2 a^{5} + 12 a^{3} + 10 a^{2} + 8 a + 21\right)\cdot 23^{31} + \left(18 a^{14} + 11 a^{13} + 4 a^{12} + 8 a^{11} + 2 a^{10} + 18 a^{9} + 11 a^{8} + 10 a^{7} + 9 a^{6} + 14 a^{5} + 13 a^{4} + 4 a^{3} + 2 a^{2} + 16 a + 11\right)\cdot 23^{32} + \left(2 a^{14} + 13 a^{13} + 5 a^{12} + 19 a^{11} + 9 a^{10} + 8 a^{9} + 19 a^{8} + 12 a^{7} + 17 a^{6} + 3 a^{5} + 15 a^{4} + 12 a^{3} + 7 a^{2} + 11 a + 6\right)\cdot 23^{33} + \left(17 a^{14} + 9 a^{13} + 19 a^{12} + 7 a^{11} + 10 a^{10} + 20 a^{9} + 8 a^{8} + 11 a^{7} + 15 a^{6} + 12 a^{5} + 10 a^{4} + 15 a^{3} + 9 a^{2} + 13 a + 2\right)\cdot 23^{34} + \left(3 a^{14} + 15 a^{13} + 4 a^{12} + 14 a^{11} + 2 a^{10} + 16 a^{9} + 22 a^{8} + 19 a^{7} + 2 a^{6} + 19 a^{5} + 6 a^{4} + 7 a^{3} + 16 a^{2} + 4 a + 5\right)\cdot 23^{35} + \left(14 a^{14} + 17 a^{13} + 10 a^{12} + 4 a^{11} + 17 a^{10} + 5 a^{8} + 7 a^{7} + 12 a^{6} + 3 a^{5} + 15 a^{4} + 16 a^{3} + 5 a^{2} + 6 a + 18\right)\cdot 23^{36} + \left(19 a^{14} + 15 a^{13} + 21 a^{12} + 6 a^{11} + 8 a^{10} + 15 a^{9} + 19 a^{8} + 22 a^{7} + 17 a^{6} + 2 a^{5} + 11 a^{4} + 14 a^{3} + 10 a^{2} + 5 a + 2\right)\cdot 23^{37} + \left(5 a^{14} + 10 a^{13} + a^{12} + 3 a^{11} + 14 a^{10} + 5 a^{9} + a^{8} + 4 a^{7} + 7 a^{6} + 14 a^{5} + 13 a^{4} + 4 a^{3} + 20 a^{2} + 20 a + 4\right)\cdot 23^{38} + \left(16 a^{14} + 21 a^{13} + 18 a^{12} + 9 a^{11} + 3 a^{10} + 18 a^{9} + 16 a^{8} + 6 a^{7} + 22 a^{5} + 13 a^{4} + 21 a^{3} + 11 a^{2} + 7 a + 9\right)\cdot 23^{39} + \left(14 a^{14} + 6 a^{13} + 19 a^{12} + 9 a^{11} + 18 a^{10} + 18 a^{9} + 7 a^{8} + 15 a^{7} + 6 a^{6} + 3 a^{5} + 18 a^{4} + 2 a^{3} + 5 a^{2} + 18 a + 7\right)\cdot 23^{40} + \left(4 a^{14} + 22 a^{13} + 6 a^{12} + 2 a^{11} + 11 a^{9} + 16 a^{8} + 6 a^{7} + 15 a^{6} + 4 a^{5} + 16 a^{4} + 15 a^{3} + 8 a^{2} + 10 a + 4\right)\cdot 23^{41} + \left(21 a^{14} + 19 a^{13} + 10 a^{12} + 10 a^{11} + 4 a^{10} + 21 a^{9} + 5 a^{8} + 10 a^{7} + 8 a^{6} + 2 a^{5} + 6 a^{4} + 8 a^{3} + 9 a^{2} + 15 a + 4\right)\cdot 23^{42} + \left(17 a^{14} + 3 a^{13} + a^{11} + 17 a^{10} + 20 a^{9} + 15 a^{8} + 2 a^{7} + 9 a^{5} + 17 a^{4} + 14 a^{3} + 22 a^{2} + 10 a + 7\right)\cdot 23^{43} + \left(5 a^{14} + 21 a^{13} + 8 a^{12} + 16 a^{11} + 10 a^{10} + 8 a^{9} + 5 a^{8} + 4 a^{7} + 7 a^{6} + 8 a^{5} + 10 a^{4} + 15 a^{2} + 6 a + 4\right)\cdot 23^{44} + \left(21 a^{14} + 5 a^{13} + 5 a^{12} + 7 a^{11} + 16 a^{10} + 3 a^{9} + 2 a^{8} + 13 a^{7} + 11 a^{6} + 14 a^{5} + 9 a^{4} + 17 a^{3} + 20 a^{2} + 18 a + 5\right)\cdot 23^{45} + \left(20 a^{14} + 18 a^{13} + 20 a^{12} + 9 a^{11} + 4 a^{10} + 19 a^{9} + 12 a^{8} + 13 a^{7} + 15 a^{6} + 15 a^{5} + 16 a^{4} + 20 a^{3} + 21 a + 21\right)\cdot 23^{46} + \left(a^{14} + 21 a^{13} + 16 a^{12} + 17 a^{11} + 2 a^{10} + 15 a^{9} + 5 a^{8} + 17 a^{7} + 8 a^{6} + 22 a^{5} + 2 a^{4} + a^{3} + 13 a^{2} + 6 a + 1\right)\cdot 23^{47} + \left(a^{14} + 14 a^{13} + 20 a^{12} + 12 a^{11} + 8 a^{10} + 18 a^{9} + 5 a^{8} + 18 a^{7} + 21 a^{6} + 3 a^{5} + 7 a^{3} + 14 a + 9\right)\cdot 23^{48} + \left(2 a^{14} + 4 a^{13} + 3 a^{12} + 16 a^{10} + 15 a^{9} + 17 a^{8} + 17 a^{7} + 14 a^{6} + a^{5} + 19 a^{3} + 9 a^{2} + 16 a + 9\right)\cdot 23^{49} + \left(6 a^{14} + 18 a^{13} + 20 a^{12} + a^{11} + 18 a^{10} + 16 a^{9} + 7 a^{8} + 7 a^{7} + 18 a^{6} + 14 a^{5} + 18 a^{4} + 19 a^{3} + 22 a^{2} + 20 a + 11\right)\cdot 23^{50} + \left(11 a^{14} + 14 a^{13} + 7 a^{12} + 13 a^{11} + 7 a^{10} + a^{9} + 12 a^{8} + 8 a^{7} + 17 a^{6} + 5 a^{5} + 13 a^{4} + 9 a^{3} + 5 a^{2} + 21 a + 19\right)\cdot 23^{51} + \left(21 a^{14} + 19 a^{13} + 9 a^{12} + 21 a^{11} + 18 a^{10} + 11 a^{9} + 6 a^{8} + 3 a^{7} + a^{6} + 7 a^{5} + 13 a^{4} + 7 a^{3} + 4 a^{2} + 20 a + 16\right)\cdot 23^{52} + \left(22 a^{14} + 21 a^{13} + 14 a^{12} + 8 a^{11} + 9 a^{10} + 12 a^{9} + 4 a^{8} + a^{7} + 6 a^{6} + 7 a^{5} + 17 a^{4} + 15 a^{3} + 18 a^{2} + 21 a + 6\right)\cdot 23^{53} + \left(11 a^{14} + 19 a^{13} + 6 a^{12} + 5 a^{11} + 9 a^{9} + 11 a^{7} + 14 a^{6} + 20 a^{4} + 22 a^{3} + 4 a^{2} + 19 a + 21\right)\cdot 23^{54} + \left(17 a^{14} + 22 a^{13} + 16 a^{12} + 8 a^{11} + 3 a^{10} + 21 a^{9} + 5 a^{8} + 22 a^{7} + 19 a^{6} + 7 a^{5} + 5 a^{4} + 6 a^{3} + a^{2} + 19 a\right)\cdot 23^{55} + \left(4 a^{14} + 11 a^{13} + a^{12} + 11 a^{11} + 10 a^{10} + 17 a^{9} + 21 a^{8} + 10 a^{7} + 20 a^{6} + 7 a^{5} + 15 a^{4} + 19 a^{3} + 18 a^{2} + 22 a + 9\right)\cdot 23^{56} + \left(12 a^{14} + 22 a^{13} + 11 a^{12} + 3 a^{11} + 19 a^{10} + 7 a^{9} + 15 a^{8} + a^{6} + 9 a^{5} + 18 a^{4} + a^{3} + 9 a^{2} + 11 a\right)\cdot 23^{57} + \left(5 a^{14} + 11 a^{13} + 15 a^{12} + 4 a^{11} + 11 a^{10} + 5 a^{9} + 3 a^{8} + 12 a^{7} + 5 a^{4} + 8 a^{3} + 6 a^{2} + 19 a + 8\right)\cdot 23^{58} + \left(12 a^{14} + 9 a^{13} + 17 a^{12} + 3 a^{11} + 17 a^{10} + 16 a^{9} + 20 a^{8} + 4 a^{7} + a^{6} + 11 a^{5} + 13 a^{4} + 15 a^{3} + 8 a^{2} + 13 a + 5\right)\cdot 23^{59} + \left(20 a^{14} + 20 a^{13} + 9 a^{12} + 15 a^{11} + 15 a^{10} + 10 a^{9} + 18 a^{8} + a^{7} + 11 a^{6} + 15 a^{5} + 12 a^{4} + 9 a^{3} + 16 a^{2} + 18 a + 20\right)\cdot 23^{60} + \left(6 a^{13} + 15 a^{12} + 21 a^{11} + 12 a^{9} + 4 a^{7} + 14 a^{6} + 17 a^{5} + 3 a^{4} + 4 a^{3} + 20 a^{2} + 20 a + 9\right)\cdot 23^{61} + \left(5 a^{14} + 22 a^{13} + 7 a^{12} + 18 a^{11} + 5 a^{10} + 12 a^{9} + 12 a^{8} + 19 a^{7} + 2 a^{6} + 6 a^{5} + 17 a^{4} + 9 a^{3} + 15 a^{2} + 2 a + 11\right)\cdot 23^{62} + \left(17 a^{14} + 16 a^{13} + 9 a^{12} + a^{11} + 8 a^{10} + 9 a^{8} + 9 a^{7} + 2 a^{6} + 8 a^{5} + 5 a^{4} + 19 a^{3} + 7 a^{2} + 6 a + 22\right)\cdot 23^{63} + \left(15 a^{14} + 16 a^{13} + 2 a^{12} + 15 a^{11} + 18 a^{10} + a^{9} + 13 a^{8} + 2 a^{7} + 5 a^{6} + 17 a^{5} + 11 a^{4} + 3 a^{3} + 12 a^{2} + 21 a + 1\right)\cdot 23^{64} + \left(22 a^{14} + 15 a^{13} + 17 a^{12} + 15 a^{11} + 15 a^{10} + 17 a^{9} + 12 a^{7} + 12 a^{6} + 14 a^{5} + 12 a^{4} + 9 a^{3} + 4 a^{2} + 22 a + 6\right)\cdot 23^{65} + \left(9 a^{14} + 12 a^{13} + 11 a^{12} + 8 a^{11} + 21 a^{10} + a^{9} + 10 a^{8} + 5 a^{7} + 20 a^{6} + 16 a^{5} + 7 a^{3} + 4 a^{2} + 15 a + 2\right)\cdot 23^{66} + \left(15 a^{14} + 3 a^{13} + 3 a^{12} + 6 a^{11} + 2 a^{9} + 9 a^{8} + 5 a^{7} + 15 a^{6} + 9 a^{5} + 12 a^{4} + 22 a^{3} + 2 a^{2} + a + 20\right)\cdot 23^{67} + \left(11 a^{14} + 11 a^{13} + 22 a^{12} + 20 a^{11} + 17 a^{10} + 17 a^{9} + 12 a^{8} + 5 a^{7} + 13 a^{6} + 20 a^{5} + 20 a^{4} + 5 a^{3} + 6 a^{2} + 12 a + 19\right)\cdot 23^{68} + \left(5 a^{14} + a^{13} + 21 a^{11} + 5 a^{10} + 7 a^{8} + 21 a^{7} + 16 a^{6} + 12 a^{5} + 18 a^{4} + 14 a^{2} + 19 a + 10\right)\cdot 23^{69} + \left(12 a^{14} + 9 a^{13} + a^{12} + 3 a^{11} + 14 a^{10} + 14 a^{9} + 5 a^{8} + 19 a^{7} + 2 a^{6} + a^{5} + 19 a^{4} + 19 a^{3} + 14 a^{2} + 22 a + 20\right)\cdot 23^{70} + \left(3 a^{14} + 22 a^{13} + 6 a^{12} + 3 a^{11} + 12 a^{10} + 2 a^{9} + 9 a^{8} + 8 a^{7} + a^{6} + 20 a^{5} + 4 a^{4} + 15 a^{3} + 10 a\right)\cdot 23^{71} + \left(5 a^{14} + a^{13} + 8 a^{12} + 14 a^{11} + 4 a^{10} + 8 a^{9} + 7 a^{8} + 2 a^{7} + 2 a^{5} + 19 a^{4} + 13 a^{3} + 14 a^{2} + 13 a + 6\right)\cdot 23^{72} + \left(11 a^{14} + 16 a^{13} + 18 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23^{78} + \left(8 a^{14} + 5 a^{13} + 16 a^{12} + 2 a^{11} + 2 a^{10} + 18 a^{9} + 11 a^{8} + 3 a^{7} + 17 a^{6} + 19 a^{5} + 7 a^{4} + 10 a^{3} + 20 a^{2} + 22 a + 17\right)\cdot 23^{79} + \left(18 a^{13} + 4 a^{12} + 18 a^{11} + 9 a^{10} + 11 a^{9} + 8 a^{8} + 9 a^{7} + 2 a^{6} + 9 a^{5} + 4 a^{4} + a^{3} + 12 a^{2} + 18 a + 1\right)\cdot 23^{80} + \left(11 a^{14} + a^{13} + 21 a^{12} + 10 a^{11} + 17 a^{10} + 21 a^{9} + 22 a^{8} + 14 a^{7} + 20 a^{6} + 15 a^{5} + 9 a^{4} + 8 a^{3} + a^{2} + 12 a + 15\right)\cdot 23^{81} + \left(13 a^{14} + 20 a^{13} + 19 a^{12} + 3 a^{11} + 17 a^{10} + 22 a^{9} + 11 a^{8} + 12 a^{7} + 20 a^{6} + 15 a^{5} + 18 a^{4} + 7 a^{3} + 11 a^{2} + 3 a + 3\right)\cdot 23^{82} + \left(9 a^{14} + 6 a^{13} + 9 a^{12} + 2 a^{11} + 15 a^{10} + 10 a^{9} + 16 a^{8} + 8 a^{7} + 13 a^{6} + 10 a^{5} + 8 a^{4} + 22 a^{3} + 11 a^{2} + 5 a\right)\cdot 23^{83} + \left(6 a^{14} + 2 a^{13} + 20 a^{12} + 13 a^{11} + 14 a^{10} + 9 a^{9} + 21 a^{8} + a^{7} + 20 a^{6} + 2 a^{5} + 8 a^{4} + 13 a^{2} + 19 a + 1\right)\cdot 23^{84} + \left(11 a^{14} + 8 a^{13} + 22 a^{12} + 17 a^{11} + 19 a^{10} + 7 a^{9} + 13 a^{8} + 11 a^{7} + 11 a^{6} + 6 a^{5} + 2 a^{4} + a^{3} + 2 a^{2} + 22 a\right)\cdot 23^{85} + \left(18 a^{14} + 18 a^{13} + a^{12} + 18 a^{11} + 22 a^{9} + 13 a^{8} + 13 a^{7} + 17 a^{6} + 15 a^{5} + 9 a^{4} + 11 a^{3} + 14 a^{2} + a + 15\right)\cdot 23^{86} + \left(8 a^{14} + 11 a^{13} + 8 a^{12} + 17 a^{11} + 2 a^{10} + 22 a^{9} + 21 a^{8} + 13 a^{5} + 2 a^{4} + 12 a^{2} + 20 a + 22\right)\cdot 23^{87} + \left(14 a^{14} + a^{13} + 5 a^{11} + 12 a^{10} + 20 a^{9} + 22 a^{8} + a^{7} + 14 a^{6} + a^{5} + 5 a^{4} + 17 a^{3} + 8 a^{2} + 9 a + 6\right)\cdot 23^{88} + \left(3 a^{14} + 11 a^{13} + 9 a^{12} + 6 a^{11} + 11 a^{10} + 20 a^{9} + a^{8} + 8 a^{7} + 13 a^{6} + 2 a^{5} + 22 a^{3} + 3 a^{2} + 21 a + 11\right)\cdot 23^{89} + \left(20 a^{14} + 5 a^{12} + 21 a^{10} + 11 a^{9} + 6 a^{8} + 10 a^{7} + 9 a^{6} + 11 a^{5} + 17 a^{4} + 17 a^{3} + 3 a^{2} + 5 a + 16\right)\cdot 23^{90} + \left(12 a^{14} + 8 a^{13} + 14 a^{12} + 4 a^{11} + 2 a^{10} + 12 a^{9} + 8 a^{8} + 18 a^{7} + 16 a^{6} + 4 a^{5} + 10 a^{4} + 8 a^{3} + 14 a^{2} + 18\right)\cdot 23^{91} + \left(5 a^{14} + 5 a^{13} + 3 a^{12} + 12 a^{11} + 10 a^{10} + 18 a^{9} + a^{8} + 13 a^{7} + 18 a^{6} + 6 a^{5} + 19 a^{4} + 19 a^{3} + 11 a^{2} + 3 a + 6\right)\cdot 23^{92} + \left(10 a^{13} + 14 a^{12} + 13 a^{10} + 16 a^{9} + 2 a^{8} + 6 a^{7} + 6 a^{6} + 22 a^{5} + 6 a^{4} + 13 a^{3} + 4 a^{2} + 10 a + 20\right)\cdot 23^{93} + \left(18 a^{14} + 8 a^{13} + 7 a^{12} + 10 a^{11} + 15 a^{10} + 15 a^{9} + 15 a^{8} + 6 a^{7} + 10 a^{6} + 11 a^{5} + 9 a^{4} + 14 a^{3} + 11 a^{2} + 21 a + 17\right)\cdot 23^{94} + \left(a^{14} + 3 a^{13} + 6 a^{12} + 19 a^{11} + 15 a^{10} + 8 a^{9} + 17 a^{8} + 21 a^{7} + 21 a^{6} + 8 a^{5} + 10 a^{4} + 11 a^{3} + 22 a^{2} + 13 a + 18\right)\cdot 23^{95} + \left(15 a^{14} + 6 a^{13} + 9 a^{12} + 2 a^{11} + 12 a^{10} + 14 a^{9} + 4 a^{8} + 13 a^{7} + 20 a^{6} + 17 a^{5} + 22 a^{4} + 14 a^{3} + 7 a^{2} + 13 a + 6\right)\cdot 23^{96} + \left(4 a^{14} + 10 a^{13} + 3 a^{12} + 10 a^{10} + 4 a^{9} + 6 a^{8} + 12 a^{7} + 19 a^{6} + 16 a^{5} + 7 a^{4} + 3 a^{3} + 2 a^{2} + 18 a + 6\right)\cdot 23^{97} + \left(19 a^{14} + 20 a^{13} + 15 a^{12} + 19 a^{11} + 14 a^{10} + 11 a^{9} + 5 a^{8} + 3 a^{7} + 22 a^{6} + a^{5} + 7 a^{4} + 2 a^{3} + 14 a^{2} + 2 a + 19\right)\cdot 23^{98} + \left(14 a^{14} + 8 a^{13} + 18 a^{12} + 14 a^{11} + 21 a^{10} + 8 a^{9} + 9 a^{8} + 7 a^{7} + 5 a^{6} + 12 a^{5} + 13 a^{4} + 11 a^{3} + 14 a + 21\right)\cdot 23^{99} + \left(16 a^{13} + 7 a^{12} + 14 a^{11} + 8 a^{10} + 2 a^{9} + 2 a^{8} + 11 a^{7} + 15 a^{6} + 7 a^{5} + 8 a^{4} + 20 a^{3} + 4 a^{2} + 20 a + 14\right)\cdot 23^{100} + \left(15 a^{14} + 7 a^{13} + 10 a^{12} + 21 a^{11} + 11 a^{10} + 16 a^{9} + a^{8} + 15 a^{7} + 3 a^{5} + 16 a^{4} + 12 a^{3} + 19 a + 10\right)\cdot 23^{101} + \left(14 a^{14} + 5 a^{13} + 20 a^{12} + 10 a^{11} + 13 a^{10} + 21 a^{9} + 5 a^{8} + 22 a^{7} + 6 a^{6} + 2 a^{5} + 6 a^{4} + 17 a^{3} + 9 a + 18\right)\cdot 23^{102} + \left(15 a^{14} + 16 a^{13} + 8 a^{12} + 10 a^{11} + 4 a^{10} + 19 a^{9} + 8 a^{8} + 4 a^{7} + 2 a^{6} + a^{5} + 19 a^{4} + 7 a^{3} + 16 a^{2} + 8 a + 17\right)\cdot 23^{103} + \left(2 a^{14} + 7 a^{13} + 16 a^{10} + 9 a^{9} + 22 a^{8} + 2 a^{7} + 8 a^{6} + 13 a^{5} + 7 a^{4} + 20 a^{3} + 10 a^{2} + 5 a + 15\right)\cdot 23^{104} + \left(18 a^{14} + 10 a^{13} + 3 a^{12} + 2 a^{11} + 19 a^{9} + 19 a^{8} + 5 a^{7} + 9 a^{6} + 19 a^{5} + a^{4} + 6 a^{3} + a + 22\right)\cdot 23^{105} + \left(15 a^{14} + 18 a^{13} + 15 a^{12} + 5 a^{11} + 19 a^{10} + 20 a^{9} + 18 a^{8} + a^{7} + 18 a^{6} + 7 a^{5} + 6 a^{4} + 13 a^{3} + 8 a^{2} + 13 a + 10\right)\cdot 23^{106} + \left(22 a^{14} + 4 a^{13} + 5 a^{12} + 20 a^{11} + 13 a^{10} + 11 a^{9} + 13 a^{8} + 3 a^{6} + 18 a^{5} + a^{4} + 2 a^{3} + 4 a^{2} + 21 a + 7\right)\cdot 23^{107} + \left(12 a^{14} + 13 a^{13} + 18 a^{12} + 19 a^{11} + 14 a^{10} + 18 a^{9} + 17 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3\right)\cdot 23^{125} + \left(20 a^{14} + 15 a^{13} + 16 a^{12} + a^{11} + 11 a^{10} + 6 a^{9} + 3 a^{8} + 4 a^{7} + 7 a^{6} + 11 a^{5} + 11 a^{4} + 8 a^{3} + 9 a^{2} + a + 15\right)\cdot 23^{126} + \left(5 a^{14} + 4 a^{13} + 18 a^{12} + 9 a^{11} + 22 a^{10} + 7 a^{9} + 12 a^{8} + 6 a^{7} + 22 a^{6} + 11 a^{5} + 17 a^{4} + 15 a^{3} + 8 a^{2} + 2 a + 5\right)\cdot 23^{127} + \left(10 a^{14} + 16 a^{13} + 13 a^{12} + 16 a^{11} + 7 a^{10} + a^{9} + 19 a^{8} + 9 a^{6} + 6 a^{5} + 11 a^{3} + 2 a^{2} + 2 a + 22\right)\cdot 23^{128} + \left(6 a^{14} + 16 a^{13} + 8 a^{12} + 19 a^{11} + 15 a^{10} + a^{9} + 20 a^{8} + 22 a^{7} + 12 a^{6} + 16 a^{5} + 8 a^{4} + 4 a^{3} + 13 a^{2} + 5 a + 2\right)\cdot 23^{129} + \left(11 a^{14} + 4 a^{13} + 18 a^{11} + 6 a^{10} + 8 a^{9} + 7 a^{8} + 22 a^{6} + 20 a^{5} + 18 a^{4} + 20 a^{3} + 2 a^{2} + 18 a + 4\right)\cdot 23^{130} + \left(4 a^{14} + 14 a^{12} + 15 a^{11} + 4 a^{10} + 16 a^{9} + 6 a^{8} + 19 a^{7} + 11 a^{6} + 8 a^{5} + 4 a^{4} + 20 a^{3} + 7 a^{2} + a\right)\cdot 23^{131} + \left(10 a^{14} + 8 a^{13} + 4 a^{12} + 9 a^{11} + 15 a^{10} + 19 a^{9} + 21 a^{8} + 5 a^{7} + 14 a^{6} + 18 a^{5} + 15 a^{4} + 13 a^{3} + 5 a^{2} + 2 a + 12\right)\cdot 23^{132} + \left(20 a^{14} + 4 a^{13} + 8 a^{12} + 22 a^{11} + 7 a^{10} + 18 a^{9} + 7 a^{8} + 11 a^{7} + 16 a^{6} + 22 a^{5} + 22 a^{4} + 16 a^{3} + 16 a^{2} + 19 a\right)\cdot 23^{133} + \left(20 a^{13} + 12 a^{12} + 14 a^{11} + 11 a^{10} + 20 a^{9} + 10 a^{8} + 17 a^{7} + 5 a^{6} + 14 a^{5} + 2 a^{4} + 9 a^{3} + 2 a^{2} + 20 a + 13\right)\cdot 23^{134} + \left(19 a^{14} + 16 a^{13} + 20 a^{12} + 16 a^{11} + 11 a^{10} + 19 a^{8} + 2 a^{7} + 18 a^{6} + 3 a^{5} + 22 a^{4} + 2 a^{3} + 7 a^{2} + 14 a + 21\right)\cdot 23^{135} + \left(22 a^{14} + 4 a^{13} + a^{12} + 2 a^{11} + 2 a^{10} + 21 a^{9} + 22 a^{8} + 14 a^{7} + 2 a^{6} + 4 a^{5} + a^{4} + 12 a^{3} + 11 a^{2} + 2 a + 2\right)\cdot 23^{136} + \left(12 a^{14} + 8 a^{13} + 13 a^{12} + 5 a^{11} + 19 a^{10} + 19 a^{9} + 15 a^{8} + 6 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10\right)\cdot 23^{148} + \left(16 a^{14} + 8 a^{13} + 5 a^{12} + 8 a^{11} + 10 a^{10} + 6 a^{9} + 19 a^{8} + 16 a^{7} + 19 a^{6} + 15 a^{5} + 18 a^{4} + 2 a^{3} + 18 a^{2} + 8 a + 13\right)\cdot 23^{149} + \left(4 a^{14} + 21 a^{13} + 2 a^{12} + 15 a^{11} + a^{9} + 6 a^{8} + 17 a^{7} + 17 a^{6} + 22 a^{5} + 18 a^{4} + 10 a^{3} + 13 a^{2} + 19 a + 6\right)\cdot 23^{150} + \left(22 a^{14} + 21 a^{13} + 15 a^{12} + 3 a^{11} + 5 a^{10} + a^{9} + 8 a^{8} + 7 a^{7} + 14 a^{6} + 7 a^{5} + 5 a^{4} + a^{3} + 14 a^{2} + 8 a + 10\right)\cdot 23^{151} + \left(15 a^{14} + 7 a^{12} + 12 a^{9} + 16 a^{8} + 16 a^{7} + 2 a^{6} + 11 a^{5} + 20 a^{4} + 4 a^{3} + 17 a^{2} + 9 a + 17\right)\cdot 23^{152} + \left(20 a^{14} + 5 a^{13} + 20 a^{12} + 2 a^{11} + 3 a^{10} + 15 a^{9} + 11 a^{8} + 15 a^{7} + 20 a^{6} + 18 a^{5} + 8 a^{4} + 16 a^{3} + 9 a^{2} + 12 a + 22\right)\cdot 23^{153} + \left(12 a^{14} + 10 a^{13} + 9 a^{12} + 3 a^{11} + 10 a^{10} + 9 a^{9} + 2 a^{8} + 9 a^{7} + 5 a^{6} + 7 a^{5} + 5 a^{4} + 11 a^{3} + 20 a^{2} + 19 a + 8\right)\cdot 23^{154} + \left(22 a^{14} + 11 a^{13} + 14 a^{12} + 4 a^{11} + 11 a^{10} + 4 a^{9} + 22 a^{8} + 11 a^{7} + 15 a^{6} + 3 a^{5} + a^{4} + 19 a^{3} + 6 a^{2} + 7 a + 18\right)\cdot 23^{155} + \left(22 a^{14} + 14 a^{13} + 6 a^{12} + 17 a^{11} + 10 a^{10} + 20 a^{9} + 16 a^{8} + 4 a^{7} + a^{6} + 17 a^{4} + 11 a^{3} + 4 a^{2} + 3 a + 22\right)\cdot 23^{156} + \left(6 a^{14} + 22 a^{13} + 13 a^{12} + 17 a^{11} + 3 a^{10} + 15 a^{9} + 9 a^{8} + 18 a^{7} + 5 a^{6} + 5 a^{5} + 21 a^{4} + 10 a^{3} + 12 a^{2} + 2 a + 1\right)\cdot 23^{157} + \left(15 a^{14} + 19 a^{12} + 12 a^{11} + 6 a^{10} + 15 a^{9} + 13 a^{8} + 10 a^{6} + 17 a^{5} + 15 a^{4} + 2 a^{3} + 2 a^{2} + 13 a + 7\right)\cdot 23^{158} + \left(17 a^{14} + a^{13} + 20 a^{12} + 18 a^{11} + 12 a^{10} + 17 a^{9} + 3 a^{8} + 9 a^{7} + 18 a^{6} + 12 a^{5} + 19 a^{4} + 2 a^{3} + a^{2} + 19 a + 10\right)\cdot 23^{159} + \left(7 a^{14} + 2 a^{13} + 19 a^{12} + 15 a^{11} + 9 a^{10} + 4 a^{9} + 21 a^{8} + 13 a^{7} + 4 a^{6} + 17 a^{5} + 10 a^{4} + 6 a^{3} + 16 a^{2} + 3 a + 22\right)\cdot 23^{160} + \left(18 a^{14} + a^{13} + 18 a^{12} + 15 a^{11} + 2 a^{10} + 13 a^{9} + 10 a^{8} + 14 a^{7} + 2 a^{6} + 6 a^{5} + 8 a^{4} + 3 a^{3} + 3 a^{2} + 14 a + 9\right)\cdot 23^{161} + \left(11 a^{14} + 21 a^{12} + 14 a^{11} + 19 a^{10} + 5 a^{9} + 13 a^{8} + 3 a^{7} + 22 a^{6} + 13 a^{5} + 2 a^{3} + 4 a^{2} + 20 a + 1\right)\cdot 23^{162} + \left(19 a^{14} + 18 a^{13} + 17 a^{12} + 3 a^{11} + 13 a^{10} + 3 a^{9} + 9 a^{8} + 18 a^{7} + 19 a^{6} + 16 a^{5} + 12 a^{4} + 2 a^{3} + 21 a^{2} + 22 a + 8\right)\cdot 23^{163} + \left(17 a^{14} + a^{13} + 6 a^{12} + 19 a^{11} + 19 a^{10} + 22 a^{9} + 22 a^{8} + 15 a^{7} + 8 a^{5} + 18 a^{4} + 2 a^{3} + 9 a^{2} + 22 a + 4\right)\cdot 23^{164} + \left(11 a^{14} + 19 a^{13} + a^{12} + 20 a^{11} + 20 a^{10} + 4 a^{8} + 7 a^{7} + 3 a^{6} + 12 a^{5} + 22 a^{4} + 5 a^{3} + 21 a^{2} + 13 a + 21\right)\cdot 23^{165} + \left(a^{14} + 9 a^{13} + 5 a^{12} + 21 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a^{5} + 15 a^{4} + 20 a^{3} + 22 a^{2} + 2 a + 14\right)\cdot 23^{183} + \left(14 a^{14} + 17 a^{13} + 19 a^{12} + 21 a^{11} + 19 a^{10} + 7 a^{9} + 17 a^{8} + a^{7} + 13 a^{6} + 9 a^{5} + 13 a^{4} + 15 a^{3} + a^{2} + 16 a + 15\right)\cdot 23^{184} + \left(13 a^{14} + a^{13} + 11 a^{12} + 21 a^{11} + 4 a^{10} + 2 a^{9} + 22 a^{8} + 7 a^{7} + 6 a^{6} + 18 a^{5} + 21 a^{3} + 19 a^{2} + 21 a + 7\right)\cdot 23^{185} + \left(18 a^{14} + 3 a^{13} + 2 a^{12} + 12 a^{11} + a^{10} + 9 a^{9} + 8 a^{8} + 4 a^{7} + 21 a^{6} + 13 a^{5} + 3 a^{4} + 5 a^{3} + 15 a^{2} + 8\right)\cdot 23^{186} + \left(a^{14} + 11 a^{13} + 22 a^{12} + 19 a^{11} + 8 a^{10} + 4 a^{9} + 22 a^{8} + 15 a^{7} + 19 a^{6} + 12 a^{5} + 4 a^{4} + 2 a^{3} + 6 a^{2} + 18 a + 15\right)\cdot 23^{187} + \left(22 a^{14} + 15 a^{13} + 20 a^{12} + 11 a^{11} + 18 a^{10} + 14 a^{9} + 5 a^{8} + 4 a^{7} + 10 a^{6} + 18 a^{5} + a^{4} + 4 a^{3} + 7 a^{2} + 20 a + 1\right)\cdot 23^{188} + \left(19 a^{14} + 19 a^{13} + 11 a^{12} + 11 a^{11} + 21 a^{10} + 18 a^{8} + 11 a^{7} + 21 a^{5} + 18 a^{4} + 9 a^{3} + 13 a^{2} + 5 a + 15\right)\cdot 23^{189} + \left(4 a^{14} + 16 a^{13} + 19 a^{12} + 21 a^{11} + 17 a^{10} + 20 a^{9} + 5 a^{8} + 13 a^{7} + 7 a^{6} + 17 a^{5} + 15 a^{4} + 13 a^{3} + 4 a^{2} + 3\right)\cdot 23^{190} + \left(19 a^{14} + 10 a^{13} + 4 a^{12} + 19 a^{11} + 8 a^{10} + 13 a^{9} + 11 a^{8} + 12 a^{7} + 10 a^{6} + 5 a^{5} + 5 a^{4} + a^{3} + 10 a^{2} + 11 a + 6\right)\cdot 23^{191} + \left(14 a^{14} + 6 a^{13} + 5 a^{12} + 6 a^{11} + 14 a^{10} + 15 a^{9} + 11 a^{8} + 16 a^{7} + 5 a^{6} + 9 a^{5} + 19 a^{4} + 5 a^{3} + 8 a^{2} + 22 a + 14\right)\cdot 23^{192} + \left(10 a^{14} + 6 a^{13} + 18 a^{12} + 7 a^{11} + 10 a^{9} + 19 a^{8} + 7 a^{7} + 13 a^{6} + 4 a^{5} + 17 a^{4} + 21 a^{3} + 11 a^{2} + 21 a + 3\right)\cdot 23^{193} + \left(15 a^{14} + 13 a^{13} + a^{12} + 11 a^{11} + 15 a^{10} + 5 a^{9} + 4 a^{8} + 19 a^{7} + 13 a^{6} + 8 a^{5} + 13 a^{4} + 16 a^{3} + 10 a^{2} + 11 a + 10\right)\cdot 23^{194} + \left(3 a^{14} + 22 a^{13} + 5 a^{12} + 20 a^{11} + 11 a^{10} + 14 a^{9} + 17 a^{8} + 18 a^{7} + 13 a^{6} + 9 a^{4} + 13 a^{3} + 6 a^{2} + 5 a + 13\right)\cdot 23^{195} + \left(19 a^{14} + 18 a^{13} + 17 a^{12} + 12 a^{11} + 14 a^{10} + 21 a^{9} + 6 a^{8} + 11 a^{7} + 18 a^{6} + 2 a^{5} + 18 a^{3} + 19 a^{2} + 22 a + 13\right)\cdot 23^{196} + \left(8 a^{14} + 7 a^{12} + 13 a^{11} + 22 a^{10} + 11 a^{9} + 22 a^{8} + 10 a^{7} + 13 a^{6} + 14 a^{5} + 11 a^{4} + 3 a^{3} + 4 a^{2} + 3 a + 10\right)\cdot 23^{197} + \left(a^{14} + 4 a^{13} + 5 a^{12} + 20 a^{11} + 20 a^{10} + 9 a^{9} + 13 a^{8} + 20 a^{7} + 10 a^{6} + 2 a^{5} + 20 a^{4} + 18 a^{3} + 12 a^{2} + a + 21\right)\cdot 23^{198} + \left(5 a^{14} + a^{13} + a^{12} + 20 a^{11} + 7 a^{10} + 9 a^{9} + 7 a^{8} + 4 a^{7} + 4 a^{6} + 20 a^{5} + 13 a^{4} + 4 a^{3} + 14 a^{2} + 10 a + 6\right)\cdot 23^{199} + \left(4 a^{14} + 22 a^{13} + 5 a^{12} + 16 a^{10} + 20 a^{9} + 17 a^{8} + 22 a^{7} + 17 a^{6} + 2 a^{5} + 20 a^{4} + 14 a^{3} + 8 a^{2} + 21 a\right)\cdot 23^{200} + \left(9 a^{14} + 3 a^{13} + 4 a^{12} + 22 a^{11} + 4 a^{10} + 7 a^{9} + 12 a^{8} + 17 a^{7} + 6 a^{5} + 6 a^{4} + 19 a^{3} + 11 a^{2} + 2 a + 10\right)\cdot 23^{201} + \left(16 a^{13} + 11 a^{12} + 19 a^{11} + 11 a^{10} + 8 a^{9} + 16 a^{8} + 3 a^{7} + 15 a^{6} + 11 a^{5} + 17 a^{4} + 11 a^{3} + 6 a^{2} + 17 a + 8\right)\cdot 23^{202} + \left(22 a^{14} + 2 a^{13} + 4 a^{12} + 7 a^{11} + a^{10} + 13 a^{9} + 22 a^{8} + 15 a^{7} + 15 a^{6} + 13 a^{5} + 8 a^{4} + 21 a^{3} + 6 a^{2} + 18\right)\cdot 23^{203} + \left(22 a^{14} + 10 a^{13} + a^{12} + 5 a^{11} + 2 a^{10} + 17 a^{9} + 10 a^{8} + 5 a^{7} + 6 a^{6} + 5 a^{5} + 13 a^{4} + 15 a^{3} + 18 a^{2} + 10 a + 7\right)\cdot 23^{204} + \left(5 a^{13} + 8 a^{12} + 11 a^{11} + 17 a^{10} + 7 a^{9} + 3 a^{7} + 20 a^{6} + 4 a^{5} + 7 a^{4} + 15 a^{3} + 17 a^{2} + 4 a + 22\right)\cdot 23^{205} + \left(5 a^{14} + 19 a^{13} + 5 a^{12} + 6 a^{11} + 8 a^{10} + 2 a^{9} + 15 a^{8} + 14 a^{7} + 20 a^{6} + 9 a^{5} + 13 a^{4} + 12 a^{3} + 18 a^{2} + 18 a + 3\right)\cdot 23^{206} + \left(12 a^{14} + 20 a^{13} + 9 a^{12} + 20 a^{11} + 11 a^{10} + 17 a^{9} + 18 a^{8} + 7 a^{7} + 14 a^{6} + 8 a^{5} + 2 a^{4} + 15 a^{3} + 15 a^{2} + 15 a + 4\right)\cdot 23^{207} + \left(6 a^{14} + 13 a^{13} + 8 a^{12} + 6 a^{11} + 18 a^{10} + 2 a^{9} + a^{8} + 19 a^{7} + 9 a^{6} + 18 a^{5} + 8 a^{4} + 6 a^{3} + 16 a^{2} + 14 a\right)\cdot 23^{208} + \left(19 a^{14} + 8 a^{13} + 12 a^{12} + 15 a^{11} + 5 a^{10} + 13 a^{9} + 13 a^{8} + 16 a^{7} + 18 a^{6} + 15 a^{5} + 6 a^{4} + 17 a^{3} + 3 a^{2} + 14 a + 10\right)\cdot 23^{209} + \left(12 a^{14} + 17 a^{13} + 18 a^{12} + 14 a^{11} + 8 a^{10} + 8 a^{9} + 6 a^{7} + 9 a^{6} + 19 a^{5} + 2 a^{4} + 13 a^{3} + 19 a^{2} + 22 a + 15\right)\cdot 23^{210} + \left(13 a^{14} + 14 a^{13} + 6 a^{12} + 13 a^{11} + 15 a^{10} + 6 a^{9} + 16 a^{8} + 19 a^{7} + 22 a^{6} + 2 a^{5} + 20 a^{4} + 21 a^{2} + 16\right)\cdot 23^{211} + \left(12 a^{13} + 16 a^{12} + 3 a^{11} + 13 a^{10} + 19 a^{8} + 7 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a^{13} + 12 a^{12} + 22 a^{11} + 20 a^{10} + 9 a^{9} + 11 a^{8} + 13 a^{7} + 6 a^{6} + 14 a^{5} + 22 a^{4} + a^{3} + 17 a + 17\right)\cdot 23^{224} + \left(2 a^{14} + 15 a^{13} + 15 a^{12} + 6 a^{11} + 22 a^{10} + 16 a^{9} + 6 a^{8} + 13 a^{7} + 20 a^{6} + 18 a^{5} + 13 a^{4} + 8 a^{3} + 4 a^{2} + 3 a + 11\right)\cdot 23^{225} + \left(8 a^{14} + 18 a^{13} + 2 a^{12} + 17 a^{11} + 20 a^{10} + 22 a^{9} + 8 a^{8} + 6 a^{7} + 15 a^{6} + 15 a^{5} + 12 a^{4} + 3 a^{3} + 13 a^{2} + 10 a + 18\right)\cdot 23^{226} + \left(19 a^{14} + 21 a^{13} + 6 a^{12} + 7 a^{11} + 19 a^{10} + 18 a^{9} + 6 a^{8} + 14 a^{7} + 21 a^{6} + 8 a^{5} + 13 a^{4} + 20 a^{3} + 11 a^{2} + 20 a + 3\right)\cdot 23^{227} + \left(4 a^{14} + 15 a^{13} + a^{12} + 11 a^{11} + 4 a^{9} + 4 a^{8} + 21 a^{7} + a^{6} + 3 a^{5} + 16 a^{4} + 8 a^{3} + 16 a^{2} + 13 a + 10\right)\cdot 23^{228} + \left(a^{14} + 4 a^{13} + 18 a^{12} + 9 a^{11} + 14 a^{10} + 2 a^{9} + 15 a^{8} + 15 a^{7} + 8 a^{6} + a^{5} + 17 a^{4} + 22 a^{3} + 8 a^{2} + 8 a + 15\right)\cdot 23^{229} + \left(19 a^{14} + 11 a^{13} + 5 a^{12} + 15 a^{11} + 5 a^{10} + 18 a^{9} + 5 a^{8} + 4 a^{7} + 7 a^{6} + 15 a^{5} + 6 a^{4} + 3 a^{3} + a^{2} + 17 a + 12\right)\cdot 23^{230} + \left(2 a^{14} + 8 a^{13} + 16 a^{12} + 22 a^{11} + 10 a^{10} + 2 a^{8} + 18 a^{7} + 12 a^{6} + 19 a^{5} + 20 a^{4} + 20 a^{3} + 19 a^{2} + 21 a + 8\right)\cdot 23^{231} + \left(2 a^{14} + 3 a^{13} + a^{12} + 10 a^{11} + 7 a^{10} + 19 a^{9} + 22 a^{8} + 2 a^{7} + 22 a^{6} + 8 a^{5} + a^{4} + 7 a^{2} + 16 a + 5\right)\cdot 23^{232} + \left(17 a^{14} + 13 a^{13} + 7 a^{12} + 17 a^{11} + 8 a^{10} + 16 a^{9} + 9 a^{8} + 6 a^{7} + 16 a^{6} + 22 a^{5} + 22 a^{4} + 5 a^{3} + 7 a^{2} + 5 a + 12\right)\cdot 23^{233} + \left(12 a^{14} + 2 a^{13} + 7 a^{12} + 6 a^{11} + 14 a^{10} + 22 a^{9} + 7 a^{8} + 15 a^{7} + 15 a^{6} + 17 a^{5} + 5 a^{4} + 20 a^{3} + 16 a^{2} + 11 a + 22\right)\cdot 23^{234} + \left(17 a^{14} + 6 a^{13} + 22 a^{12} + 16 a^{11} + 17 a^{10} + 19 a^{9} + 13 a^{8} + 16 a^{7} + 4 a^{6} + 14 a^{5} + 21 a^{4} + 5 a^{2} + 2 a + 16\right)\cdot 23^{235} + \left(10 a^{14} + 3 a^{13} + 12 a^{12} + 4 a^{11} + 15 a^{10} + 14 a^{9} + 21 a^{8} + 15 a^{7} + 16 a^{6} + 8 a^{5} + 18 a^{4} + 20 a^{3} + 4 a^{2} + 10 a + 7\right)\cdot 23^{236} + \left(15 a^{14} + 12 a^{13} + 10 a^{12} + 15 a^{11} + 14 a^{10} + 8 a^{9} + 13 a^{8} + 15 a^{7} + 3 a^{6} + 21 a^{5} + 12 a^{4} + 12 a^{3} + 7 a^{2} + 22\right)\cdot 23^{237} + \left(18 a^{14} + 20 a^{13} + 9 a^{12} + 16 a^{11} + 14 a^{10} + 13 a^{9} + 5 a^{8} + 3 a^{7} + 10 a^{6} + 11 a^{5} + 11 a^{4} + 22 a^{3} + 8 a^{2} + 11 a + 9\right)\cdot 23^{238} + \left(2 a^{14} + 19 a^{13} + 21 a^{12} + 5 a^{11} + 12 a^{10} + 6 a^{9} + 12 a^{8} + 19 a^{7} + a^{6} + 5 a^{5} + 4 a^{4} + 4 a^{3} + 11 a^{2} + 22 a + 4\right)\cdot 23^{239} + \left(20 a^{14} + 13 a^{13} + 15 a^{12} + 9 a^{11} + 22 a^{10} + 17 a^{9} + 7 a^{8} + 16 a^{7} + 15 a^{6} + 13 a^{5} + 7 a^{4} + 22 a^{3} + 15 a^{2} + 21 a + 16\right)\cdot 23^{240} + \left(9 a^{14} + a^{13} + 22 a^{12} + 6 a^{11} + 6 a^{10} + 12 a^{9} + 22 a^{8} + 4 a^{7} + 4 a^{6} + 11 a^{5} + 10 a^{4} + 4 a^{3} + 7 a^{2} + 15 a + 9\right)\cdot 23^{241} + \left(7 a^{14} + 22 a^{12} + 4 a^{11} + 10 a^{10} + 18 a^{9} + 7 a^{8} + 20 a^{7} + 20 a^{6} + 2 a^{5} + 2 a^{3} + 9 a + 1\right)\cdot 23^{242} + \left(12 a^{14} + 22 a^{13} + 9 a^{12} + 6 a^{11} + 9 a^{10} + 22 a^{9} + 12 a^{8} + 11 a^{7} + 17 a^{5} + 10 a^{3} + 22 a^{2} + 13 a + 15\right)\cdot 23^{243} + \left(11 a^{14} + 17 a^{13} + 22 a^{12} + 10 a^{11} + 19 a^{10} + 8 a^{9} + 2 a^{8} + 10 a^{7} + 14 a^{6} + 7 a^{5} + 3 a^{4} + 18 a^{3} + 4 a^{2} + 11\right)\cdot 23^{244} + \left(7 a^{14} + 15 a^{12} + 19 a^{11} + 20 a^{10} + 16 a^{9} + 10 a^{8} + 3 a^{7} + 21 a^{6} + 9 a^{4} + 10 a^{3} + 3 a^{2} + 11 a\right)\cdot 23^{245} + \left(2 a^{14} + 5 a^{13} + 15 a^{12} + 3 a^{11} + 9 a^{10} + 10 a^{9} + 7 a^{8} + 17 a^{7} + 16 a^{6} + 12 a^{5} + 9 a^{4} + 22 a^{3} + 13 a^{2} + 9 a + 3\right)\cdot 23^{246} + \left(9 a^{14} + 2 a^{13} + 14 a^{12} + 15 a^{11} + 2 a^{10} + 10 a^{9} + 6 a^{8} + 19 a^{7} + 20 a^{6} + 3 a^{5} + 6 a^{4} + 8 a^{3} + 22 a^{2} + 13 a + 9\right)\cdot 23^{247} + \left(17 a^{14} + 12 a^{13} + 21 a^{12} + 3 a^{11} + 11 a^{10} + 3 a^{9} + 18 a^{8} + 12 a^{7} + 22 a^{6} + 22 a^{5} + 8 a^{4} + a^{3} + 10 a^{2} + 21 a + 22\right)\cdot 23^{248} + \left(21 a^{14} + 12 a^{13} + 9 a^{12} + 6 a^{11} + 17 a^{10} + 5 a^{9} + 10 a^{7} + 20 a^{6} + 2 a^{5} + 6 a^{3} + 22 a^{2} + 3 a + 18\right)\cdot 23^{249} + \left(18 a^{14} + 7 a^{13} + 15 a^{12} + 12 a^{11} + 18 a^{10} + 4 a^{9} + 5 a^{8} + 7 a^{7} + 3 a^{6} + 6 a^{5} + 14 a^{4} + 4 a^{3} + a^{2} + 2\right)\cdot 23^{250} + \left(19 a^{13} + 16 a^{12} + 9 a^{11} + 11 a^{10} + 2 a^{9} + 4 a^{8} + 11 a^{7} + 12 a^{6} + 21 a^{5} + 21 a^{4} + 11 a^{3} + 14 a^{2} + 20 a + 17\right)\cdot 23^{251} + \left(6 a^{14} + 5 a^{12} + 15 a^{11} + 2 a^{10} + 7 a^{9} + 15 a^{8} + 11 a^{6} + 15 a^{5} + 11 a^{4} + 18 a^{3} + 16 a + 14\right)\cdot 23^{252} + \left(5 a^{14} + 18 a^{13} + 2 a^{12} + 16 a^{11} + 14 a^{10} + 5 a^{9} + 22 a^{8} + 3 a^{7} + 6 a^{6} + 3 a^{5} + 22 a^{4} + 16 a^{3} + 6 a^{2} + 10 a + 11\right)\cdot 23^{253} + \left(6 a^{14} + 10 a^{13} + 15 a^{12} + 15 a^{11} + 15 a^{10} + 15 a^{9} + 18 a^{8} + 14 a^{7} + 8 a^{6} + 10 a^{5} + 20 a^{4} + 22 a^{3} + 15 a^{2} + 15 a + 2\right)\cdot 23^{254} + \left(21 a^{14} + a^{13} + a^{12} + 21 a^{11} + 19 a^{10} + 10 a^{9} + 15 a^{8} + 12 a^{7} + 8 a^{6} + 14 a^{5} + 18 a^{3} + 9 a^{2} + 14 a + 10\right)\cdot 23^{255} + \left(14 a^{14} + 12 a^{13} + 17 a^{12} + 19 a^{11} + 8 a^{10} + 3 a^{9} + 4 a^{8} + 22 a^{7} + 18 a^{6} + 11 a^{5} + 3 a^{4} + 3 a^{3} + 21 a^{2} + 13 a + 16\right)\cdot 23^{256} + \left(20 a^{14} + 10 a^{13} + 10 a^{12} + 2 a^{11} + a^{10} + 11 a^{9} + 16 a^{8} + 15 a^{7} + 7 a^{6} + 6 a^{5} + 14 a^{3} + 13 a^{2} + 7 a + 5\right)\cdot 23^{257} + \left(5 a^{14} + 10 a^{13} + 4 a^{12} + 18 a^{11} + 18 a^{10} + 20 a^{9} + 22 a^{8} + 16 a^{7} + 7 a^{6} + 5 a^{5} + 17 a^{4} + 14 a^{3} + 17 a^{2} + 9 a + 12\right)\cdot 23^{258} + \left(10 a^{14} + 4 a^{13} + 10 a^{12} + 13 a^{11} + 10 a^{10} + 21 a^{9} + 19 a^{8} + 8 a^{7} + 2 a^{6} + a^{4} + 22 a^{3} + 3 a^{2} + 18 a + 16\right)\cdot 23^{259} + \left(16 a^{14} + 10 a^{13} + a^{12} + 4 a^{11} + 3 a^{10} + 4 a^{9} + 8 a^{8} + 6 a^{6} + 15 a^{5} + 18 a^{4} + 4 a^{3} + 5 a^{2} + 11\right)\cdot 23^{260} + \left(13 a^{14} + 11 a^{13} + a^{12} + 2 a^{11} + 9 a^{10} + 12 a^{9} + 22 a^{8} + 7 a^{7} + 3 a^{6} + 8 a^{5} + 16 a^{4} + 21 a^{3} + 17 a^{2} + 19 a + 4\right)\cdot 23^{261} + \left(11 a^{14} + 10 a^{13} + 16 a^{12} + 7 a^{11} + 4 a^{10} + 10 a^{9} + 15 a^{8} + 20 a^{7} + 11 a^{6} + 8 a^{5} + 11 a^{4} + 20 a^{3} + 7 a^{2} + 6 a + 4\right)\cdot 23^{262} + \left(a^{14} + 17 a^{13} + 22 a^{12} + 5 a^{11} + 2 a^{10} + 6 a^{9} + 10 a^{8} + 2 a^{7} + 3 a^{6} + a^{5} + a^{4} + 3 a^{3} + 15 a^{2} + 2 a + 15\right)\cdot 23^{263} + \left(13 a^{14} + 12 a^{13} + 8 a^{12} + 15 a^{11} + 3 a^{10} + 10 a^{9} + 20 a^{8} + 9 a^{7} + 10 a^{6} + 16 a^{5} + 14 a^{4} + 22 a^{3} + 7 a^{2} + 6 a + 15\right)\cdot 23^{264} + \left(20 a^{14} + 18 a^{13} + 4 a^{12} + a^{11} + 13 a^{10} + 13 a^{9} + 7 a^{8} + 19 a^{7} + 14 a^{6} + 7 a^{5} + 18 a^{4} + 11 a^{3} + 11 a^{2} + 4 a + 5\right)\cdot 23^{265} + \left(16 a^{14} + 2 a^{13} + 3 a^{12} + 9 a^{11} + 22 a^{10} + 15 a^{9} + 11 a^{8} + 13 a^{7} + 7 a^{6} + 6 a^{5} + 22 a^{4} + 7 a^{3} + 21 a^{2} + 15 a + 7\right)\cdot 23^{266} + \left(15 a^{13} + 8 a^{12} + 4 a^{11} + 15 a^{10} + 4 a^{9} + 5 a^{8} + 20 a^{6} + 17 a^{5} + 3 a^{4} + a^{3} + 18 a^{2} + 15\right)\cdot 23^{267} + \left(20 a^{14} + 8 a^{13} + 7 a^{12} + 8 a^{11} + 15 a^{10} + 7 a^{9} + 13 a^{8} + 13 a^{7} + 7 a^{6} + 3 a^{5} + 21 a^{4} + 16 a^{3} + 15 a^{2} + 4 a + 2\right)\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 8 }$ $=$ $8 a^{14} + 2 a^{13} + 20 a^{12} + 19 a^{11} + 17 a^{10} + 17 a^{9} + 17 a^{8} + 8 a^{7} + 4 a^{6} + 3 a^{5} + 7 a^{4} + 3 a^{3} + 14 a^{2} + 12 a + \left(21 a^{14} + 15 a^{13} + 21 a^{12} + 18 a^{10} + a^{9} + 10 a^{8} + 22 a^{7} + 9 a^{6} + 2 a^{5} + 4 a^{4} + 13 a^{3} + 22 a^{2} + 10 a + 3\right)\cdot 23 + \left(10 a^{14} + 2 a^{13} + 16 a^{12} + 19 a^{11} + 6 a^{10} + 11 a^{9} + 16 a^{8} + 2 a^{6} + 8 a^{5} + 14 a^{4} + 10 a^{3} + 10 a^{2} + 6 a + 19\right)\cdot 23^{2} + \left(2 a^{14} + 13 a^{13} + 12 a^{12} + 6 a^{11} + a^{9} + 5 a^{8} + 19 a^{7} + 8 a^{6} + 21 a^{5} + 19 a^{4} + a^{3} + 2 a^{2} + 9 a + 20\right)\cdot 23^{3} + \left(18 a^{14} + 8 a^{13} + 17 a^{11} + 18 a^{10} + 13 a^{9} + 22 a^{8} + 3 a^{6} + 21 a^{5} + 14 a^{4} + 20 a^{3} + 21 a + 10\right)\cdot 23^{4} + \left(5 a^{14} + 12 a^{13} + 12 a^{12} + a^{11} + 18 a^{10} + 13 a^{9} + 15 a^{8} + 14 a^{7} + a^{6} + 11 a^{5} + 12 a^{4} + 6 a^{3} + a^{2} + 8 a + 20\right)\cdot 23^{5} + \left(11 a^{14} + 18 a^{13} + 5 a^{11} + 20 a^{10} + 5 a^{9} + 7 a^{8} + 7 a^{7} + 12 a^{6} + 7 a^{5} + 2 a^{4} + a^{3} + 12 a^{2} + 5 a + 19\right)\cdot 23^{6} + \left(11 a^{14} + 19 a^{13} + 22 a^{12} + 15 a^{11} + 17 a^{10} + 4 a^{9} + 14 a^{8} + 21 a^{7} + 21 a^{5} + 4 a^{4} + 12 a^{3} + 11 a^{2} + 18 a + 20\right)\cdot 23^{7} + \left(6 a^{14} + 18 a^{13} + 18 a^{11} + 9 a^{10} + 5 a^{9} + 16 a^{8} + 6 a^{7} + a^{6} + 6 a^{5} + 4 a^{4} + 6 a^{3} + 4 a^{2} + 9 a + 17\right)\cdot 23^{8} + \left(a^{14} + 5 a^{13} + 9 a^{12} + 10 a^{11} + 9 a^{10} + 12 a^{9} + 11 a^{8} + 3 a^{7} + a^{5} + 8 a^{4} + a^{3} + 18 a^{2} + 21 a + 15\right)\cdot 23^{9} + \left(6 a^{14} + 6 a^{13} + 14 a^{12} + 8 a^{11} + 11 a^{10} + 5 a^{9} + 18 a^{7} + 4 a^{6} + 15 a^{5} + 19 a^{4} + 19 a^{3} + 2 a^{2} + 10 a + 4\right)\cdot 23^{10} + \left(15 a^{14} + 12 a^{13} + 9 a^{12} + 11 a^{11} + 18 a^{10} + 8 a^{9} + 12 a^{8} + 20 a^{7} + 21 a^{6} + 7 a^{5} + 16 a^{4} + 20 a^{3} + 10 a^{2} + 16 a + 2\right)\cdot 23^{11} + \left(5 a^{14} + 8 a^{13} + 13 a^{11} + 16 a^{10} + 14 a^{9} + 3 a^{8} + 19 a^{7} + 13 a^{6} + 11 a^{5} + 8 a^{4} + 7 a^{3} + 13 a^{2} + 12 a + 9\right)\cdot 23^{12} + \left(4 a^{14} + 21 a^{13} + 18 a^{12} + 20 a^{10} + 14 a^{9} + 14 a^{7} + 3 a^{6} + 9 a^{5} + 8 a^{4} + a^{3} + 17 a^{2} + 10 a + 6\right)\cdot 23^{13} + \left(12 a^{14} + 11 a^{13} + 10 a^{12} + 12 a^{11} + 8 a^{10} + 12 a^{9} + 17 a^{8} + 21 a^{7} + 13 a^{6} + 4 a^{5} + 22 a^{4} + 8 a^{3} + 22 a^{2} + 7 a + 17\right)\cdot 23^{14} + \left(19 a^{14} + a^{12} + 21 a^{11} + 15 a^{10} + 2 a^{9} + 12 a^{8} + 3 a^{7} + 2 a^{6} + 19 a^{5} + 3 a^{4} + 20 a^{3} + 22 a^{2} + 16 a + 15\right)\cdot 23^{15} + \left(21 a^{14} + 4 a^{13} + 8 a^{12} + 10 a^{11} + 11 a^{10} + 10 a^{9} + 12 a^{8} + 22 a^{7} + 3 a^{6} + 9 a^{5} + 11 a^{4} + 20 a^{2} + 17 a + 6\right)\cdot 23^{16} + \left(11 a^{14} + 12 a^{13} + 11 a^{12} + 10 a^{11} + 16 a^{10} + 6 a^{9} + 17 a^{8} + 9 a^{7} + 11 a^{6} + 10 a^{5} + 12 a^{4} + 8 a^{3} + 21 a^{2} + 2 a + 2\right)\cdot 23^{17} + \left(9 a^{14} + 16 a^{13} + 5 a^{12} + 15 a^{11} + 8 a^{10} + 8 a^{9} + 3 a^{8} + 19 a^{7} + 12 a^{5} + 21 a^{4} + 5 a^{3} + 5 a^{2} + 2 a + 6\right)\cdot 23^{18} + \left(10 a^{14} + 13 a^{13} + 16 a^{12} + 19 a^{11} + 4 a^{10} + 7 a^{9} + 7 a^{8} + 8 a^{7} + 2 a^{6} + 19 a^{5} + 16 a^{4} + a^{3} + 16 a^{2} + 18 a + 17\right)\cdot 23^{19} + \left(8 a^{14} + 20 a^{13} + 6 a^{12} + 11 a^{11} + 11 a^{10} + 11 a^{9} + 11 a^{8} + 17 a^{7} + 3 a^{6} + 14 a^{5} + 9 a^{4} + 9 a^{3} + 2 a^{2} + 7 a + 13\right)\cdot 23^{20} + \left(20 a^{13} + 7 a^{11} + 15 a^{10} + 4 a^{9} + 21 a^{8} + 11 a^{7} + 8 a^{6} + 11 a^{5} + 11 a^{4} + 15 a^{3} + 17 a^{2} + 3 a\right)\cdot 23^{21} + \left(21 a^{14} + 10 a^{13} + 9 a^{12} + 15 a^{11} + 4 a^{10} + 20 a^{9} + 12 a^{8} + 21 a^{7} + 12 a^{6} + 14 a^{5} + 12 a^{4} + 8 a^{3} + 14 a^{2} + 2 a + 12\right)\cdot 23^{22} + \left(2 a^{14} + 20 a^{12} + 2 a^{11} + 13 a^{10} + 4 a^{9} + 10 a^{8} + 4 a^{7} + 11 a^{6} + 6 a^{5} + 6 a^{4} + 10 a^{3} + 18 a^{2} + 7 a + 11\right)\cdot 23^{23} + \left(11 a^{14} + 11 a^{13} + 3 a^{12} + 11 a^{11} + 8 a^{10} + 15 a^{9} + 20 a^{8} + 5 a^{7} + 22 a^{6} + 4 a^{5} + 10 a^{4} + a^{3} + 7 a^{2} + 5 a + 11\right)\cdot 23^{24} + \left(18 a^{14} + 21 a^{13} + 9 a^{12} + 17 a^{11} + 2 a^{10} + 15 a^{9} + 14 a^{8} + 22 a^{7} + 4 a^{6} + 11 a^{5} + 9 a^{4} + 5 a^{3} + 10 a^{2} + 9 a + 4\right)\cdot 23^{25} + \left(12 a^{14} + 19 a^{13} + 9 a^{12} + 7 a^{11} + 19 a^{10} + 12 a^{9} + 10 a^{8} + 15 a^{7} + 19 a^{6} + 20 a^{5} + 17 a^{4} + 7 a^{3} + 5 a^{2} + 12 a + 8\right)\cdot 23^{26} + \left(a^{14} + 7 a^{13} + 20 a^{12} + 16 a^{11} + 19 a^{10} + 8 a^{9} + a^{8} + 5 a^{7} + 22 a^{6} + 11 a^{5} + 21 a^{4} + 22 a^{3} + 5 a^{2} + 20 a + 7\right)\cdot 23^{27} + \left(22 a^{14} + 9 a^{13} + 22 a^{12} + 16 a^{11} + 11 a^{10} + 15 a^{9} + 6 a^{8} + 19 a^{7} + 8 a^{6} + 3 a^{5} + 15 a^{4} + 5 a^{2} + 13 a + 3\right)\cdot 23^{28} + \left(20 a^{14} + 18 a^{13} + 7 a^{12} + 21 a^{11} + 8 a^{10} + 14 a^{9} + 13 a^{8} + 9 a^{7} + 4 a^{6} + 18 a^{4} + 8 a^{3} + 11 a^{2} + 14 a + 17\right)\cdot 23^{29} + \left(7 a^{14} + 18 a^{13} + 9 a^{12} + 15 a^{11} + 8 a^{10} + 3 a^{9} + 20 a^{8} + 18 a^{7} + 5 a^{6} + 3 a^{5} + 6 a^{4} + 7 a^{3} + 19 a^{2} + 17 a + 19\right)\cdot 23^{30} + \left(22 a^{14} + 13 a^{13} + 20 a^{11} + 4 a^{10} + 12 a^{9} + 16 a^{8} + 8 a^{7} + 13 a^{6} + 18 a^{5} + 17 a^{4} + 17 a^{3} + 9 a^{2} + 10 a + 15\right)\cdot 23^{31} + \left(6 a^{14} + 16 a^{13} + 4 a^{12} + 4 a^{11} + 12 a^{10} + 14 a^{9} + 15 a^{8} + 4 a^{7} + a^{6} + 4 a^{5} + 15 a^{4} + 10 a^{3} + 2 a^{2} + 8 a + 10\right)\cdot 23^{32} + \left(14 a^{14} + 8 a^{13} + 13 a^{12} + 19 a^{11} + 16 a^{10} + 4 a^{9} + 21 a^{8} + 16 a^{7} + 4 a^{6} + 10 a^{5} + 13 a^{4} + 13 a^{3} + 12 a + 7\right)\cdot 23^{33} + \left(6 a^{14} + 7 a^{13} + 3 a^{12} + 11 a^{11} + 9 a^{10} + 19 a^{9} + 10 a^{8} + 8 a^{6} + 6 a^{5} + 6 a^{4} + 3 a^{3} + 22 a^{2} + 22 a + 22\right)\cdot 23^{34} + \left(3 a^{14} + 10 a^{13} + 19 a^{12} + 7 a^{11} + 15 a^{10} + 5 a^{8} + 21 a^{7} + a^{6} + 6 a^{5} + 21 a^{4} + 18 a^{3} + 12 a^{2} + 20 a + 22\right)\cdot 23^{35} + \left(9 a^{14} + 3 a^{13} + 5 a^{12} + 2 a^{11} + a^{10} + 4 a^{9} + 19 a^{7} + 19 a^{6} + 10 a^{5} + 17 a^{4} + 11 a^{3} + 5 a^{2} + 22 a + 12\right)\cdot 23^{36} + \left(20 a^{14} + 12 a^{13} + 10 a^{12} + 21 a^{11} + 19 a^{10} + 19 a^{9} + 5 a^{8} + 12 a^{7} + 3 a^{6} + 12 a^{5} + 15 a^{4} + 7 a^{3} + 20 a^{2} + 6 a + 8\right)\cdot 23^{37} + \left(19 a^{14} + 6 a^{13} + 13 a^{12} + 22 a^{11} + 18 a^{10} + 10 a^{9} + 6 a^{8} + a^{7} + 4 a^{6} + 18 a^{5} + 22 a^{4} + 8 a^{3} + 18 a^{2} + 13 a + 2\right)\cdot 23^{38} + \left(21 a^{14} + 5 a^{13} + 12 a^{12} + 19 a^{11} + 21 a^{10} + 13 a^{8} + 9 a^{7} + 15 a^{5} + 9 a^{4} + 9 a^{3} + 7 a^{2} + 19 a + 15\right)\cdot 23^{39} + \left(14 a^{14} + 13 a^{13} + 21 a^{12} + 9 a^{11} + 8 a^{10} + 19 a^{9} + 18 a^{8} + 6 a^{7} + 5 a^{6} + 13 a^{5} + 19 a^{4} + 9 a^{3} + 12 a^{2} + 12 a + 15\right)\cdot 23^{40} + \left(22 a^{14} + 7 a^{13} + 9 a^{12} + 17 a^{11} + 16 a^{10} + 5 a^{9} + 7 a^{8} + 6 a^{7} + 12 a^{6} + 18 a^{5} + 22 a^{4} + 15 a^{3} + 21 a^{2} + 8 a + 8\right)\cdot 23^{41} + \left(18 a^{14} + 18 a^{13} + 6 a^{12} + a^{11} + a^{10} + 13 a^{9} + 11 a^{8} + 14 a^{7} + 6 a^{6} + 4 a^{4} + 12 a^{3} + 2 a^{2} + 12 a + 14\right)\cdot 23^{42} + \left(10 a^{14} + 10 a^{13} + 20 a^{12} + 6 a^{10} + 14 a^{9} + 7 a^{8} + 3 a^{7} + 7 a^{6} + 8 a^{5} + 10 a^{4} + 19 a^{3} + 7 a + 21\right)\cdot 23^{43} + \left(5 a^{14} + 22 a^{13} + 12 a^{12} + 6 a^{11} + 17 a^{10} + 18 a^{9} + a^{8} + 11 a^{7} + a^{6} + 7 a^{5} + 4 a^{4} + 7 a^{3} + 5 a^{2} + 20 a + 6\right)\cdot 23^{44} + \left(21 a^{14} + 4 a^{13} + 12 a^{12} + 19 a^{11} + 11 a^{10} + 9 a^{9} + 15 a^{8} + 15 a^{7} + 21 a^{6} + 9 a^{5} + 6 a^{4} + 2 a^{3} + 10 a^{2} + 20 a + 18\right)\cdot 23^{45} + \left(20 a^{14} + 14 a^{13} + 4 a^{12} + 22 a^{11} + 12 a^{10} + 7 a^{9} + 4 a^{8} + 5 a^{7} + 10 a^{6} + 10 a^{4} + 13 a^{2} + 5 a + 13\right)\cdot 23^{46} + \left(9 a^{14} + 6 a^{13} + a^{12} + 2 a^{11} + 5 a^{10} + 3 a^{9} + 9 a^{8} + 12 a^{7} + 11 a^{5} + 17 a^{4} + 11 a^{3} + 17 a^{2} + 9 a + 5\right)\cdot 23^{47} + \left(4 a^{14} + 12 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\left(15 a^{14} + 18 a^{13} + 15 a^{12} + 4 a^{11} + 17 a^{10} + 18 a^{9} + 20 a^{8} + 9 a^{7} + 5 a^{6} + 16 a^{5} + 11 a^{4} + a^{3} + 21 a + 14\right)\cdot 23^{71} + \left(6 a^{14} + 17 a^{13} + 20 a^{12} + 7 a^{11} + 17 a^{10} + 16 a^{9} + 16 a^{7} + 6 a^{6} + 18 a^{5} + 9 a^{4} + 13 a^{3} + 17 a^{2} + 20 a + 9\right)\cdot 23^{72} + \left(7 a^{14} + 15 a^{13} + 10 a^{12} + 9 a^{10} + 19 a^{9} + 8 a^{8} + 9 a^{7} + 20 a^{6} + 6 a^{5} + 18 a^{4} + 20 a^{3} + 18 a^{2} + 16 a + 18\right)\cdot 23^{73} + \left(15 a^{14} + 13 a^{13} + 4 a^{12} + 12 a^{11} + 13 a^{10} + 7 a^{9} + 8 a^{8} + 15 a^{7} + 10 a^{6} + 22 a^{5} + a^{4} + 3 a^{3} + 8 a^{2} + a + 21\right)\cdot 23^{74} + \left(4 a^{14} + 13 a^{13} + 11 a^{12} + a^{11} + 17 a^{10} + 17 a^{9} + 7 a^{8} + 17 a^{7} + 12 a^{6} + 6 a^{5} + a^{4} + 17 a^{3} + 20 a^{2} + 3 a + 14\right)\cdot 23^{75} + \left(22 a^{14} + 6 a^{13} + 15 a^{12} + 3 a^{11} + 20 a^{10} + 7 a^{9} + 2 a^{8} + 10 a^{7} + 2 a^{6} + 15 a^{5} + a^{4} + 9 a^{3} + 4 a^{2} + 22 a + 14\right)\cdot 23^{76} + \left(7 a^{14} + 9 a^{13} + 16 a^{12} + a^{11} + 17 a^{10} + a^{9} + 8 a^{8} + 8 a^{7} + 2 a^{6} + 17 a^{5} + 13 a^{4} + 10 a^{3} + a^{2} + 19 a + 7\right)\cdot 23^{77} + \left(5 a^{14} + 12 a^{13} + 19 a^{12} + 18 a^{11} + 8 a^{10} + 3 a^{9} + 13 a^{8} + 19 a^{7} + 15 a^{6} + 21 a^{4} + 8 a^{3} + 15 a^{2} + a + 18\right)\cdot 23^{78} + \left(21 a^{14} + 15 a^{13} + 4 a^{12} + 16 a^{11} + 5 a^{10} + 12 a^{9} + 17 a^{8} + 19 a^{7} + 6 a^{6} + 11 a^{5} + 9 a^{4} + 2 a^{3} + 6 a^{2} + 14 a + 5\right)\cdot 23^{79} + \left(5 a^{14} + 20 a^{13} + 6 a^{12} + 6 a^{11} + 14 a^{10} + 4 a^{9} + 10 a^{8} + 6 a^{7} + 14 a^{6} + 19 a^{5} + 9 a^{4} + 3 a^{3} + 13 a^{2} + 18 a + 20\right)\cdot 23^{80} + \left(21 a^{14} + 10 a^{13} + 17 a^{12} + 2 a^{11} + 21 a^{10} + 12 a^{9} + 10 a^{8} + a^{7} + 3 a^{6} + 12 a^{5} + 4 a^{4} + 4 a^{3} + 22 a^{2} + a + 13\right)\cdot 23^{81} + \left(17 a^{13} + 9 a^{12} + 13 a^{11} + 15 a^{10} + 21 a^{9} + 9 a^{8} + 16 a^{7} + 16 a^{6} + 16 a^{5} + 14 a^{4} + 7 a^{3} + 20 a + 20\right)\cdot 23^{82} + \left(8 a^{14} + 5 a^{13} + 4 a^{12} + 8 a^{11} + 15 a^{10} + 7 a^{9} + 15 a^{8} + 22 a^{7} + 22 a^{5} + 21 a^{4} + 21 a^{3} + 4 a^{2} + 11 a\right)\cdot 23^{83} + \left(6 a^{14} + 5 a^{13} + 12 a^{12} + 12 a^{11} + a^{10} + a^{9} + 10 a^{8} + 8 a^{7} + 18 a^{6} + 8 a^{5} + 2 a^{4} + 2 a^{3} + 15 a^{2} + 14 a + 14\right)\cdot 23^{84} + \left(17 a^{14} + 11 a^{13} + 18 a^{11} + 8 a^{10} + 22 a^{9} + 9 a^{8} + 15 a^{7} + 21 a^{5} + 21 a^{3} + 17 a^{2} + 19 a + 6\right)\cdot 23^{85} + \left(16 a^{14} + 9 a^{13} + 3 a^{12} + 4 a^{11} + 16 a^{10} + 15 a^{9} + 18 a^{8} + 19 a^{7} + 18 a^{6} + a^{5} + 17 a^{4} + 10 a^{3} + 10 a^{2} + 13 a + 17\right)\cdot 23^{86} + \left(15 a^{14} + 20 a^{13} + 16 a^{12} + 11 a^{11} + 17 a^{10} + 18 a^{9} + 12 a^{8} + 20 a^{7} + 18 a^{6} + 14 a^{3} + 4 a^{2} + 16 a + 15\right)\cdot 23^{87} + \left(20 a^{14} + 4 a^{13} + 10 a^{12} + 11 a^{11} + 7 a^{10} + 19 a^{9} + 22 a^{8} + 16 a^{7} + 6 a^{6} + 11 a^{5} + 7 a^{4} + 21 a^{3} + 15 a^{2} + 3 a + 14\right)\cdot 23^{88} + \left(7 a^{14} + 10 a^{13} + 10 a^{12} + 21 a^{11} + 12 a^{10} + 10 a^{9} + 17 a^{8} + 7 a^{7} + 19 a^{5} + 3 a^{4} + 11 a^{3} + a^{2} + 7 a + 7\right)\cdot 23^{89} + \left(6 a^{14} + 19 a^{13} + 17 a^{12} + 11 a^{11} + 22 a^{10} + 8 a^{9} + 15 a^{8} + 3 a^{7} + 16 a^{6} + 15 a^{5} + 19 a^{4} + 2 a^{3} + 10 a^{2} + 11 a + 16\right)\cdot 23^{90} + \left(20 a^{14} + 2 a^{13} + 18 a^{12} + 11 a^{11} + 18 a^{10} + 22 a^{9} + 21 a^{8} + 15 a^{7} + 12 a^{6} + 21 a^{5} + 19 a^{4} + 12 a^{3} + 4 a^{2} + 8 a + 3\right)\cdot 23^{91} + \left(12 a^{14} + 13 a^{13} + 12 a^{12} + 16 a^{11} + 15 a^{10} + 13 a^{9} + 18 a^{8} + 18 a^{7} + 12 a^{6} + 5 a^{5} + 20 a^{4} + 17 a^{3} + 10 a^{2} + a + 15\right)\cdot 23^{92} + \left(16 a^{14} + 6 a^{13} + 19 a^{11} + 5 a^{10} + 15 a^{9} + 20 a^{8} + 14 a^{7} + 6 a^{6} + 3 a^{5} + 3 a^{3} + 22 a^{2} + 12 a\right)\cdot 23^{93} + \left(4 a^{14} + 10 a^{13} + 20 a^{12} + 6 a^{11} + 17 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a^{2} + 20 a + 22\right)\cdot 23^{111} + \left(11 a^{14} + 8 a^{13} + 18 a^{12} + 20 a^{11} + 12 a^{10} + 20 a^{9} + 3 a^{8} + 19 a^{7} + 6 a^{6} + 15 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} + 4 a + 1\right)\cdot 23^{112} + \left(14 a^{14} + a^{13} + 13 a^{12} + 21 a^{11} + 19 a^{10} + 7 a^{9} + 10 a^{8} + 16 a^{7} + 7 a^{6} + 15 a^{5} + a^{4} + 17 a^{3} + 8 a^{2} + 20 a + 1\right)\cdot 23^{113} + \left(12 a^{14} + 20 a^{13} + 18 a^{12} + 22 a^{11} + 20 a^{10} + 17 a^{9} + 6 a^{8} + 3 a^{7} + 21 a^{6} + 6 a^{5} + 9 a^{4} + 13 a^{3} + 12 a^{2} + 15 a + 21\right)\cdot 23^{114} + \left(15 a^{14} + 4 a^{13} + 15 a^{12} + 5 a^{11} + 11 a^{10} + 12 a^{9} + 18 a^{8} + 9 a^{7} + 10 a^{6} + 18 a^{5} + 6 a^{4} + 11 a^{3} + 22 a^{2} + 17 a + 7\right)\cdot 23^{115} + \left(7 a^{14} + 5 a^{13} + 4 a^{12} + 16 a^{11} + 20 a^{10} + 16 a^{9} + 7 a^{8} + 6 a^{7} + 14 a^{6} + 2 a^{5} + 4 a^{4} + 3 a^{3} + 22 a^{2} + 9 a + 9\right)\cdot 23^{116} + \left(14 a^{14} + 19 a^{12} + 11 a^{11} + 14 a^{10} + a^{9} + 7 a^{7} + 20 a^{6} + 2 a^{5} + 18 a^{4} + 4 a^{3} + 5 a^{2} + 22 a + 4\right)\cdot 23^{117} + \left(11 a^{14} + 17 a^{13} + 10 a^{12} + 19 a^{11} + 21 a^{10} + 11 a^{9} + 11 a^{7} + 14 a^{6} + 9 a^{5} + 11 a^{4} + a^{3} + 10 a^{2} + 7 a + 22\right)\cdot 23^{118} + \left(5 a^{14} + 10 a^{13} + 12 a^{12} + 11 a^{11} + 11 a^{10} + 15 a^{9} + 4 a^{8} + 14 a^{7} + 5 a^{6} + 10 a^{5} + a^{4} + 6 a^{3} + 20 a^{2} + 21 a + 5\right)\cdot 23^{119} + \left(10 a^{14} + 19 a^{13} + 21 a^{12} + 19 a^{11} + 8 a^{10} + 5 a^{9} + 5 a^{8} + 20 a^{7} + 5 a^{6} + 9 a^{5} + 22 a^{4} + 13 a^{3} + 3 a^{2} + 3 a + 5\right)\cdot 23^{120} + \left(20 a^{14} + 13 a^{13} + a^{12} + 12 a^{11} + 2 a^{10} + 20 a^{9} + 3 a^{8} + 15 a^{7} + 21 a^{6} + 5 a^{5} + 19 a^{4} + 5 a^{3} + 21 a^{2} + 9 a + 2\right)\cdot 23^{121} + \left(3 a^{14} + 19 a^{13} + 11 a^{12} + 22 a^{11} + 4 a^{10} + 8 a^{9} + 19 a^{8} + 18 a^{7} + 20 a^{6} + 13 a^{5} + 20 a^{4} + 13 a^{3} + 9 a^{2} + 21 a + 21\right)\cdot 23^{122} + \left(9 a^{14} + 22 a^{13} + 9 a^{11} + 3 a^{10} + 20 a^{9} + 15 a^{8} + 22 a^{7} + 10 a^{6} + 13 a^{5} + 20 a^{4} + 3 a^{3} + 5 a^{2} + 13 a + 9\right)\cdot 23^{123} + \left(8 a^{14} + 3 a^{13} + 5 a^{12} + 2 a^{11} + 22 a^{10} + 9 a^{9} + 9 a^{8} + 12 a^{7} + 4 a^{6} + 5 a^{5} + 22 a^{4} + 22 a^{3} + 13 a^{2} + 14 a + 11\right)\cdot 23^{124} + \left(8 a^{14} + 2 a^{13} + 17 a^{12} + 7 a^{11} + 3 a^{10} + 21 a^{9} + 11 a^{8} + 22 a^{7} + 2 a^{6} + 4 a^{5} + 14 a^{4} + 15 a^{3} + 8 a^{2} + 20 a + 10\right)\cdot 23^{125} + \left(19 a^{14} + 6 a^{13} + 19 a^{12} + 5 a^{11} + 2 a^{10} + 8 a^{9} + 2 a^{8} + 9 a^{7} + 5 a^{6} + 8 a^{5} + 5 a^{4} + 19 a^{3} + 3 a^{2} + 7 a + 18\right)\cdot 23^{126} + \left(18 a^{14} + 9 a^{13} + 16 a^{12} + 15 a^{11} + 17 a^{10} + 7 a^{9} + 15 a^{8} + 16 a^{7} + 2 a^{6} + 21 a^{5} + 2 a^{4} + 13 a^{3} + 18 a^{2} + 18 a + 14\right)\cdot 23^{127} + \left(6 a^{14} + 18 a^{13} + 11 a^{12} + 13 a^{11} + 17 a^{10} + 3 a^{9} + 7 a^{8} + 10 a^{7} + 18 a^{6} + 11 a^{4} + 5 a^{3} + 3 a^{2} + 8 a + 2\right)\cdot 23^{128} + \left(4 a^{14} + 6 a^{13} + 9 a^{12} + 19 a^{11} + 7 a^{10} + 9 a^{9} + 22 a^{8} + 13 a^{7} + 3 a^{6} + 9 a^{5} + 9 a^{4} + 7 a^{3} + 15 a + 10\right)\cdot 23^{129} + \left(3 a^{14} + 14 a^{13} + 15 a^{12} + 14 a^{11} + 16 a^{10} + 10 a^{9} + 22 a^{8} + 8 a^{7} + 2 a^{6} + 14 a^{5} + 18 a^{4} + 17 a^{3} + 20 a^{2} + 12 a + 3\right)\cdot 23^{130} + \left(5 a^{14} + 6 a^{12} + 20 a^{11} + 6 a^{10} + 16 a^{9} + 9 a^{8} + 19 a^{7} + 2 a^{6} + a^{5} + 2 a^{4} + 4 a^{3} + 11 a^{2} + 22 a + 19\right)\cdot 23^{131} + \left(5 a^{14} + 13 a^{13} + 21 a^{12} + 9 a^{11} + 12 a^{10} + 17 a^{9} + 20 a^{8} + 14 a^{7} + 7 a^{6} + 18 a^{5} + 22 a^{4} + 13 a^{3} + 9 a^{2} + 9 a + 9\right)\cdot 23^{132} + \left(13 a^{14} + 21 a^{13} + 9 a^{12} + 19 a^{11} + 5 a^{10} + 3 a^{9} + 15 a^{7} + 12 a^{6} + 9 a^{5} + 17 a^{4} + 2 a^{3} + 19 a^{2} + 21 a + 7\right)\cdot 23^{133} + \left(9 a^{13} + 9 a^{12} + 20 a^{11} + 8 a^{10} + 11 a^{9} + 10 a^{8} + 6 a^{7} + 7 a^{6} + 5 a^{5} + 22 a^{4} + 17 a^{3} + 11 a^{2} + 15 a + 22\right)\cdot 23^{134} + \left(9 a^{14} + 22 a^{13} + 16 a^{12} + 17 a^{11} + a^{10} + a^{9} + 14 a^{8} + 15 a^{7} + 15 a^{6} + 11 a^{5} + a^{4} + 8 a^{3} + 2 a^{2} + 20 a + 10\right)\cdot 23^{135} + \left(5 a^{14} + 12 a^{13} + 19 a^{12} + 7 a^{11} + a^{10} + 3 a^{9} + 4 a^{8} + 20 a^{7} + 18 a^{6} + 6 a^{5} + 18 a^{4} + 11 a^{3} + 14 a^{2} + 22 a + 20\right)\cdot 23^{136} + \left(19 a^{14} + 19 a^{13} + 6 a^{12} + 14 a^{11} + 16 a^{10} + 17 a^{9} + 13 a^{8} + 22 a^{7} + 5 a^{6} + 16 a^{5} + 19 a^{4} + 19 a^{3} + 17 a^{2} + 9 a + 6\right)\cdot 23^{137} + \left(2 a^{14} + 4 a^{13} + 4 a^{12} + 14 a^{11} + 15 a^{10} + 11 a^{9} + 17 a^{8} + 3 a^{7} + 6 a^{6} + 19 a^{5} + 19 a^{4} + 14 a^{3} + 22 a^{2} + 4 a + 18\right)\cdot 23^{138} + \left(8 a^{14} + 10 a^{13} + 5 a^{12} + 8 a^{11} + 15 a^{10} + 16 a^{9} + 5 a^{8} + 5 a^{7} + 15 a^{6} + 8 a^{5} + 13 a^{4} + 5 a^{3} + 13 a^{2} + 15 a + 9\right)\cdot 23^{139} + \left(11 a^{14} + 19 a^{13} + 21 a^{11} + a^{10} + 3 a^{9} + 5 a^{8} + 20 a^{7} + 5 a^{6} + 8 a^{5} + 8 a^{4} + 5 a^{3} + 9 a^{2} + 5 a + 15\right)\cdot 23^{140} + \left(11 a^{14} + 13 a^{13} + 20 a^{12} + 18 a^{11} + 7 a^{10} + 16 a^{9} + 18 a^{8} + 12 a^{7} + 2 a^{6} + 7 a^{5} + 18 a^{4} + 17 a^{3} + 5 a^{2} + 2 a + 20\right)\cdot 23^{141} + \left(18 a^{14} + 11 a^{13} + 3 a^{11} + 8 a^{10} + 17 a^{9} + 11 a^{8} + 8 a^{7} + 9 a^{6} + 7 a^{5} + 11 a^{4} + 3 a^{3} + 10 a^{2} + 17 a + 16\right)\cdot 23^{142} + \left(6 a^{14} + 15 a^{13} + 17 a^{12} + 4 a^{11} + a^{10} + 12 a^{9} + 21 a^{6} + 2 a^{5} + 4 a^{4} + 15 a^{3} + 21 a^{2} + 18 a + 2\right)\cdot 23^{143} + \left(17 a^{14} + a^{13} + 12 a^{11} + 18 a^{10} + 4 a^{8} + 10 a^{7} + 12 a^{6} + 8 a^{5} + 14 a^{4} + 12 a^{3} + 14 a^{2} + 4 a + 21\right)\cdot 23^{144} + \left(a^{14} + 22 a^{13} + 17 a^{12} + 12 a^{11} + 5 a^{10} + 16 a^{9} + 3 a^{8} + 20 a^{7} + 18 a^{6} + 17 a^{5} + 16 a^{4} + 15 a^{3} + 19 a^{2} + 11 a + 10\right)\cdot 23^{145} + \left(18 a^{14} + 8 a^{13} + 3 a^{12} + 10 a^{11} + 19 a^{10} + 8 a^{9} + 15 a^{8} + 14 a^{7} + a^{6} + 5 a^{5} + 6 a^{4} + 19 a^{3} + 4 a^{2} + 19 a + 19\right)\cdot 23^{146} + \left(6 a^{14} + 18 a^{12} + 21 a^{10} + 10 a^{9} + 17 a^{8} + 17 a^{7} + 21 a^{6} + 22 a^{5} + 12 a^{4} + 17 a^{3} + 13 a^{2} + 5 a + 9\right)\cdot 23^{147} + \left(9 a^{14} + 6 a^{13} + 19 a^{12} + 10 a^{11} + 14 a^{10} + 22 a^{9} + 16 a^{8} + 12 a^{7} + 22 a^{6} + 7 a^{5} + 15 a^{4} + 16 a^{3} + 11 a + 13\right)\cdot 23^{148} + \left(22 a^{14} + 14 a^{13} + 2 a^{12} + 9 a^{11} + 20 a^{9} + 18 a^{8} + 6 a^{7} + 12 a^{6} + 16 a^{5} + 18 a^{4} + 15 a^{3} + 11 a^{2} + 14 a + 5\right)\cdot 23^{149} + \left(a^{14} + 20 a^{13} + 15 a^{12} + 9 a^{11} + 6 a^{10} + 9 a^{9} + 10 a^{8} + 20 a^{7} + 9 a^{6} + 7 a^{5} + 14 a^{4} + 16 a^{3} + 15 a^{2} + 19 a + 1\right)\cdot 23^{150} + \left(2 a^{14} + 18 a^{13} + 10 a^{12} + 8 a^{11} + a^{10} + 12 a^{9} + 12 a^{8} + a^{7} + 8 a^{6} + 14 a^{5} + 12 a^{3} + 8 a^{2} + 19 a + 8\right)\cdot 23^{151} + \left(13 a^{14} + 5 a^{13} + 20 a^{12} + a^{11} + 10 a^{10} + 19 a^{9} + 21 a^{8} + 18 a^{7} + 8 a^{6} + 4 a^{5} + 11 a^{4} + 2 a^{3} + 11 a^{2} + 15\right)\cdot 23^{152} + \left(3 a^{14} + 10 a^{13} + 21 a^{12} + 9 a^{11} + 21 a^{10} + 4 a^{9} + 6 a^{8} + 2 a^{7} + 19 a^{6} + 7 a^{4} + 11 a^{3} + 16 a^{2} + 8 a + 20\right)\cdot 23^{153} + \left(18 a^{14} + 14 a^{13} + 22 a^{12} + 3 a^{11} + 6 a^{10} + 19 a^{9} + 3 a^{8} + 14 a^{7} + 12 a^{6} + 3 a^{5} + 16 a^{4} + 4 a^{3} + 12 a^{2} + 3 a + 13\right)\cdot 23^{154} + \left(a^{14} + 3 a^{12} + 5 a^{11} + 10 a^{10} + 18 a^{9} + 7 a^{8} + 20 a^{7} + 6 a^{6} + 15 a^{5} + 22 a^{4} + 17 a^{3} + 6 a^{2} + 11 a + 12\right)\cdot 23^{155} + \left(16 a^{14} + 13 a^{13} + 4 a^{12} + a^{11} + 15 a^{10} + 4 a^{9} + 3 a^{8} + 2 a^{7} + 5 a^{6} + 19 a^{5} + 13 a^{3} + 12 a^{2} + 14 a + 5\right)\cdot 23^{156} + \left(3 a^{14} + 3 a^{13} + 6 a^{12} + 19 a^{11} + 14 a^{9} + 20 a^{8} + 20 a^{7} + 20 a^{6} + 18 a^{5} + 17 a^{4} + 14 a^{3} + a^{2} + 12 a + 7\right)\cdot 23^{157} + \left(15 a^{14} + 2 a^{13} + 11 a^{12} + 8 a^{11} + 12 a^{10} + a^{9} + 8 a^{8} + 13 a^{7} + 8 a^{6} + 22 a^{5} + 2 a^{4} + 5 a^{3} + 16 a^{2} + 19 a + 4\right)\cdot 23^{158} + \left(17 a^{14} + 13 a^{13} + 20 a^{12} + 20 a^{11} + 15 a^{10} + 8 a^{8} + 13 a^{7} + a^{6} + a^{5} + 21 a^{4} + 15 a^{3} + 17 a^{2} + 3 a + 11\right)\cdot 23^{159} + \left(10 a^{14} + 2 a^{13} + 20 a^{12} + 18 a^{11} + 2 a^{10} + 18 a^{9} + 16 a^{8} + 8 a^{7} + 9 a^{6} + 19 a^{5} + 14 a^{4} + 9 a^{3} + 17 a^{2} + 5 a + 13\right)\cdot 23^{160} + \left(6 a^{14} + 4 a^{13} + 12 a^{12} + 14 a^{11} + 13 a^{10} + 18 a^{9} + 17 a^{8} + 17 a^{7} + 8 a^{6} + 21 a^{5} + 11 a^{4} + 12 a^{3} + 7 a^{2} + 11 a + 16\right)\cdot 23^{161} + \left(8 a^{14} + 8 a^{13} + a^{12} + 10 a^{11} + 15 a^{10} + 15 a^{9} + 11 a^{8} + 2 a^{7} + 8 a^{6} + 6 a^{5} + 8 a^{4} + 7 a^{3} + 20 a^{2} + 20 a + 18\right)\cdot 23^{162} + \left(8 a^{14} + 8 a^{13} + 19 a^{12} + 17 a^{11} + a^{10} + 3 a^{9} + 15 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7 a^{13} + 10 a^{12} + 8 a^{11} + 4 a^{10} + 18 a^{9} + 22 a^{8} + 13 a^{7} + 13 a^{6} + 16 a^{5} + 22 a^{4} + 12 a^{3} + 9 a^{2} + 21\right)\cdot 23^{169} + \left(5 a^{14} + a^{13} + 20 a^{12} + 14 a^{11} + 21 a^{10} + 15 a^{9} + 16 a^{8} + 3 a^{7} + 4 a^{6} + 17 a^{5} + 18 a^{4} + 8 a^{3} + 22 a^{2} + 10 a + 17\right)\cdot 23^{170} + \left(3 a^{14} + 10 a^{13} + a^{12} + 6 a^{11} + 18 a^{10} + 6 a^{9} + 10 a^{8} + 2 a^{7} + 21 a^{6} + 6 a^{5} + 22 a^{3} + 22 a^{2} + 2 a + 6\right)\cdot 23^{171} + \left(19 a^{14} + 10 a^{13} + 22 a^{10} + 21 a^{9} + 14 a^{8} + 6 a^{7} + 10 a^{6} + 16 a^{5} + 6 a^{4} + 12 a^{3} + 4 a^{2} + 17 a\right)\cdot 23^{172} + \left(20 a^{13} + 11 a^{12} + 4 a^{11} + 10 a^{10} + 11 a^{9} + 16 a^{8} + 18 a^{7} + 12 a^{6} + 20 a^{5} + 22 a^{4} + 5 a^{3} + 3 a^{2} + 21 a + 3\right)\cdot 23^{173} + \left(12 a^{14} + 10 a^{13} + a^{12} + 2 a^{11} + 18 a^{9} + 12 a^{8} + 5 a^{7} + a^{6} + 17 a^{5} + 4 a^{4} + 3 a^{3} + 8 a^{2} + 5 a + 5\right)\cdot 23^{174} + \left(7 a^{14} + 4 a^{13} + 3 a^{12} + 8 a^{11} + 7 a^{10} + 10 a^{9} + 16 a^{7} + 5 a^{6} + 19 a^{5} + 22 a^{4} + 12 a^{3} + 21 a^{2} + 5 a + 18\right)\cdot 23^{175} + \left(8 a^{14} + 5 a^{13} + 10 a^{12} + 14 a^{11} + 16 a^{10} + 20 a^{9} + 8 a^{8} + 8 a^{7} + 3 a^{6} + 15 a^{5} + 18 a^{4} + 11 a^{3} + 16 a^{2} + 15 a + 20\right)\cdot 23^{176} + \left(20 a^{13} + 10 a^{12} + 13 a^{11} + 9 a^{10} + 17 a^{9} + 14 a^{8} + 2 a^{7} + 2 a^{6} + 9 a^{5} + 8 a^{4} + 16 a^{3} + a^{2} + 3 a + 20\right)\cdot 23^{177} + \left(a^{14} + 22 a^{13} + 18 a^{12} + 20 a^{11} + 17 a^{10} + 22 a^{9} + 22 a^{7} + 14 a^{6} + 20 a^{5} + 3 a^{4} + 19 a^{3} + 6 a^{2} + 5 a + 3\right)\cdot 23^{178} + \left(19 a^{14} + 16 a^{13} + 22 a^{12} + 8 a^{11} + 2 a^{10} + 5 a^{9} + 15 a^{7} + 5 a^{6} + 13 a^{5} + 8 a^{4} + 13 a^{3} + 12 a^{2} + 10 a + 13\right)\cdot 23^{179} + \left(11 a^{14} + 18 a^{13} + 19 a^{11} + a^{10} + 7 a^{9} + 16 a^{8} + 20 a^{7} + a^{6} + 5 a^{5} + 9 a^{4} + 15 a^{3} + 2 a^{2} + 9 a + 6\right)\cdot 23^{180} + \left(17 a^{14} + 20 a^{13} + 22 a^{12} + 2 a^{11} + 3 a^{10} + 11 a^{9} + 18 a^{8} + 7 a^{7} + 3 a^{6} + 11 a^{5} + 6 a^{4} + 20 a^{3} + 16 a^{2} + 21 a + 1\right)\cdot 23^{181} + \left(2 a^{14} + 9 a^{13} + 18 a^{12} + 8 a^{11} + 17 a^{10} + 10 a^{9} + 12 a^{8} + 15 a^{7} + 20 a^{6} + 13 a^{5} + 4 a^{4} + 19 a^{3} + 18 a^{2} + 15 a + 11\right)\cdot 23^{182} + \left(14 a^{14} + a^{13} + 13 a^{12} + 7 a^{11} + 13 a^{10} + 16 a^{9} + 20 a^{8} + 9 a^{7} + 15 a^{6} + 3 a^{5} + 2 a^{4} + 15 a^{3} + 14 a^{2} + 7 a + 22\right)\cdot 23^{183} + \left(14 a^{14} + 10 a^{13} + 4 a^{12} + 13 a^{11} + 17 a^{10} + 4 a^{9} + 19 a^{8} + 15 a^{7} + 13 a^{6} + 7 a^{5} + 8 a^{4} + 12 a^{3} + 7 a^{2} + 11 a + 15\right)\cdot 23^{184} + \left(14 a^{14} + 6 a^{13} + 11 a^{12} + 4 a^{11} + 19 a^{10} + 18 a^{9} + 7 a^{8} + 9 a^{7} + 6 a^{6} + 12 a^{5} + 6 a^{4} + 8 a^{3} + 14 a^{2} + 14 a + 1\right)\cdot 23^{185} + \left(6 a^{14} + 9 a^{13} + 2 a^{11} + 7 a^{10} + 20 a^{9} + 3 a^{8} + 9 a^{7} + 4 a^{6} + 14 a^{5} + 3 a^{4} + 16 a^{3} + 8 a + 10\right)\cdot 23^{186} + \left(6 a^{14} + a^{13} + 20 a^{12} + 17 a^{11} + 10 a^{10} + 13 a^{9} + 11 a^{8} + 3 a^{7} + 12 a^{6} + 10 a^{5} + 22 a^{4} + 8 a^{3} + 8 a^{2} + 7 a + 8\right)\cdot 23^{187} + \left(11 a^{14} + 3 a^{13} + 5 a^{12} + 17 a^{11} + 15 a^{10} + 7 a^{9} + 13 a^{8} + 9 a^{5} + 19 a^{4} + 21 a^{3} + 2 a^{2} + 5 a + 13\right)\cdot 23^{188} + \left(13 a^{14} + 15 a^{13} + 5 a^{11} + 21 a^{10} + 4 a^{9} + 21 a^{8} + 20 a^{7} + 12 a^{6} + 9 a^{5} + 3 a^{4} + 22 a^{3} + 18 a^{2} + 18 a + 14\right)\cdot 23^{189} + \left(7 a^{14} + 8 a^{13} + 14 a^{12} + 14 a^{11} + 18 a^{10} + 12 a^{9} + 11 a^{8} + 17 a^{7} + 3 a^{6} + 6 a^{5} + 22 a^{4} + 13 a^{2} + 21 a + 15\right)\cdot 23^{190} + \left(a^{14} + 14 a^{13} + 13 a^{12} + 6 a^{11} + 22 a^{10} + 7 a^{9} + 6 a^{8} + 19 a^{7} + 6 a^{6} + 3 a^{5} + 20 a^{4} + a^{3} + 2 a^{2} + 7 a + 7\right)\cdot 23^{191} + \left(20 a^{14} + 6 a^{13} + 15 a^{12} + a^{11} + 15 a^{10} + 9 a^{9} + 20 a^{8} + 5 a^{7} + 10 a^{6} + 14 a^{5} + 21 a^{4} + 2 a^{3} + 7 a^{2} + 19 a + 22\right)\cdot 23^{192} + \left(9 a^{14} + 13 a^{12} + 13 a^{11} + 8 a^{10} + 4 a^{9} + 19 a^{8} + 2 a^{7} + 5 a^{6} + 5 a^{5} + 16 a^{4} + 18 a^{3} + 20 a^{2} + 18 a + 1\right)\cdot 23^{193} + \left(16 a^{14} + 18 a^{13} + 8 a^{12} + 19 a^{11} + 14 a^{10} + 11 a^{8} + 7 a^{7} + 22 a^{6} + 9 a^{5} + 17 a^{4} + 2 a^{3} + 17 a^{2} + 6 a + 1\right)\cdot 23^{194} + \left(12 a^{13} + 13 a^{12} + 18 a^{11} + 9 a^{10} + a^{9} + 7 a^{8} + 14 a^{7} + 12 a^{6} + 17 a^{5} + 17 a^{4} + a^{3} + 16 a^{2} + 22\right)\cdot 23^{195} + \left(15 a^{14} + 12 a^{13} + 5 a^{12} + 2 a^{11} + 3 a^{10} + 22 a^{9} + 3 a^{8} + 22 a^{7} + 10 a^{6} + 7 a^{5} + a^{4} + 16 a^{3} + 6 a^{2} + 18 a + 16\right)\cdot 23^{196} + \left(21 a^{14} + 5 a^{13} + 22 a^{12} + 8 a^{11} + 5 a^{9} + 22 a^{6} + 19 a^{5} + 5 a^{3} + 19 a^{2} + 18 a + 18\right)\cdot 23^{197} + \left(19 a^{14} + 19 a^{13} + 16 a^{12} + 18 a^{11} + 11 a^{10} + 12 a^{9} + 17 a^{8} + 15 a^{7} + 4 a^{6} + 13 a^{5} + 15 a^{4} + 17 a^{3} + 11 a^{2} + 3\right)\cdot 23^{198} + \left(8 a^{14} + 19 a^{13} + 13 a^{12} + 13 a^{11} + 8 a^{10} + 20 a^{9} + 17 a^{8} + 7 a^{7} + 11 a^{6} + 4 a^{5} + 8 a^{4} + 17 a^{3} + 3 a^{2} + 7 a + 16\right)\cdot 23^{199} + \left(14 a^{14} + 22 a^{13} + 12 a^{12} + 17 a^{11} + 20 a^{10} + 2 a^{9} + 6 a^{8} + 5 a^{7} + 12 a^{6} + 17 a^{5} + 13 a^{4} + 15 a^{3} + 20 a^{2} + 21 a + 15\right)\cdot 23^{200} + \left(8 a^{14} + 14 a^{13} + 10 a^{12} + 7 a^{11} + 15 a^{10} + 14 a^{9} + 13 a^{8} + 3 a^{7} + 5 a^{6} + 5 a^{5} + 3 a^{4} + 19 a^{3} + 11 a^{2} + a + 21\right)\cdot 23^{201} + \left(16 a^{14} + 2 a^{13} + 3 a^{11} + 6 a^{10} + 10 a^{9} + 6 a^{8} + 15 a^{7} + 17 a^{6} + 6 a^{5} + 4 a^{4} + 16 a^{3} + 6 a^{2} + 9 a + 6\right)\cdot 23^{202} + \left(3 a^{14} + 21 a^{13} + 11 a^{12} + 15 a^{11} + 3 a^{10} + 16 a^{9} + 14 a^{8} + 6 a^{7} + 9 a^{6} + 18 a^{5} + 22 a^{3} + 9 a^{2} + 16 a + 1\right)\cdot 23^{203} + \left(20 a^{14} + 9 a^{12} + 9 a^{11} + 5 a^{10} + 8 a^{8} + 12 a^{7} + 14 a^{6} + 3 a^{5} + 11 a^{4} + 3 a^{3} + 3 a^{2} + 16 a + 19\right)\cdot 23^{204} + \left(a^{14} + 2 a^{13} + 8 a^{12} + 3 a^{11} + 12 a^{10} + 11 a^{9} + 20 a^{8} + 22 a^{7} + 5 a^{6} + 18 a^{5} + 22 a^{4} + 17 a^{3} + 21 a^{2} + 16 a + 13\right)\cdot 23^{205} + \left(a^{14} + 16 a^{13} + 8 a^{12} + 3 a^{11} + 7 a^{10} + 14 a^{9} + 13 a^{8} + 22 a^{7} + 7 a^{6} + 22 a^{5} + 6 a^{3} + 8 a^{2} + 10 a + 12\right)\cdot 23^{206} + \left(2 a^{14} + 13 a^{12} + 15 a^{11} + 21 a^{10} + 18 a^{8} + 17 a^{7} + 18 a^{6} + 20 a^{5} + 17 a^{4} + 21 a^{3} + 15 a^{2} + 19 a + 16\right)\cdot 23^{207} + \left(19 a^{14} + 16 a^{13} + 18 a^{12} + 10 a^{11} + 13 a^{10} + 9 a^{9} + a^{8} + 21 a^{7} + 17 a^{6} + 20 a^{4} + 18 a^{3} + 20 a^{2} + 3 a + 7\right)\cdot 23^{208} + \left(14 a^{14} + 2 a^{13} + 17 a^{12} + 5 a^{11} + 9 a^{10} + 3 a^{9} + 21 a^{8} + 8 a^{7} + 2 a^{6} + 17 a^{5} + 18 a^{4} + 11 a^{3} + 14 a^{2} + 6 a + 18\right)\cdot 23^{209} + \left(8 a^{14} + 5 a^{13} + 7 a^{12} + 21 a^{11} + 3 a^{10} + 19 a^{9} + 6 a^{7} + a^{6} + 14 a^{5} + 22 a^{4} + 9 a^{3} + 15 a^{2} + 13 a + 11\right)\cdot 23^{210} + \left(7 a^{14} + 4 a^{13} + 11 a^{12} + 17 a^{11} + 10 a^{10} + 2 a^{9} + 4 a^{8} + 5 a^{6} + 9 a^{5} + 18 a^{4} + 17 a^{3} + 17 a^{2} + 7 a + 5\right)\cdot 23^{211} + \left(10 a^{14} + 18 a^{13} + 20 a^{12} + 19 a^{11} + 4 a^{10} + 7 a^{9} + 22 a^{8} + 21 a^{7} + 10 a^{6} + 21 a^{5} + 5 a^{4} + 4 a^{3} + 8 a^{2} + 4 a + 18\right)\cdot 23^{212} + \left(18 a^{14} + 22 a^{13} + 5 a^{12} + 14 a^{11} + 20 a^{10} + 4 a^{9} + 19 a^{8} + 18 a^{7} + 3 a^{6} + 10 a^{5} + 5 a^{4} + 2 a^{3} + 17 a^{2} + 8 a + 9\right)\cdot 23^{213} + \left(11 a^{14} + 6 a^{13} + 10 a^{12} + 20 a^{11} + 13 a^{10} + 5 a^{9} + 22 a^{8} + 9 a^{7} + 15 a^{5} + 13 a^{4} + 17 a^{3} + 19\right)\cdot 23^{214} + \left(18 a^{14} + 9 a^{13} + 9 a^{12} + 14 a^{11} + 2 a^{10} + 9 a^{9} + 3 a^{6} + 17 a^{5} + 15 a^{4} + a^{3} + 21 a^{2} + 14 a + 14\right)\cdot 23^{215} + \left(17 a^{14} + 12 a^{13} + 21 a^{12} + 4 a^{11} + 4 a^{10} + 2 a^{9} + 8 a^{8} + 9 a^{7} + 19 a^{6} + 15 a^{5} + 14 a^{4} + 13 a^{3} + 10 a^{2} + a + 4\right)\cdot 23^{216} + \left(15 a^{14} + 3 a^{13} + 16 a^{12} + 10 a^{11} + 6 a^{10} + 15 a^{9} + 3 a^{8} + 12 a^{7} + 18 a^{6} + 7 a^{5} + 6 a^{4} + 14 a^{3} + 8 a^{2} + 5 a + 3\right)\cdot 23^{217} + \left(13 a^{14} + 16 a^{13} + 3 a^{12} + 18 a^{11} + 22 a^{10} + 18 a^{8} + 22 a^{7} + 12 a^{6} + 9 a^{5} + 20 a^{4} + 21 a^{3} + 18 a^{2} + 6 a + 13\right)\cdot 23^{218} + \left(13 a^{14} + 9 a^{13} + 18 a^{12} + 11 a^{11} + 8 a^{9} + 19 a^{8} + 13 a^{7} + 20 a^{6} + 16 a^{5} + 14 a^{4} + 9 a^{3} + 19 a + 11\right)\cdot 23^{219} + \left(10 a^{14} + 17 a^{13} + 9 a^{12} + 4 a^{11} + 11 a^{10} + 14 a^{9} + 14 a^{8} + 2 a^{7} + 3 a^{6} + 3 a^{5} + 20 a^{4} + 14 a^{3} + 17 a^{2} + 22 a + 3\right)\cdot 23^{220} + \left(21 a^{14} + 4 a^{13} + 15 a^{12} + 7 a^{10} + 20 a^{9} + 6 a^{8} + 5 a^{7} + 11 a^{6} + 19 a^{5} + 22 a^{4} + 22 a^{2} + 19 a + 20\right)\cdot 23^{221} + \left(11 a^{14} + 9 a^{13} + 3 a^{12} + 12 a^{11} + 8 a^{10} + 3 a^{9} + 9 a^{8} + 17 a^{7} + a^{6} + 13 a^{5} + 11 a^{4} + 21 a^{3} + 4 a^{2} + 13\right)\cdot 23^{222} + \left(15 a^{14} + 4 a^{13} + 7 a^{12} + 15 a^{11} + 2 a^{10} + 10 a^{9} + 2 a^{8} + 21 a^{7} + 17 a^{6} + 9 a^{5} + 8 a^{4} + 5 a^{3} + 12 a^{2} + 7 a + 6\right)\cdot 23^{223} + \left(12 a^{14} + 12 a^{13} + 4 a^{12} + 8 a^{11} + 12 a^{10} + 17 a^{9} + 15 a^{7} + 22 a^{6} + 14 a^{5} + 17 a^{4} + 12 a^{3} + 12 a^{2} + 18 a + 4\right)\cdot 23^{224} + \left(9 a^{14} + 8 a^{13} + 18 a^{12} + 5 a^{11} + 22 a^{10} + 20 a^{9} + 2 a^{8} + 10 a^{6} + 16 a^{5} + 2 a^{4} + 12 a^{3} + 5 a^{2} + 11 a\right)\cdot 23^{225} + \left(7 a^{14} + 11 a^{13} + 7 a^{12} + 20 a^{11} + 3 a^{10} + 8 a^{9} + 2 a^{8} + a^{7} + 17 a^{6} + 12 a^{5} + 16 a^{4} + 14 a^{3} + 14 a^{2} + 10\right)\cdot 23^{226} + \left(7 a^{14} + 19 a^{13} + 16 a^{12} + 18 a^{11} + 4 a^{10} + 4 a^{9} + 9 a^{8} + 5 a^{7} + 5 a^{6} + 7 a^{5} + 10 a^{4} + 3 a^{3} + 17 a^{2} + 3 a + 9\right)\cdot 23^{227} + \left(12 a^{14} + 22 a^{13} + 14 a^{12} + 14 a^{11} + 5 a^{10} + 13 a^{8} + 18 a^{7} + 5 a^{6} + a^{5} + 3 a^{4} + 21 a^{3} + 2 a^{2} + 2 a + 17\right)\cdot 23^{228} + \left(16 a^{14} + 20 a^{13} + a^{12} + 14 a^{11} + 19 a^{10} + 12 a^{9} + 12 a^{8} + 13 a^{7} + 19 a^{6} + 8 a^{5} + 22 a^{4} + 17 a^{3} + 20 a^{2} + 9 a + 7\right)\cdot 23^{229} + \left(6 a^{14} + 7 a^{13} + 9 a^{12} + 9 a^{11} + a^{10} + 8 a^{9} + 4 a^{8} + 6 a^{7} + 3 a^{6} + a^{5} + 19 a^{4} + a^{3} + 17 a^{2} + 7 a + 5\right)\cdot 23^{230} + \left(4 a^{14} + 4 a^{13} + 7 a^{12} + 5 a^{11} + 15 a^{10} + 12 a^{9} + 14 a^{8} + 21 a^{7} + 11 a^{6} + 12 a^{5} + 14 a^{4} + 2 a^{3} + 4 a^{2} + 10 a + 4\right)\cdot 23^{231} + \left(10 a^{14} + 22 a^{13} + 11 a^{12} + 22 a^{11} + 5 a^{9} + 12 a^{8} + 17 a^{7} + 17 a^{6} + 6 a^{5} + 12 a^{4} + 2 a^{3} + 8 a^{2} + 11 a + 9\right)\cdot 23^{232} + \left(16 a^{13} + 4 a^{12} + 9 a^{11} + 21 a^{10} + 18 a^{8} + 9 a^{7} + 17 a^{6} + 8 a^{5} + a^{4} + 22 a^{3} + 22 a^{2} + 10 a + 4\right)\cdot 23^{233} + \left(18 a^{14} + 11 a^{13} + 4 a^{12} + 18 a^{11} + 20 a^{10} + 2 a^{9} + 19 a^{8} + 22 a^{6} + 11 a^{5} + 2 a^{4} + 16 a^{3} + 19 a^{2} + a + 22\right)\cdot 23^{234} + \left(a^{14} + 22 a^{13} + 10 a^{12} + 6 a^{11} + 11 a^{10} + 3 a^{9} + 10 a^{8} + 15 a^{7} + 22 a^{6} + 5 a^{5} + 22 a^{4} + 6 a^{3} + 17 a^{2} + 14 a + 22\right)\cdot 23^{235} + \left(12 a^{14} + 4 a^{13} + 2 a^{12} + 20 a^{11} + 8 a^{10} + 14 a^{9} + 3 a^{8} + 18 a^{7} + 14 a^{6} + 9 a^{5} + 15 a^{4} + 21 a^{3} + 10 a^{2} + 19 a + 11\right)\cdot 23^{236} + \left(9 a^{14} + 13 a^{13} + 5 a^{12} + 13 a^{11} + 21 a^{10} + 9 a^{9} + 19 a^{8} + 17 a^{7} + 6 a^{6} + 18 a^{5} + 3 a^{4} + 9 a^{3} + 9 a^{2} + 12 a + 5\right)\cdot 23^{237} + \left(9 a^{14} + 18 a^{12} + 7 a^{11} + 20 a^{10} + 13 a^{9} + 2 a^{8} + 14 a^{7} + 19 a^{6} + 11 a^{5} + 20 a^{4} + 16 a^{3} + 2 a^{2} + 5 a + 20\right)\cdot 23^{238} + \left(16 a^{14} + 19 a^{13} + 7 a^{12} + 9 a^{11} + 8 a^{10} + 6 a^{9} + 5 a^{8} + 13 a^{7} + 9 a^{6} + 16 a^{5} + 7 a^{4} + 22 a^{3} + 11 a^{2} + 5 a + 12\right)\cdot 23^{239} + \left(22 a^{14} + 9 a^{13} + 6 a^{12} + 14 a^{11} + 10 a^{10} + 12 a^{9} + 18 a^{8} + 7 a^{7} + 13 a^{6} + 5 a^{5} + 7 a^{4} + 9 a^{3} + 4 a^{2} + 2 a + 22\right)\cdot 23^{240} + \left(4 a^{14} + 10 a^{13} + 11 a^{12} + 15 a^{11} + 2 a^{10} + 7 a^{9} + 14 a^{8} + 4 a^{7} + 14 a^{6} + 21 a^{5} + 16 a^{4} + 12 a^{3} + 4 a^{2} + 12 a + 15\right)\cdot 23^{241} + \left(7 a^{14} + 2 a^{13} + 4 a^{12} + 7 a^{11} + 18 a^{10} + 14 a^{9} + 3 a^{8} + 3 a^{7} + a^{5} + 6 a^{4} + 4 a^{3} + 22 a^{2} + 8 a + 3\right)\cdot 23^{242} + \left(18 a^{14} + a^{13} + 21 a^{12} + 16 a^{11} + a^{9} + 14 a^{8} + 7 a^{7} + a^{6} + 18 a^{5} + 9 a^{4} + 19 a^{3} + 21 a^{2} + 19 a + 22\right)\cdot 23^{243} + \left(14 a^{14} + 19 a^{13} + 19 a^{12} + 9 a^{11} + 10 a^{10} + 8 a^{9} + 7 a^{8} + 20 a^{7} + 20 a^{5} + 3 a^{4} + 20 a^{3} + 11 a^{2} + 19 a + 9\right)\cdot 23^{244} + \left(18 a^{14} + 22 a^{13} + 9 a^{12} + 9 a^{11} + 2 a^{10} + 6 a^{9} + 5 a^{8} + 21 a^{7} + 21 a^{6} + 18 a^{5} + 15 a^{4} + 3 a^{3} + 7 a^{2} + 13 a + 9\right)\cdot 23^{245} + \left(17 a^{14} + 9 a^{13} + 10 a^{12} + 20 a^{11} + a^{10} + 11 a^{9} + 21 a^{8} + 2 a^{7} + 17 a^{6} + 21 a^{5} + 13 a^{4} + 9 a^{3} + 18 a^{2} + 16 a + 6\right)\cdot 23^{246} + \left(21 a^{14} + 10 a^{13} + 15 a^{12} + 6 a^{11} + 20 a^{10} + 6 a^{9} + 20 a^{8} + 22 a^{7} + 6 a^{6} + 4 a^{5} + 12 a^{4} + 5 a^{3} + 9 a^{2} + 2 a + 10\right)\cdot 23^{247} + \left(18 a^{14} + 11 a^{13} + 16 a^{12} + 20 a^{11} + 10 a^{10} + 16 a^{9} + 19 a^{8} + 6 a^{7} + 21 a^{5} + 11 a^{4} + 9 a^{3} + 6 a^{2} + 2 a + 20\right)\cdot 23^{248} + \left(10 a^{14} + 18 a^{13} + 9 a^{12} + 17 a^{11} + 4 a^{10} + a^{9} + 12 a^{8} + 4 a^{7} + 7 a^{6} + 2 a^{5} + 11 a^{4} + 5 a^{3} + 10 a^{2} + 18 a + 15\right)\cdot 23^{249} + \left(21 a^{14} + 10 a^{13} + 14 a^{12} + 7 a^{11} + 19 a^{10} + 11 a^{9} + 2 a^{8} + 12 a^{7} + 20 a^{6} + 6 a^{5} + 11 a^{4} + 15 a^{3} + 19 a^{2} + 8 a + 9\right)\cdot 23^{250} + \left(a^{14} + 18 a^{13} + 5 a^{12} + 6 a^{11} + 19 a^{10} + 12 a^{9} + 18 a^{8} + 8 a^{7} + 16 a^{6} + 17 a^{5} + 12 a^{4} + 15 a^{3} + 18 a^{2} + 17 a + 12\right)\cdot 23^{251} + \left(8 a^{14} + 21 a^{13} + 21 a^{12} + 15 a^{11} + 4 a^{10} + 7 a^{9} + 5 a^{8} + 12 a^{7} + 3 a^{6} + 8 a^{5} + 16 a^{4} + 15 a^{3} + 22 a^{2} + 8 a + 20\right)\cdot 23^{252} + \left(7 a^{14} + 22 a^{13} + 15 a^{12} + 10 a^{11} + a^{10} + 12 a^{9} + 16 a^{8} + 6 a^{7} + 3 a^{6} + 8 a^{5} + 4 a^{4} + 21 a^{3} + 20 a^{2} + a + 17\right)\cdot 23^{253} + \left(14 a^{14} + 7 a^{13} + 18 a^{12} + 8 a^{11} + 9 a^{10} + 10 a^{9} + 11 a^{8} + 18 a^{7} + 5 a^{6} + 11 a^{5} + 21 a^{4} + 21 a^{3} + 9 a^{2} + 14 a + 20\right)\cdot 23^{254} + \left(17 a^{14} + 20 a^{13} + 15 a^{12} + 20 a^{11} + 10 a^{10} + 17 a^{9} + 12 a^{8} + 18 a^{7} + 14 a^{6} + a^{5} + 21 a^{4} + 14 a^{3} + 8 a^{2} + 14 a + 15\right)\cdot 23^{255} + \left(19 a^{14} + 9 a^{12} + 16 a^{11} + 16 a^{10} + 12 a^{9} + 6 a^{8} + 8 a^{7} + 12 a^{6} + 5 a^{5} + 16 a^{4} + 14 a^{3} + 6 a^{2} + 10 a + 11\right)\cdot 23^{256} + \left(5 a^{14} + 4 a^{13} + 12 a^{12} + 22 a^{11} + 11 a^{10} + 5 a^{9} + 3 a^{8} + 12 a^{7} + 2 a^{6} + 13 a^{5} + a^{4} + 18 a^{3} + 14 a^{2} + 12 a + 1\right)\cdot 23^{257} + \left(2 a^{14} + 22 a^{13} + 7 a^{12} + 19 a^{11} + 7 a^{10} + 12 a^{9} + 5 a^{7} + a^{6} + 7 a^{5} + 6 a^{4} + a^{3} + 22 a^{2} + 17 a + 20\right)\cdot 23^{258} + \left(17 a^{14} + 2 a^{13} + a^{12} + 21 a^{11} + 14 a^{10} + 19 a^{9} + 13 a^{8} + 9 a^{7} + 17 a^{6} + 5 a^{5} + 16 a^{4} + 5 a^{3} + 10 a^{2} + a + 17\right)\cdot 23^{259} + \left(11 a^{14} + 4 a^{13} + a^{12} + 9 a^{11} + 22 a^{10} + 3 a^{9} + 19 a^{8} + 3 a^{7} + 12 a^{5} + 22 a^{4} + 17 a^{3} + 11 a^{2} + a + 3\right)\cdot 23^{260} + \left(3 a^{14} + 5 a^{13} + 12 a^{12} + 13 a^{11} + 19 a^{10} + 10 a^{9} + 14 a^{8} + 8 a^{7} + 22 a^{6} + 9 a^{5} + 2 a^{4} + 6 a^{3} + 15 a^{2} + 16 a + 13\right)\cdot 23^{261} + \left(4 a^{14} + 8 a^{13} + 14 a^{12} + 2 a^{11} + 21 a^{10} + 7 a^{9} + 14 a^{8} + 16 a^{7} + 19 a^{6} + 22 a^{5} + 13 a^{4} + 16 a^{3} + 16 a^{2} + 16 a + 1\right)\cdot 23^{262} + \left(20 a^{14} + 9 a^{13} + 13 a^{12} + 22 a^{11} + 18 a^{10} + 18 a^{9} + 8 a^{8} + 16 a^{7} + 17 a^{6} + 21 a^{4} + 22 a^{3} + 3 a^{2} + 3 a\right)\cdot 23^{263} + \left(21 a^{14} + 9 a^{13} + 20 a^{12} + 13 a^{11} + 16 a^{10} + 4 a^{9} + 15 a^{8} + 11 a^{7} + 3 a^{6} + 20 a^{5} + 12 a^{4} + 10 a^{3} + 12 a^{2} + 22\right)\cdot 23^{264} + \left(21 a^{13} + 6 a^{12} + 12 a^{11} + 21 a^{10} + 20 a^{9} + 2 a^{8} + 7 a^{7} + 10 a^{6} + 3 a^{5} + 16 a^{4} + 13 a^{3} + 2 a^{2} + 5 a + 6\right)\cdot 23^{265} + \left(19 a^{14} + 21 a^{13} + 17 a^{12} + a^{11} + 6 a^{10} + 3 a^{9} + 17 a^{8} + 17 a^{7} + 5 a^{6} + 15 a^{5} + 22 a^{4} + 4 a^{3} + 11 a^{2} + 13\right)\cdot 23^{266} + \left(8 a^{14} + a^{13} + 22 a^{12} + 9 a^{11} + 13 a^{10} + 12 a^{9} + 10 a^{8} + 16 a^{7} + 7 a^{6} + 16 a^{5} + 5 a^{4} + 22 a^{3} + 6 a + 9\right)\cdot 23^{267} + \left(6 a^{14} + 12 a^{13} + 3 a^{12} + 21 a^{11} + 2 a^{10} + 15 a^{9} + 7 a^{8} + 2 a^{7} + 10 a^{6} + 13 a^{5} + 19 a^{4} + 22 a^{3} + 9 a^{2} + 2 a + 2\right)\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 9 }$ $=$ $8 a^{13} + 16 a^{12} + 15 a^{11} + 2 a^{10} + 18 a^{9} + 4 a^{8} + 3 a^{7} + 17 a^{6} + 9 a^{5} + 18 a^{4} + 9 a^{3} + 5 a^{2} + 8 + \left(9 a^{14} + 13 a^{13} + 21 a^{12} + 2 a^{11} + 19 a^{10} + 12 a^{9} + 11 a^{8} + 10 a^{7} + 14 a^{6} + 4 a^{5} + 7 a^{4} + 20 a^{3} + 6 a^{2} + 21 a + 21\right)\cdot 23 + \left(5 a^{14} + 13 a^{13} + 4 a^{12} + 10 a^{11} + 10 a^{10} + 20 a^{9} + 5 a^{8} + 15 a^{7} + 7 a^{5} + 19 a^{4} + 14 a^{3} + 11 a^{2} + 3 a + 16\right)\cdot 23^{2} + \left(4 a^{14} + 15 a^{13} + 2 a^{12} + 21 a^{11} + 14 a^{10} + 21 a^{9} + 5 a^{8} + 18 a^{7} + a^{6} + 14 a^{5} + 6 a^{4} + 2 a^{3} + 17 a^{2} + 11 a + 9\right)\cdot 23^{3} + \left(16 a^{14} + 6 a^{13} + 6 a^{12} + 10 a^{11} + 14 a^{10} + 8 a^{9} + 14 a^{8} + 3 a^{7} + 17 a^{6} + 16 a^{5} + 4 a^{4} + 4 a^{3} + 17 a^{2} + 20 a + 1\right)\cdot 23^{4} + \left(9 a^{14} + 14 a^{13} + 13 a^{12} + 11 a^{11} + 7 a^{10} + 18 a^{9} + 11 a^{8} + 3 a^{7} + 20 a^{6} + 8 a^{5} + 12 a^{4} + 10 a^{3} + 15 a^{2} + 6 a + 19\right)\cdot 23^{5} + \left(3 a^{13} + 5 a^{12} + 6 a^{11} + 16 a^{10} + 18 a^{9} + 5 a^{8} + 15 a^{7} + 12 a^{6} + 11 a^{5} + 13 a^{4} + 8 a^{3} + 8 a^{2} + 22 a + 16\right)\cdot 23^{6} + \left(10 a^{14} + 14 a^{13} + 16 a^{12} + 16 a^{11} + 11 a^{10} + 18 a^{9} + 10 a^{8} + a^{7} + 4 a^{6} + 7 a^{5} + 14 a^{4} + 4 a^{3} + 4 a^{2} + 2 a\right)\cdot 23^{7} + \left(14 a^{14} + 15 a^{13} + 10 a^{11} + 10 a^{10} + 2 a^{9} + 12 a^{8} + 9 a^{6} + 12 a^{5} + 11 a^{4} + 20 a^{3} + 18 a^{2} + 6 a + 8\right)\cdot 23^{8} + \left(a^{14} + 3 a^{13} + 7 a^{12} + 16 a^{11} + 13 a^{10} + 9 a^{9} + 11 a^{8} + 21 a^{7} + 4 a^{6} + 21 a^{5} + 21 a^{4} + 21 a^{3} + 5 a^{2} + 2 a + 17\right)\cdot 23^{9} + \left(10 a^{14} + 2 a^{13} + 3 a^{12} + 17 a^{10} + 8 a^{9} + a^{7} + 20 a^{6} + 8 a^{5} + 22 a^{4} + a^{3} + 16 a^{2} + 8 a + 19\right)\cdot 23^{10} + \left(15 a^{14} + 11 a^{12} + 9 a^{11} + 6 a^{10} + 15 a^{8} + 15 a^{7} + 12 a^{6} + 10 a^{5} + 3 a^{4} + 2 a^{3} + 3 a^{2} + 4 a\right)\cdot 23^{11} + \left(7 a^{14} + 9 a^{13} + 21 a^{12} + 15 a^{11} + 4 a^{10} + 9 a^{9} + 4 a^{8} + 5 a^{7} + 14 a^{6} + 20 a^{5} + 19 a^{4} + 10 a^{3} + 17 a^{2} + 7 a + 4\right)\cdot 23^{12} + \left(12 a^{14} + 5 a^{13} + a^{12} + 22 a^{11} + 12 a^{10} + 21 a^{9} + 13 a^{8} + 13 a^{7} + 19 a^{6} + 4 a^{5} + 15 a^{4} + 18 a^{3} + a^{2} + 21 a + 6\right)\cdot 23^{13} + \left(a^{14} + 20 a^{13} + 22 a^{12} + 17 a^{11} + 15 a^{10} + 7 a^{9} + 10 a^{8} + 4 a^{7} + 15 a^{6} + 10 a^{5} + 2 a^{4} + 9 a^{3} + 14 a^{2} + 6 a + 13\right)\cdot 23^{14} + \left(a^{14} + 13 a^{13} + 7 a^{12} + a^{11} + 9 a^{10} + 14 a^{9} + 16 a^{8} + 13 a^{7} + 5 a^{6} + 2 a^{5} + 10 a^{4} + 12 a^{3} + 19 a^{2} + 11 a + 21\right)\cdot 23^{15} + \left(11 a^{14} + 9 a^{13} + 7 a^{12} + 4 a^{11} + 16 a^{10} + 8 a^{9} + 7 a^{8} + 22 a^{7} + 10 a^{6} + a^{5} + a^{4} + 13 a^{3} + 21 a^{2} + 18 a + 5\right)\cdot 23^{16} + \left(2 a^{14} + 16 a^{13} + 2 a^{12} + 13 a^{11} + 6 a^{10} + 17 a^{9} + 12 a^{8} + 4 a^{7} + 11 a^{6} + 21 a^{5} + 3 a^{4} + 21 a^{2} + 6 a + 12\right)\cdot 23^{17} + \left(10 a^{14} + 10 a^{13} + 11 a^{12} + 3 a^{11} + 20 a^{10} + 16 a^{9} + 11 a^{8} + 3 a^{7} + 11 a^{6} + 12 a^{5} + 5 a^{4} + 3 a^{3} + 14 a^{2} + 4 a + 20\right)\cdot 23^{18} + \left(20 a^{14} + 7 a^{13} + 13 a^{12} + a^{11} + 17 a^{10} + 2 a^{9} + 2 a^{8} + 19 a^{7} + 10 a^{6} + 20 a^{5} + a^{4} + 9 a^{3} + 7 a^{2} + 13 a + 22\right)\cdot 23^{19} + \left(10 a^{14} + 20 a^{13} + 10 a^{12} + 15 a^{11} + 6 a^{10} + 11 a^{9} + 14 a^{8} + 12 a^{7} + 4 a^{6} + 2 a^{5} + 19 a^{4} + 21 a^{3} + 18 a^{2} + 3 a + 7\right)\cdot 23^{20} + \left(20 a^{14} + 4 a^{13} + 22 a^{12} + 13 a^{11} + 8 a^{10} + 6 a^{9} + 19 a^{8} + 8 a^{7} + 17 a^{6} + 14 a^{5} + 19 a^{4} + 3 a^{3} + 12 a^{2} + 19 a + 15\right)\cdot 23^{21} + \left(21 a^{14} + 19 a^{13} + 3 a^{12} + 4 a^{11} + 6 a^{10} + a^{9} + 5 a^{8} + 2 a^{7} + 20 a^{6} + 18 a^{5} + 8 a^{4} + 15 a^{3} + 19 a^{2} + 20 a + 5\right)\cdot 23^{22} + \left(11 a^{14} + 13 a^{13} + 4 a^{12} + 17 a^{11} + 18 a^{10} + 12 a^{9} + 17 a^{8} + 21 a^{7} + 18 a^{6} + 2 a^{5} + 8 a^{4} + 21 a^{3} + 3 a + 4\right)\cdot 23^{23} + \left(9 a^{14} + 12 a^{13} + 20 a^{12} + 12 a^{11} + 4 a^{10} + 10 a^{9} + 20 a^{8} + 19 a^{7} + 13 a^{6} + 16 a^{5} + 3 a^{4} + 7 a^{3} + 21 a^{2} + 20 a + 16\right)\cdot 23^{24} + \left(22 a^{13} + 14 a^{12} + 22 a^{11} + 9 a^{10} + 9 a^{9} + 22 a^{8} + 4 a^{7} + 3 a^{6} + 20 a^{4} + 16 a^{3} + 12 a^{2} + 10 a + 12\right)\cdot 23^{25} + \left(5 a^{14} + 7 a^{13} + 15 a^{11} + 7 a^{10} + 2 a^{9} + 8 a^{8} + 18 a^{7} + 5 a^{6} + 18 a^{5} + 15 a^{4} + 21 a^{3} + 4 a^{2} + 13 a + 10\right)\cdot 23^{26} + \left(10 a^{14} + 6 a^{13} + 3 a^{12} + 16 a^{11} + 20 a^{10} + 2 a^{9} + a^{8} + 11 a^{7} + 10 a^{6} + 9 a^{5} + 4 a^{4} + 6 a^{3} + 12 a^{2} + 2 a\right)\cdot 23^{27} + \left(20 a^{14} + 11 a^{13} + 9 a^{12} + 16 a^{11} + 18 a^{10} + 18 a^{9} + 22 a^{8} + 10 a^{7} + a^{6} + 17 a^{5} + 14 a^{4} + 22 a^{3} + 18 a^{2} + 14 a + 17\right)\cdot 23^{28} + \left(13 a^{14} + 8 a^{13} + 11 a^{12} + 5 a^{11} + 15 a^{10} + 7 a^{9} + 17 a^{8} + 7 a^{7} + 12 a^{6} + 12 a^{5} + 2 a^{4} + 21 a^{3} + 16 a^{2} + 4 a + 8\right)\cdot 23^{29} + \left(4 a^{14} + a^{13} + a^{12} + 22 a^{11} + 10 a^{10} + 16 a^{9} + 12 a^{8} + 21 a^{7} + 11 a^{6} + 13 a^{5} + 8 a^{4} + 5 a^{3} + 12 a^{2} + 5 a + 15\right)\cdot 23^{30} + \left(21 a^{14} + 20 a^{13} + 11 a^{12} + 19 a^{11} + 12 a^{10} + 20 a^{9} + 6 a^{8} + 4 a^{7} + 19 a^{6} + 8 a^{5} + 22 a^{4} + 6 a^{3} + 9 a^{2} + 22 a + 15\right)\cdot 23^{31} + \left(12 a^{14} + 17 a^{13} + 11 a^{12} + 11 a^{11} + 7 a^{10} + 11 a^{9} + 4 a^{8} + 8 a^{7} + 7 a^{6} + 9 a^{5} + 3 a^{4} + 6 a^{3} + 22 a^{2} + a + 7\right)\cdot 23^{32} + \left(16 a^{14} + 2 a^{13} + 21 a^{12} + 4 a^{10} + 10 a^{9} + 11 a^{8} + 4 a^{7} + 22 a^{6} + a^{5} + 3 a^{4} + 22 a^{2} + 19 a\right)\cdot 23^{33} + \left(4 a^{14} + 18 a^{13} + 10 a^{12} + 16 a^{11} + 18 a^{10} + 14 a^{9} + 18 a^{8} + 11 a^{7} + 15 a^{6} + 21 a^{5} + 19 a^{4} + 15 a^{3} + 3 a^{2} + 20 a + 7\right)\cdot 23^{34} + \left(11 a^{14} + 19 a^{13} + 8 a^{12} + 4 a^{11} + 10 a^{10} + 5 a^{9} + 6 a^{8} + 15 a^{7} + 10 a^{6} + 12 a^{5} + 7 a^{4} + 15 a^{2} + 11 a + 5\right)\cdot 23^{35} + \left(16 a^{14} + 10 a^{13} + 16 a^{12} + 15 a^{11} + 17 a^{10} + 6 a^{9} + 5 a^{8} + 5 a^{7} + 21 a^{5} + 13 a^{4} + 21 a^{3} + 22 a^{2} + 22 a + 1\right)\cdot 23^{36} + \left(8 a^{14} + 9 a^{13} + 16 a^{12} + 14 a^{11} + 22 a^{10} + 10 a^{9} + 19 a^{8} + 12 a^{7} + 12 a^{6} + 2 a^{5} + 9 a^{4} + 20 a^{3} + 22 a^{2} + 10 a + 3\right)\cdot 23^{37} + \left(6 a^{14} + 18 a^{13} + 10 a^{12} + 9 a^{11} + 4 a^{10} + a^{9} + 17 a^{8} + 22 a^{7} + 14 a^{6} + a^{5} + 9 a^{4} + 5 a^{3} + 6 a^{2} + 3\right)\cdot 23^{38} + \left(4 a^{14} + 11 a^{13} + 21 a^{12} + 17 a^{11} + 16 a^{10} + 14 a^{9} + 11 a^{8} + 22 a^{7} + 3 a^{5} + 16 a^{4} + 4 a^{3} + 2 a^{2} + 19 a + 8\right)\cdot 23^{39} + \left(20 a^{14} + 14 a^{13} + 18 a^{11} + 22 a^{10} + 14 a^{9} + a^{8} + 9 a^{7} + 11 a^{6} + 13 a^{5} + 20 a^{4} + 14 a^{3} + 7 a^{2} + 3 a + 15\right)\cdot 23^{40} + \left(9 a^{14} + 4 a^{13} + 19 a^{12} + 22 a^{11} + 4 a^{9} + 13 a^{8} + 15 a^{7} + 4 a^{6} + 10 a^{5} + 15 a^{4} + 7 a^{3} + 21 a^{2} + 11 a + 10\right)\cdot 23^{41} + \left(13 a^{14} + 11 a^{13} + 18 a^{12} + 7 a^{11} + 5 a^{10} + 12 a^{9} + 12 a^{8} + 19 a^{7} + 3 a^{6} + 3 a^{5} + 21 a^{4} + 21 a^{3} + 7 a^{2} + 21 a + 21\right)\cdot 23^{42} + \left(13 a^{13} + 6 a^{12} + 4 a^{11} + 2 a^{9} + 14 a^{8} + 5 a^{7} + 15 a^{6} + 6 a^{5} + 12 a^{4} + 21 a^{3} + 5 a^{2} + 18 a + 7\right)\cdot 23^{43} + \left(8 a^{14} + 16 a^{13} + 7 a^{12} + 6 a^{11} + 4 a^{10} + 22 a^{9} + 12 a^{7} + 14 a^{6} + 21 a^{5} + 14 a^{4} + 13 a^{3} + 2 a^{2} + 8 a + 18\right)\cdot 23^{44} + \left(18 a^{14} + 20 a^{13} + 15 a^{12} + 5 a^{11} + 12 a^{10} + 13 a^{9} + 21 a^{8} + 7 a^{7} + 14 a^{6} + 5 a^{5} + 14 a^{4} + 7 a^{3} + 17 a^{2} + 12 a + 22\right)\cdot 23^{45} + \left(4 a^{14} + 18 a^{13} + 13 a^{12} + 12 a^{11} + 19 a^{10} + 19 a^{9} + 12 a^{8} + 7 a^{7} + 13 a^{6} + 15 a^{5} + 2 a^{4} + 7 a^{3} + 8 a^{2} + 16 a + 9\right)\cdot 23^{46} + \left(7 a^{14} + 6 a^{13} + 14 a^{12} + a^{11} + 14 a^{10} + 15 a^{9} + 19 a^{8} + 19 a^{7} + 7 a^{6} + 13 a^{5} + 7 a^{4} + 6 a^{3} + a^{2} + 9 a + 11\right)\cdot 23^{47} + \left(14 a^{14} + 15 a^{13} + 16 a^{12} + 14 a^{11} + 5 a^{10} + 3 a^{9} + 9 a^{8} + 19 a^{7} + 22 a^{6} + 7 a^{5} + 11 a^{4} + 22 a^{3} + 13 a^{2} + 10 a + 12\right)\cdot 23^{48} + \left(16 a^{14} + 5 a^{13} + 18 a^{12} + 22 a^{11} + 15 a^{9} + 10 a^{8} + 5 a^{7} + 22 a^{6} + 4 a^{5} + 13 a^{4} + 7 a^{3} + 13 a^{2} + 11 a + 15\right)\cdot 23^{49} + \left(12 a^{14} + a^{13} + 15 a^{12} + 8 a^{11} + 22 a^{10} + 7 a^{9} + 14 a^{7} + a^{6} + 9 a^{4} + 2 a^{3} + 12 a^{2} + 1\right)\cdot 23^{50} + \left(22 a^{14} + 6 a^{13} + 3 a^{12} + 5 a^{11} + 6 a^{10} + 3 a^{9} + 8 a^{8} + 13 a^{7} + 17 a^{6} + 4 a^{5} + 19 a^{4} + 14 a^{2} + 14 a + 13\right)\cdot 23^{51} + \left(21 a^{14} + 18 a^{13} + 20 a^{12} + 10 a^{11} + 22 a^{9} + 12 a^{8} + a^{7} + 20 a^{6} + 17 a^{5} + 17 a^{4} + 21 a^{3} + 21 a^{2} + a + 3\right)\cdot 23^{52} + \left(6 a^{14} + 17 a^{13} + 6 a^{12} + 7 a^{11} + 15 a^{10} + 2 a^{9} + 2 a^{8} + 7 a^{7} + 4 a^{6} + 13 a^{4} + 12 a^{3} + 18 a^{2} + a + 15\right)\cdot 23^{53} + \left(4 a^{14} + 3 a^{13} + 18 a^{12} + 15 a^{11} + 7 a^{10} + 20 a^{9} + 5 a^{8} + 22 a^{7} + 12 a^{6} + 20 a^{5} + 14 a^{4} + a^{3} + 2 a^{2} + 20 a + 1\right)\cdot 23^{54} + \left(11 a^{14} + 20 a^{13} + 2 a^{12} + 13 a^{11} + 16 a^{10} + 17 a^{9} + 17 a^{8} + 3 a^{7} + 16 a^{6} + 18 a^{5} + 11 a^{4} + 5 a^{3} + 17 a^{2} + 14 a + 4\right)\cdot 23^{55} + \left(22 a^{14} + 17 a^{13} + 16 a^{12} + 5 a^{11} + 18 a^{10} + 4 a^{9} + 3 a^{8} + 17 a^{7} + 17 a^{6} + 11 a^{5} + 20 a^{4} + 16 a^{3} + 20 a^{2} + 8 a + 14\right)\cdot 23^{56} + \left(14 a^{14} + 18 a^{13} + 5 a^{12} + 17 a^{11} + 12 a^{10} + 4 a^{9} + 3 a^{8} + 14 a^{7} + 7 a^{6} + 18 a^{5} + a^{4} + 10 a^{3} + 13 a^{2} + 5 a + 3\right)\cdot 23^{57} + \left(a^{14} + 13 a^{13} + 18 a^{12} + 8 a^{11} + 12 a^{10} + 11 a^{9} + 14 a^{8} + a^{6} + 13 a^{5} + 9 a^{4} + a^{3} + 6 a^{2} + 5 a + 6\right)\cdot 23^{58} + \left(10 a^{14} + 11 a^{13} + 10 a^{12} + 5 a^{11} + 4 a^{10} + 3 a^{9} + 2 a^{8} + 4 a^{7} + 2 a^{6} + 14 a^{5} + 10 a^{4} + 2 a^{3} + 14 a^{2} + 9 a + 11\right)\cdot 23^{59} + \left(2 a^{14} + 8 a^{13} + 16 a^{12} + 20 a^{11} + 6 a^{10} + 2 a^{9} + 21 a^{8} + 4 a^{7} + 2 a^{6} + 18 a^{5} + 20 a^{4} + 7 a^{3} + 18 a^{2} + 20 a + 7\right)\cdot 23^{60} + \left(18 a^{14} + 15 a^{13} + 15 a^{12} + 8 a^{11} + 6 a^{10} + 18 a^{9} + 15 a^{8} + 14 a^{7} + a^{6} + 20 a^{5} + 21 a^{4} + 17 a^{3} + 6 a^{2} + a + 6\right)\cdot 23^{61} + \left(17 a^{14} + 21 a^{13} + 17 a^{12} + 7 a^{11} + 22 a^{10} + 2 a^{9} + 6 a^{8} + 10 a^{7} + a^{6} + 7 a^{5} + 11 a^{4} + 19 a^{3} + 13 a^{2} + 3 a + 6\right)\cdot 23^{62} + \left(13 a^{13} + 5 a^{12} + 2 a^{11} + 19 a^{10} + 8 a^{9} + 8 a^{8} + 4 a^{7} + a^{6} + 15 a^{5} + 15 a^{4} + 12 a^{3} + 17 a^{2} + 18 a + 3\right)\cdot 23^{63} + \left(22 a^{14} + 22 a^{13} + 16 a^{12} + 8 a^{10} + 16 a^{9} + 21 a^{8} + 5 a^{7} + 3 a^{6} + 5 a^{5} + 8 a^{4} + 12 a^{3} + 11 a + 12\right)\cdot 23^{64} + \left(20 a^{14} + 21 a^{13} + 5 a^{12} + 3 a^{11} + a^{10} + 5 a^{9} + 14 a^{7} + 17 a^{6} + 10 a^{5} + 7 a^{4} + 20 a^{3} + 8 a^{2} + 11\right)\cdot 23^{65} + \left(7 a^{14} + 19 a^{13} + 14 a^{12} + 6 a^{11} + 19 a^{10} + 2 a^{9} + 14 a^{8} + 11 a^{7} + a^{6} + 17 a^{5} + 5 a^{4} + 2 a^{3} + 8 a^{2} + 12 a + 6\right)\cdot 23^{66} + \left(7 a^{14} + 16 a^{13} + 11 a^{12} + 5 a^{11} + 7 a^{10} + 21 a^{9} + a^{8} + 7 a^{7} + 18 a^{6} + 10 a^{5} + 3 a^{4} + 13 a^{3} + 19 a^{2} + a + 6\right)\cdot 23^{67} + \left(18 a^{14} + 17 a^{13} + 5 a^{12} + 12 a^{11} + 11 a^{10} + 21 a^{9} + 19 a^{8} + 8 a^{7} + 13 a^{6} + 12 a^{5} + 10 a^{4} + 8 a^{3} + 13 a^{2} + 18 a + 21\right)\cdot 23^{68} + \left(11 a^{14} + 10 a^{13} + 14 a^{12} + 16 a^{11} + 13 a^{10} + 3 a^{9} + 7 a^{8} + 9 a^{7} + 15 a^{6} + 21 a^{5} + 18 a^{4} + 5 a^{3} + 11 a^{2} + 11 a + 7\right)\cdot 23^{69} + \left(12 a^{14} + 13 a^{13} + 4 a^{12} + 8 a^{11} + 20 a^{10} + 15 a^{9} + 8 a^{8} + 13 a^{7} + 21 a^{6} + 20 a^{5} + 18 a^{4} + 8 a^{3} + 18 a^{2} + 3 a + 13\right)\cdot 23^{70} + \left(15 a^{14} + 15 a^{13} + 8 a^{12} + 11 a^{11} + 3 a^{10} + 8 a^{8} + a^{7} + 11 a^{6} + 3 a^{5} + 13 a^{4} + 16 a^{3} + 4 a^{2} + a\right)\cdot 23^{71} + \left(14 a^{14} + 22 a^{13} + 13 a^{11} + 17 a^{10} + 3 a^{9} + 8 a^{8} + 7 a^{7} + 17 a^{6} + 21 a^{5} + 20 a^{4} + 2 a^{3} + 13 a^{2} + 21 a + 22\right)\cdot 23^{72} + \left(21 a^{14} + a^{13} + 11 a^{12} + 13 a^{11} + 18 a^{10} + 5 a^{9} + 8 a^{8} + 3 a^{7} + 19 a^{6} + 6 a^{5} + 12 a^{4} + 22 a^{3} + 10 a^{2} + a + 4\right)\cdot 23^{73} + \left(6 a^{14} + 17 a^{13} + 18 a^{12} + 8 a^{11} + 13 a^{10} + 12 a^{8} + 20 a^{7} + 2 a^{6} + 17 a^{5} + 5 a^{4} + 6 a^{3} + 13 a^{2} + a + 22\right)\cdot 23^{74} + \left(10 a^{14} + 19 a^{13} + 6 a^{12} + 12 a^{11} + 10 a^{10} + 21 a^{9} + 4 a^{8} + 2 a^{7} + 4 a^{6} + a^{5} + 7 a^{4} + 15 a^{3} + 10 a^{2} + 14 a\right)\cdot 23^{75} + \left(8 a^{13} + a^{12} + 17 a^{11} + 6 a^{10} + 14 a^{9} + 15 a^{7} + 16 a^{6} + 12 a^{5} + 20 a^{4} + 9 a^{3} + 21 a^{2} + 10 a + 11\right)\cdot 23^{76} + \left(13 a^{14} + 9 a^{13} + 15 a^{12} + a^{11} + 3 a^{9} + 19 a^{8} + 9 a^{7} + 15 a^{6} + 17 a^{5} + 14 a^{4} + 10 a^{3} + 16 a^{2} + 15 a + 9\right)\cdot 23^{77} + \left(12 a^{14} + 8 a^{13} + 2 a^{12} + 16 a^{11} + a^{10} + 9 a^{9} + 16 a^{8} + 21 a^{7} + 5 a^{6} + 6 a^{5} + 13 a^{4} + 13 a^{3} + 10 a^{2} + 13 a + 16\right)\cdot 23^{78} + \left(3 a^{14} + 6 a^{13} + 3 a^{12} + 11 a^{11} + 11 a^{10} + 6 a^{9} + a^{8} + 20 a^{7} + 2 a^{6} + 7 a^{5} + 11 a^{4} + 2 a^{3} + 18 a^{2} + 7 a + 15\right)\cdot 23^{79} + \left(6 a^{14} + 3 a^{13} + 18 a^{12} + 22 a^{11} + a^{10} + 2 a^{9} + 17 a^{8} + 7 a^{7} + 5 a^{6} + 2 a^{5} + 15 a^{4} + 18 a^{3} + 5 a^{2} + 9 a + 10\right)\cdot 23^{80} + \left(12 a^{14} + 21 a^{13} + a^{12} + 17 a^{11} + 13 a^{10} + 15 a^{9} + 9 a^{8} + 21 a^{7} + 4 a^{6} + 21 a^{5} + 6 a^{4} + 9 a^{3} + 18 a^{2} + 19 a + 14\right)\cdot 23^{81} + \left(10 a^{14} + 11 a^{13} + 9 a^{12} + 9 a^{11} + a^{10} + 17 a^{9} + 22 a^{8} + 6 a^{7} + 8 a^{6} + 18 a^{5} + 7 a^{4} + 11 a^{3} + 19 a^{2} + 2 a + 22\right)\cdot 23^{82} + \left(15 a^{14} + 8 a^{13} + 12 a^{12} + 19 a^{11} + 14 a^{10} + 19 a^{9} + 18 a^{8} + 19 a^{7} + 4 a^{6} + 7 a^{5} + 6 a^{4} + 4 a^{3} + 21 a^{2} + 14 a + 20\right)\cdot 23^{83} + \left(a^{14} + 21 a^{13} + 16 a^{12} + 2 a^{11} + 8 a^{10} + 7 a^{8} + 12 a^{7} + 13 a^{6} + 21 a^{5} + 11 a^{4} + 19 a^{3} + 10 a^{2} + 16 a + 9\right)\cdot 23^{84} + \left(8 a^{14} + 5 a^{13} + 21 a^{12} + 17 a^{11} + 5 a^{10} + 19 a^{9} + 2 a^{8} + 20 a^{7} + 3 a^{5} + 22 a^{4} + 14 a^{3} + 9 a^{2} + 6 a + 12\right)\cdot 23^{85} + \left(12 a^{14} + 14 a^{13} + 16 a^{12} + 14 a^{10} + 22 a^{9} + 17 a^{8} + 9 a^{7} + 14 a^{6} + 3 a^{4} + 13 a^{3} + 3 a^{2} + 20 a + 12\right)\cdot 23^{86} + \left(7 a^{14} + 3 a^{13} + 6 a^{12} + 9 a^{11} + 4 a^{10} + 8 a^{9} + 6 a^{8} + a^{6} + 3 a^{5} + 4 a^{4} + 3 a^{3} + 11 a^{2} + 9 a + 9\right)\cdot 23^{87} + \left(17 a^{14} + 3 a^{13} + 6 a^{11} + 19 a^{10} + 10 a^{9} + 15 a^{8} + 3 a^{7} + 11 a^{6} + 17 a^{5} + 10 a^{4} + 2 a^{3} + 9 a^{2} + 12 a + 1\right)\cdot 23^{88} + \left(19 a^{14} + 13 a^{13} + 21 a^{12} + 19 a^{11} + 14 a^{10} + 14 a^{9} + 20 a^{8} + 8 a^{7} + 7 a^{6} + 10 a^{5} + 22 a^{4} + 16 a^{3} + 8 a^{2} + 2 a + 13\right)\cdot 23^{89} + \left(5 a^{14} + 8 a^{13} + 2 a^{12} + 10 a^{11} + 6 a^{10} + 17 a^{9} + 15 a^{8} + 7 a^{7} + 2 a^{6} + 20 a^{5} + 16 a^{4} + 16 a^{3} + a^{2} + 5 a + 21\right)\cdot 23^{90} + \left(18 a^{13} + 18 a^{12} + 15 a^{11} + 2 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14 a^{13} + a^{12} + 6 a^{11} + 15 a^{10} + 17 a^{9} + 17 a^{8} + 15 a^{7} + 5 a^{6} + 19 a^{4} + 13 a^{3} + 14 a^{2} + 18 a + 7\right)\cdot 23^{97} + \left(11 a^{14} + 22 a^{13} + 21 a^{12} + 21 a^{11} + 14 a^{10} + 15 a^{9} + 12 a^{7} + 16 a^{6} + 19 a^{5} + 14 a^{4} + 16 a^{3} + a^{2} + 7 a + 18\right)\cdot 23^{98} + \left(8 a^{14} + 18 a^{13} + 17 a^{12} + 7 a^{11} + 16 a^{10} + 5 a^{9} + 15 a^{8} + 4 a^{7} + 10 a^{6} + 2 a^{5} + 4 a^{4} + 5 a^{3} + 12 a^{2} + 20 a + 18\right)\cdot 23^{99} + \left(20 a^{13} + 6 a^{12} + 9 a^{11} + 17 a^{10} + 21 a^{9} + 3 a^{8} + a^{7} + 15 a^{6} + 8 a^{5} + 12 a^{2} + a + 8\right)\cdot 23^{100} + \left(19 a^{14} + 11 a^{13} + 5 a^{12} + a^{11} + 11 a^{10} + 8 a^{9} + a^{8} + 8 a^{7} + 18 a^{6} + 14 a^{5} + 13 a^{4} + 22 a^{3} + 14 a + 9\right)\cdot 23^{101} + \left(12 a^{14} + 20 a^{13} + 21 a^{12} + 22 a^{11} + 18 a^{10} + 22 a^{9} + 5 a^{8} + 3 a^{7} + 21 a^{6} + 18 a^{5} + 14 a^{4} + 17 a^{3} + 21 a^{2} + 10 a + 5\right)\cdot 23^{102} + 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19\right)\cdot 23^{108} + \left(12 a^{14} + 2 a^{13} + 8 a^{12} + 13 a^{11} + 3 a^{10} + 20 a^{9} + a^{8} + 11 a^{7} + 8 a^{6} + 21 a^{5} + 17 a^{4} + 8 a^{3} + 8 a^{2} + 14 a + 22\right)\cdot 23^{109} + \left(3 a^{14} + 20 a^{13} + a^{12} + 2 a^{11} + 15 a^{10} + 11 a^{9} + a^{8} + 10 a^{7} + 10 a^{6} + 17 a^{5} + 3 a^{4} + 6 a^{3} + 22 a^{2} + 9 a + 8\right)\cdot 23^{110} + \left(5 a^{14} + 19 a^{13} + 11 a^{12} + 8 a^{11} + 5 a^{10} + 14 a^{9} + 7 a^{8} + 6 a^{7} + 2 a^{6} + 12 a^{5} + 18 a^{4} + 5 a^{3} + 16 a^{2} + 7 a + 19\right)\cdot 23^{111} + \left(22 a^{14} + 10 a^{13} + 10 a^{12} + 3 a^{11} + 8 a^{10} + 8 a^{9} + 21 a^{8} + 6 a^{7} + 11 a^{6} + 10 a^{5} + 6 a^{4} + 7 a^{3} + 9 a^{2} + 16 a + 16\right)\cdot 23^{112} + \left(2 a^{13} + 21 a^{12} + 7 a^{11} + 16 a^{10} + 22 a^{9} + 7 a^{8} + 11 a^{7} + 14 a^{6} + 17 a^{5} + 15 a^{4} + a^{3} + 16 a^{2} + 22 a + 15\right)\cdot 23^{113} + \left(4 a^{14} + 15 a^{13} + 9 a^{12} + 19 a^{11} + 13 a^{10} + 3 a^{9} + 20 a^{8} + 18 a^{7} + 4 a^{6} + 3 a^{5} + 15 a^{4} + 5 a^{3} + 6 a^{2} + 4 a + 13\right)\cdot 23^{114} + \left(2 a^{14} + 22 a^{13} + 21 a^{12} + 12 a^{10} + 3 a^{9} + 16 a^{8} + 4 a^{7} + 17 a^{6} + 9 a^{5} + 3 a^{4} + 6 a^{3} + 17 a^{2} + 6\right)\cdot 23^{115} + \left(14 a^{14} + 14 a^{13} + 11 a^{12} + 9 a^{11} + 13 a^{10} + 2 a^{9} + 4 a^{8} + 21 a^{7} + 12 a^{6} + 2 a^{5} + 11 a^{4} + 18 a^{3} + 19 a^{2} + a + 7\right)\cdot 23^{116} + \left(11 a^{13} + 11 a^{12} + 7 a^{11} + 12 a^{10} + 17 a^{9} + 12 a^{8} + 15 a^{7} + a^{6} + 21 a^{5} + 5 a^{4} + 11 a^{3} + 9 a^{2} + 14 a + 10\right)\cdot 23^{117} + \left(2 a^{14} + 9 a^{13} + 16 a^{12} + 10 a^{11} + 5 a^{10} + 14 a^{9} + 21 a^{8} + 18 a^{7} + 13 a^{6} + 14 a^{5} + 3 a^{4} + a^{3} + 2 a^{2} + 4 a + 6\right)\cdot 23^{118} + \left(8 a^{14} + 21 a^{13} + 9 a^{12} + 19 a^{11} + 21 a^{10} + 16 a^{9} + 12 a^{8} + 11 a^{7} + 8 a^{6} + 19 a^{5} + 2 a^{4} + 10 a^{3} + 17 a^{2} + 14 a + 17\right)\cdot 23^{119} + \left(11 a^{14} + 20 a^{13} + 2 a^{12} + 21 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23^{131} + \left(7 a^{14} + 9 a^{13} + 10 a^{12} + 11 a^{11} + 19 a^{10} + 3 a^{9} + 22 a^{8} + 9 a^{7} + 22 a^{6} + 12 a^{5} + 5 a^{4} + 21 a^{3} + 16 a^{2} + 3 a + 7\right)\cdot 23^{132} + \left(19 a^{14} + 5 a^{13} + 11 a^{12} + 21 a^{11} + 5 a^{10} + 2 a^{9} + a^{8} + 14 a^{7} + 15 a^{6} + a^{5} + 8 a^{4} + 14 a^{3} + 7 a^{2} + 2 a + 11\right)\cdot 23^{133} + \left(15 a^{14} + 7 a^{13} + 21 a^{12} + 3 a^{11} + 16 a^{10} + a^{9} + 16 a^{8} + 3 a^{7} + 5 a^{6} + 9 a^{5} + 15 a^{4} + 4 a^{3} + 16 a^{2} + a + 14\right)\cdot 23^{134} + \left(11 a^{14} + 10 a^{13} + 15 a^{12} + 12 a^{11} + 3 a^{9} + 20 a^{8} + 15 a^{7} + 14 a^{6} + 12 a^{5} + 20 a^{4} + 15 a^{3} + 15 a^{2} + 3 a + 7\right)\cdot 23^{135} + \left(5 a^{14} + 2 a^{13} + 4 a^{12} + 12 a^{11} + 14 a^{10} + 4 a^{9} + 3 a^{8} + a^{7} + 18 a^{6} + 6 a^{5} + 19 a^{3} + 6 a^{2} + 21 a + 5\right)\cdot 23^{136} + \left(19 a^{14} + 12 a^{13} + 18 a^{12} + 10 a^{10} + 13 a^{9} + 21 a^{8} + 9 a^{7} + 22 a^{6} + 7 a^{5} + 20 a^{4} + 11 a^{3} + a^{2} + 4 a + 8\right)\cdot 23^{137} + \left(18 a^{14} + 12 a^{13} + 8 a^{12} + 2 a^{11} + 5 a^{10} + 14 a^{8} + 3 a^{7} + 13 a^{6} + 5 a^{5} + 9 a^{4} + 12 a^{3} + 17 a^{2} + 22 a + 18\right)\cdot 23^{138} + \left(18 a^{14} + 5 a^{13} + 22 a^{12} + 8 a^{11} + 7 a^{10} + 12 a^{9} + 4 a^{8} + 10 a^{7} + 19 a^{6} + 18 a^{5} + 18 a^{4} + 15 a^{3} + 4 a^{2} + 19 a + 10\right)\cdot 23^{139} + \left(10 a^{14} + 5 a^{13} + 14 a^{12} + 3 a^{11} + 8 a^{10} + 15 a^{9} + 14 a^{8} + 18 a^{7} + 17 a^{6} + 14 a^{5} + 16 a^{4} + 18 a^{3} + 2 a^{2} + 21 a + 3\right)\cdot 23^{140} + \left(6 a^{14} + 13 a^{13} + 10 a^{12} + 4 a^{11} + 3 a^{10} + 11 a^{9} + 13 a^{8} + 16 a^{7} + 21 a^{6} + 8 a^{5} + 13 a^{4} + 12 a^{3} + 20 a^{2} + 8 a + 13\right)\cdot 23^{141} + \left(6 a^{14} + 4 a^{12} + 9 a^{11} + 5 a^{10} + 5 a^{9} + 4 a^{8} + 5 a^{7} + 9 a^{6} + 22 a^{5} + 19 a^{4} + a^{3} + 4 a^{2} + 5 a + 6\right)\cdot 23^{142} + \left(5 a^{14} + 17 a^{13} + 14 a^{12} + 20 a^{11} + 4 a^{9} + 10 a^{8} + 5 a^{7} + 17 a^{6} + 15 a^{5} + 15 a^{4} + 9 a^{3} + 6 a^{2} + 15 a + 2\right)\cdot 23^{143} + \left(3 a^{14} + 16 a^{13} + 21 a^{12} + 19 a^{11} + 13 a^{10} + 4 a^{9} + 19 a^{8} + 19 a^{7} + 8 a^{6} + 2 a^{5} + 5 a^{4} + 9 a^{3} + 16 a^{2} + 4 a + 15\right)\cdot 23^{144} + \left(5 a^{13} + 5 a^{12} + 2 a^{11} + 5 a^{10} + 7 a^{9} + 18 a^{7} + 10 a^{6} + 15 a^{5} + 12 a^{4} + 4 a^{3} + 8 a^{2} + 10 a + 5\right)\cdot 23^{145} + \left(14 a^{14} + 22 a^{13} + 9 a^{12} + 17 a^{11} + 5 a^{10} + 4 a^{8} + 14 a^{7} + 11 a^{6} + 3 a^{5} + 2 a^{4} + 20 a^{3} + 22 a^{2} + 18 a + 22\right)\cdot 23^{146} + \left(19 a^{13} + 10 a^{12} + 7 a^{11} + a^{10} + 17 a^{9} + 22 a^{8} + 9 a^{7} + 22 a^{6} + 9 a^{4} + 12 a^{3} + 14 a^{2} + 2 a + 6\right)\cdot 23^{147} + \left(8 a^{14} + 4 a^{13} + 18 a^{12} + 2 a^{11} + 17 a^{10} + 8 a^{9} + 2 a^{8} + 18 a^{7} + 13 a^{6} + 2 a^{5} + 21 a^{4} + 20 a^{3} + 11 a^{2} + 16 a + 2\right)\cdot 23^{148} + \left(21 a^{14} + 2 a^{13} + a^{12} + a^{11} + 10 a^{9} + 7 a^{8} + 2 a^{7} + 17 a^{6} + 13 a^{5} + 2 a^{4} + 14 a^{3} + 5 a^{2} + 22 a + 4\right)\cdot 23^{149} + \left(10 a^{14} + 15 a^{13} + 15 a^{12} + a^{11} + 19 a^{10} + a^{9} + 20 a^{8} + 17 a^{7} + 6 a^{6} + 4 a^{5} + 12 a^{4} + 6 a^{3} + 15 a^{2} + 15 a + 22\right)\cdot 23^{150} + \left(15 a^{14} + 19 a^{13} + 8 a^{12} + a^{11} + 5 a^{10} + 11 a^{9} + a^{8} + 16 a^{7} + 18 a^{6} + 17 a^{5} + 13 a^{4} + 2 a^{2} + 10 a + 6\right)\cdot 23^{151} + \left(13 a^{14} + 3 a^{13} + 6 a^{12} + 3 a^{11} + 9 a^{10} + 3 a^{9} + 15 a^{8} + 7 a^{7} + 15 a^{6} + 12 a^{5} + 7 a^{4} + 5 a^{3} + 3 a^{2} + 6 a + 3\right)\cdot 23^{152} + \left(15 a^{14} + 15 a^{13} + 14 a^{12} + 12 a^{11} + 8 a^{10} + 6 a^{9} + a^{8} + 19 a^{7} + 21 a^{6} + 22 a^{5} + 13 a^{4} + 14 a^{3} + 7 a^{2} + 18 a + 7\right)\cdot 23^{153} + \left(21 a^{14} + 2 a^{13} + 9 a^{12} + 8 a^{11} + 10 a^{10} + 16 a^{9} + 21 a^{8} + 13 a^{7} + 19 a^{6} + 13 a^{5} + 21 a^{3} + 11 a^{2} + 6 a + 21\right)\cdot 23^{154} + \left(15 a^{14} + 7 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21 a^{13} + 12 a^{12} + a^{11} + 5 a^{10} + 20 a^{9} + 18 a^{8} + 18 a^{7} + 14 a^{6} + 22 a^{5} + 6 a^{4} + 12 a^{3} + 10 a^{2} + 18 a + 18\right)\cdot 23^{161} + \left(5 a^{14} + 12 a^{13} + 8 a^{12} + 2 a^{11} + 5 a^{10} + 22 a^{9} + 17 a^{8} + 22 a^{7} + 11 a^{6} + 10 a^{5} + 15 a^{4} + a^{3} + 12 a^{2} + 10 a\right)\cdot 23^{162} + \left(12 a^{14} + 14 a^{13} + 6 a^{12} + 18 a^{11} + 10 a^{10} + 8 a^{9} + 7 a^{7} + 7 a^{6} + 13 a^{5} + 15 a^{4} + 18 a^{3} + 15 a^{2} + 2 a + 17\right)\cdot 23^{163} + \left(21 a^{14} + 3 a^{13} + 4 a^{12} + 21 a^{11} + 15 a^{10} + 14 a^{9} + 14 a^{8} + 6 a^{7} + 13 a^{5} + 9 a^{4} + 3 a^{3} + 8 a^{2} + 2 a + 4\right)\cdot 23^{164} + \left(15 a^{13} + 6 a^{12} + 7 a^{11} + 20 a^{10} + 7 a^{9} + 6 a^{8} + 22 a^{7} + 9 a^{6} + 11 a^{5} + 15 a^{4} + 6 a^{3} + 12 a^{2} + 10 a + 10\right)\cdot 23^{165} + \left(a^{14} + 11 a^{13} + 21 a^{12} + 2 a^{11} + 14 a^{10} + 17 a^{9} + 2 a^{8} + a^{7} + 6 a^{6} + 18 a^{5} + 2 a^{4} + 7 a^{3} + 5 a^{2} + 16 a + 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4 a^{3} + 16 a^{2} + 8 a + 1\right)\cdot 23^{172} + \left(21 a^{14} + 9 a^{13} + 3 a^{12} + 7 a^{11} + 4 a^{10} + 20 a^{9} + 20 a^{8} + 7 a^{7} + 20 a^{6} + 2 a^{5} + 10 a^{4} + a^{3} + 9 a + 22\right)\cdot 23^{173} + \left(7 a^{14} + 3 a^{13} + 11 a^{12} + 8 a^{11} + 8 a^{10} + 14 a^{9} + 13 a^{8} + 22 a^{7} + 19 a^{6} + 8 a^{5} + 19 a^{4} + 9 a^{3} + 19 a + 6\right)\cdot 23^{174} + \left(22 a^{14} + a^{13} + 12 a^{12} + 21 a^{11} + 17 a^{10} + 18 a^{9} + 22 a^{8} + 5 a^{7} + 2 a^{6} + a^{5} + 18 a^{4} + 17 a^{3} + 5 a^{2} + 8 a + 16\right)\cdot 23^{175} + \left(10 a^{14} + 6 a^{13} + 18 a^{12} + 15 a^{11} + 4 a^{10} + 20 a^{9} + a^{7} + 4 a^{6} + 16 a^{5} + 6 a^{4} + 15 a^{3} + 2 a^{2} + 4 a + 22\right)\cdot 23^{176} + \left(10 a^{14} + 4 a^{13} + 13 a^{12} + 20 a^{11} + 6 a^{10} + 21 a^{9} + 8 a^{8} + 12 a^{7} + 6 a^{6} + a^{5} + 14 a^{3} + 14 a^{2} + 9 a + 14\right)\cdot 23^{177} + \left(18 a^{14} + 7 a^{13} + 7 a^{12} + 22 a^{11} + 7 a^{10} + 2 a^{9} + 13 a^{8} + 20 a^{7} + 22 a^{5} + 9 a^{4} + 21 a^{3} + a^{2} + 6 a + 9\right)\cdot 23^{178} + \left(17 a^{14} + 7 a^{13} + 10 a^{12} + 18 a^{11} + 12 a^{10} + 3 a^{9} + 12 a^{8} + 11 a^{7} + 10 a^{6} + 14 a^{5} + 18 a^{4} + 12 a^{3} + 5 a^{2} + 7 a + 2\right)\cdot 23^{179} + \left(14 a^{14} + 15 a^{13} + 8 a^{12} + 9 a^{11} + 4 a^{10} + 16 a^{9} + 16 a^{8} + 20 a^{7} + 19 a^{6} + 14 a^{5} + 3 a^{4} + 14 a^{3} + 16 a^{2} + 17 a + 8\right)\cdot 23^{180} + \left(3 a^{14} + 8 a^{13} + 15 a^{12} + 22 a^{11} + a^{10} + 7 a^{9} + 13 a^{8} + 17 a^{7} + 22 a^{6} + 2 a^{5} + 12 a^{4} + 22 a^{3} + a^{2} + 3 a + 16\right)\cdot 23^{181} + \left(8 a^{14} + 3 a^{13} + 20 a^{12} + 10 a^{11} + 5 a^{10} + 9 a^{9} + 17 a^{8} + 7 a^{7} + 20 a^{6} + 6 a^{5} + 22 a^{4} + 17 a^{3} + 15 a^{2} + 2 a + 4\right)\cdot 23^{182} + \left(18 a^{14} + 16 a^{13} + 19 a^{12} + 16 a^{11} + 19 a^{10} + 15 a^{9} + 21 a^{8} + 22 a^{7} + 4 a^{6} + 8 a^{5} + 9 a^{4} + 21 a^{3} + 6 a^{2} + 12 a + 22\right)\cdot 23^{183} + \left(18 a^{14} + 3 a^{13} + 3 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23^{189} + \left(17 a^{14} + 5 a^{13} + 20 a^{12} + 17 a^{11} + 19 a^{10} + 12 a^{9} + 3 a^{8} + a^{7} + 7 a^{6} + 21 a^{5} + 3 a^{4} + 17 a^{3} + 16 a^{2} + 21 a + 18\right)\cdot 23^{190} + \left(a^{14} + 8 a^{13} + 20 a^{12} + 11 a^{11} + 2 a^{10} + 6 a^{9} + 17 a^{8} + 8 a^{7} + 5 a^{6} + 8 a^{5} + a^{4} + 18 a^{3} + 17 a^{2} + 3 a + 12\right)\cdot 23^{191} + \left(16 a^{14} + 9 a^{13} + 2 a^{12} + a^{11} + 6 a^{10} + 15 a^{9} + 14 a^{8} + 21 a^{7} + 22 a^{6} + 13 a^{5} + 2 a^{4} + 6 a^{3} + 19 a^{2} + 13 a + 15\right)\cdot 23^{192} + \left(22 a^{14} + 15 a^{12} + 6 a^{10} + 12 a^{9} + 19 a^{8} + 21 a^{7} + 6 a^{6} + 8 a^{5} + 8 a^{4} + 16 a^{3} + 18 a^{2} + 22 a + 13\right)\cdot 23^{193} + \left(16 a^{14} + a^{13} + 17 a^{12} + 19 a^{11} + 3 a^{10} + 22 a^{9} + 9 a^{8} + 17 a^{7} + 5 a^{6} + 7 a^{5} + 10 a^{4} + 10 a^{3} + 17 a^{2} + 12 a + 7\right)\cdot 23^{194} + \left(4 a^{14} + 16 a^{13} + 10 a^{12} + 2 a^{11} + 13 a^{9} + 4 a^{8} + 5 a^{7} + 4 a^{6} + 17 a^{5} + 2 a^{4} + 5 a^{3} + 19 a^{2} + 4 a + 14\right)\cdot 23^{195} + \left(19 a^{14} + 5 a^{13} + 20 a^{12} + 14 a^{11} + 10 a^{10} + 7 a^{9} + 3 a^{8} + 2 a^{7} + 13 a^{6} + 11 a^{5} + 5 a^{4} + 10 a^{3} + 4 a^{2} + 12 a + 9\right)\cdot 23^{196} + \left(9 a^{14} + 4 a^{13} + 11 a^{12} + 19 a^{11} + 5 a^{10} + 7 a^{9} + 19 a^{8} + 2 a^{7} + 13 a^{6} + 10 a^{5} + 11 a^{4} + 5 a^{3} + 19 a^{2} + 11 a + 4\right)\cdot 23^{197} + \left(20 a^{14} + 10 a^{13} + 3 a^{12} + 17 a^{11} + 14 a^{10} + 20 a^{9} + 3 a^{8} + 15 a^{7} + 6 a^{6} + 12 a^{5} + 13 a^{4} + 21 a^{3} + 11 a^{2} + 6 a + 1\right)\cdot 23^{198} + \left(12 a^{14} + 2 a^{13} + 4 a^{12} + 4 a^{11} + 3 a^{10} + 16 a^{9} + 22 a^{8} + 19 a^{7} + 6 a^{6} + 10 a^{4} + 21 a^{3} + 10 a + 5\right)\cdot 23^{199} + \left(14 a^{14} + 8 a^{13} + 14 a^{12} + 2 a^{11} + 4 a^{10} + 12 a^{9} + 7 a^{8} + 6 a^{7} + 2 a^{6} + 12 a^{5} + 2 a^{4} + 4 a^{3} + 10 a^{2} + 8 a + 3\right)\cdot 23^{200} + \left(11 a^{14} + 16 a^{13} + 12 a^{12} + 12 a^{11} + 15 a^{10} + 2 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+ 17 a^{13} + 6 a^{12} + 15 a^{11} + 6 a^{10} + 2 a^{9} + 21 a^{8} + 20 a^{7} + 11 a^{6} + 9 a^{5} + 2 a^{4} + 18 a^{3} + 17 a^{2} + 14 a + 5\right)\cdot 23^{207} + \left(20 a^{13} + 11 a^{12} + 21 a^{11} + 17 a^{10} + 13 a^{9} + 4 a^{8} + 2 a^{7} + a^{6} + 10 a^{5} + 8 a^{4} + 12 a^{3} + 11 a^{2} + 22 a + 3\right)\cdot 23^{208} + \left(11 a^{13} + 15 a^{12} + 2 a^{11} + 10 a^{10} + 18 a^{9} + 3 a^{8} + 22 a^{7} + 14 a^{6} + 16 a^{5} + a^{4} + 20 a^{3} + 22 a^{2} + 4 a\right)\cdot 23^{209} + \left(22 a^{14} + 22 a^{13} + 19 a^{12} + 21 a^{11} + 14 a^{10} + 6 a^{9} + 15 a^{8} + 15 a^{7} + a^{5} + 8 a^{4} + 19 a^{3} + 8 a^{2} + 3 a + 18\right)\cdot 23^{210} + \left(12 a^{14} + 7 a^{13} + 17 a^{12} + 11 a^{11} + 4 a^{10} + 16 a^{9} + 10 a^{8} + 11 a^{7} + 12 a^{6} + 10 a^{5} + a^{4} + a^{3} + 10 a^{2} + 4 a + 3\right)\cdot 23^{211} + \left(2 a^{14} + 13 a^{13} + 2 a^{12} + 17 a^{10} + 10 a^{9} + 5 a^{8} + 22 a^{7} + 17 a^{6} + 12 a^{5} + 11 a^{4} + 11 a^{3} + 3 a^{2} + 13 a + 13\right)\cdot 23^{212} + \left(8 a^{13} + 7 a^{12} + 3 a^{11} + 12 a^{10} + 15 a^{9} + 17 a^{8} + 17 a^{7} + 2 a^{5} + 18 a^{4} + 13 a^{3} + 7 a^{2} + 9 a + 7\right)\cdot 23^{213} + \left(9 a^{14} + a^{13} + 15 a^{12} + 8 a^{11} + 6 a^{10} + 15 a^{8} + 12 a^{7} + 15 a^{6} + 16 a^{5} + 10 a^{4} + 19 a^{3} + 6 a^{2} + 12 a + 10\right)\cdot 23^{214} + \left(22 a^{14} + 22 a^{13} + 4 a^{11} + 5 a^{10} + 3 a^{9} + 22 a^{8} + 6 a^{7} + 4 a^{6} + 21 a^{5} + 21 a^{4} + 17 a^{3} + 7 a^{2} + 14 a + 3\right)\cdot 23^{215} + \left(9 a^{14} + 3 a^{13} + 22 a^{12} + 18 a^{11} + 11 a^{10} + 15 a^{9} + 19 a^{8} + 16 a^{7} + 4 a^{6} + 13 a^{5} + 16 a^{4} + 7 a^{2} + a + 13\right)\cdot 23^{216} + \left(6 a^{14} + 17 a^{12} + 4 a^{11} + 7 a^{10} + 8 a^{9} + 6 a^{8} + a^{7} + 20 a^{6} + 2 a^{5} + 13 a^{4} + 21 a^{3} + 10 a^{2} + 22 a + 7\right)\cdot 23^{217} + \left(10 a^{14} + 7 a^{13} + 13 a^{11} + 14 a^{10} + 5 a^{9} + 6 a^{8} + 7 a^{7} + 5 a^{6} + 11 a^{5} + 18 a^{4} + 13 a^{3} + 3 a^{2} + 11 a + 13\right)\cdot 23^{218} + \left(4 a^{14} + 16 a^{13} + 3 a^{12} + 10 a^{11} + 18 a^{10} + 16 a^{9} + 13 a^{8} + 5 a^{7} + 18 a^{6} + 17 a^{5} + 7 a^{4} + a^{3} + 13 a^{2} + 3 a + 20\right)\cdot 23^{219} + \left(21 a^{14} + 5 a^{13} + 18 a^{12} + 22 a^{10} + 17 a^{9} + 3 a^{8} + a^{6} + 7 a^{4} + 6 a^{3} + a^{2} + 17 a + 11\right)\cdot 23^{220} + \left(10 a^{14} + 8 a^{12} + 21 a^{11} + 12 a^{10} + 9 a^{9} + 14 a^{8} + 5 a^{7} + a^{6} + 7 a^{4} + 22 a^{3} + 8 a^{2} + 8 a + 19\right)\cdot 23^{221} + \left(20 a^{14} + 2 a^{13} + 12 a^{12} + 2 a^{11} + 22 a^{10} + 14 a^{9} + 11 a^{8} + 20 a^{7} + 19 a^{6} + 9 a^{5} + 17 a^{4} + 4 a^{2} + 8 a + 20\right)\cdot 23^{222} + \left(7 a^{14} + 8 a^{13} + a^{12} + 4 a^{11} + 19 a^{10} + 18 a^{9} + 21 a^{7} + 13 a^{6} + 8 a^{5} + 11 a^{4} + 15 a^{3} + 15 a^{2} + 10 a + 13\right)\cdot 23^{223} + \left(4 a^{14} + 6 a^{13} + 8 a^{12} + 4 a^{11} + 11 a^{10} + 18 a^{9} + 9 a^{8} + 6 a^{7} + 21 a^{6} + 8 a^{5} + 8 a^{3} + 7 a^{2} + 17 a + 1\right)\cdot 23^{224} + \left(15 a^{14} + 12 a^{13} + 18 a^{12} + 3 a^{11} + 21 a^{10} + 18 a^{9} + 16 a^{8} + 10 a^{7} + 21 a^{6} + 13 a^{5} + 4 a^{4} + 22 a^{3} + 19 a^{2} + 6 a + 1\right)\cdot 23^{225} + \left(4 a^{14} + 13 a^{13} + 9 a^{12} + 20 a^{11} + a^{10} + 15 a^{9} + 21 a^{7} + 21 a^{6} + 11 a^{5} + 11 a^{4} + 18 a^{3} + 9 a^{2} + 3 a + 6\right)\cdot 23^{226} + \left(6 a^{14} + 15 a^{13} + 14 a^{12} + 5 a^{11} + 8 a^{10} + 8 a^{9} + 10 a^{8} + 2 a^{7} + 13 a^{6} + 21 a^{5} + 13 a^{4} + 6 a^{3} + 22 a^{2} + 3\right)\cdot 23^{227} + \left(13 a^{14} + 21 a^{13} + 13 a^{12} + 4 a^{10} + 13 a^{9} + 19 a^{8} + 3 a^{7} + 10 a^{6} + 8 a^{5} + 16 a^{4} + 20 a^{3} + 13 a^{2} + 6\right)\cdot 23^{228} + \left(13 a^{14} + 5 a^{13} + 3 a^{12} + 2 a^{11} + 6 a^{10} + 19 a^{9} + 16 a^{8} + a^{7} + 11 a^{6} + 12 a^{5} + 12 a^{4} + 22 a^{3} + 14 a^{2} + 15 a + 20\right)\cdot 23^{229} + \left(13 a^{14} + 21 a^{12} + 11 a^{11} + 6 a^{10} + 8 a^{9} + 5 a^{8} + 3 a^{7} + 2 a^{6} + 17 a^{5} + 12 a^{4} + 3 a^{2} + 12 a + 18\right)\cdot 23^{230} + \left(11 a^{13} + 5 a^{12} + 3 a^{11} + 7 a^{10} + 4 a^{9} + 14 a^{8} + 8 a^{7} + 2 a^{6} + 14 a^{5} + 12 a^{4} + 4 a^{3} + 12 a^{2} + 14 a + 12\right)\cdot 23^{231} + \left(2 a^{14} + 6 a^{13} + 11 a^{12} + 21 a^{11} + 7 a^{10} + 7 a^{9} + 10 a^{8} + 8 a^{7} + 20 a^{6} + 17 a^{5} + 10 a^{4} + 16 a^{3} + 13 a^{2} + 15 a + 16\right)\cdot 23^{232} + \left(4 a^{14} + 10 a^{13} + 14 a^{12} + 6 a^{11} + 22 a^{10} + 13 a^{9} + 12 a^{8} + 8 a^{7} + 20 a^{6} + 2 a^{5} + 7 a^{4} + a^{3} + 22 a^{2} + 7 a + 21\right)\cdot 23^{233} + \left(20 a^{14} + 13 a^{13} + 16 a^{12} + 9 a^{11} + 9 a^{10} + 14 a^{9} + 7 a^{8} + 6 a^{7} + 15 a^{6} + 2 a^{5} + 15 a^{4} + 7 a^{3} + 13 a^{2} + 14 a + 6\right)\cdot 23^{234} + \left(12 a^{14} + 9 a^{12} + 14 a^{11} + 19 a^{10} + 19 a^{9} + 17 a^{8} + 2 a^{7} + 18 a^{6} + 17 a^{5} + 15 a^{4} + 4 a^{3} + 21 a^{2} + 15 a + 17\right)\cdot 23^{235} + \left(19 a^{14} + 13 a^{12} + 11 a^{11} + 11 a^{10} + a^{9} + 15 a^{7} + 7 a^{6} + 20 a^{5} + 20 a^{4} + 4 a^{3} + 16 a^{2} + 3 a + 20\right)\cdot 23^{236} + \left(5 a^{14} + 5 a^{13} + 6 a^{12} + 17 a^{11} + 12 a^{10} + 19 a^{9} + 15 a^{8} + 18 a^{7} + 20 a^{6} + 15 a^{5} + a^{4} + 19 a^{3} + 13 a^{2} + 12 a + 15\right)\cdot 23^{237} + \left(13 a^{14} + 6 a^{13} + a^{12} + 9 a^{11} + 6 a^{10} + 5 a^{9} + 17 a^{8} + 17 a^{7} + 22 a^{6} + 19 a^{5} + 5 a^{4} + 9 a^{3} + 5 a^{2} + 4 a + 14\right)\cdot 23^{238} + \left(20 a^{14} + 12 a^{13} + 14 a^{12} + 20 a^{10} + 14 a^{9} + 21 a^{8} + 21 a^{7} + 8 a^{6} + 5 a^{5} + 22 a^{4} + 18 a^{3} + 22 a^{2} + 10 a\right)\cdot 23^{239} + \left(15 a^{14} + 18 a^{13} + 19 a^{12} + 9 a^{11} + 22 a^{10} + 11 a^{9} + 3 a^{8} + 9 a^{7} + 13 a^{6} + 7 a^{5} + a^{4} + 19 a^{3} + 9 a^{2} + 14 a + 20\right)\cdot 23^{240} + \left(4 a^{13} + 4 a^{12} + 7 a^{11} + 9 a^{10} + 15 a^{9} + 16 a^{8} + 11 a^{7} + 15 a^{6} + 22 a^{5} + 6 a^{4} + 16 a^{3} + 4 a^{2} + 18 a + 15\right)\cdot 23^{241} + \left(8 a^{14} + 20 a^{13} + a^{12} + 22 a^{11} + 7 a^{10} + 4 a^{9} + 15 a^{8} + 12 a^{7} + 18 a^{6} + 14 a^{5} + 9 a^{4} + 6 a^{3} + 11 a^{2} + 15 a + 17\right)\cdot 23^{242} + \left(19 a^{13} + 3 a^{12} + 3 a^{11} + 9 a^{10} + 6 a^{9} + a^{8} + 20 a^{7} + a^{6} + 14 a^{5} + 9 a^{4} + 18 a^{3} + 8 a^{2} + 18 a + 15\right)\cdot 23^{243} + \left(15 a^{14} + 22 a^{13} + 13 a^{12} + 15 a^{11} + 21 a^{10} + 10 a^{9} + 22 a^{8} + 3 a^{7} + 8 a^{6} + 9 a^{5} + 12 a^{4} + 19 a^{3} + 17 a^{2} + 9 a + 14\right)\cdot 23^{244} + \left(8 a^{14} + 4 a^{13} + 11 a^{12} + 22 a^{11} + 6 a^{10} + 18 a^{9} + 16 a^{7} + 7 a^{6} + 18 a^{5} + 7 a^{4} + a^{3} + 21 a + 15\right)\cdot 23^{245} + \left(3 a^{13} + 13 a^{12} + 13 a^{11} + 4 a^{10} + 21 a^{9} + 5 a^{8} + 10 a^{7} + 22 a^{6} + 11 a^{4} + 12 a^{2} + 3 a + 4\right)\cdot 23^{246} + \left(a^{14} + 20 a^{13} + 3 a^{12} + 6 a^{11} + 14 a^{10} + 5 a^{9} + 6 a^{8} + 16 a^{7} + 8 a^{6} + 19 a^{5} + 20 a^{4} + 4 a^{3} + 15 a^{2} + 5 a\right)\cdot 23^{247} + \left(3 a^{14} + 6 a^{13} + 12 a^{12} + 19 a^{11} + 20 a^{10} + 10 a^{9} + 17 a^{8} + 11 a^{7} + 20 a^{6} + 10 a^{5} + 15 a^{4} + 14 a^{3} + 20 a^{2} + 14 a + 21\right)\cdot 23^{248} + \left(19 a^{13} + 19 a^{12} + a^{11} + a^{10} + 2 a^{9} + 14 a^{8} + 20 a^{7} + 5 a^{6} + 4 a^{5} + 19 a^{4} + 19 a^{3} + 22 a^{2} + 12 a + 22\right)\cdot 23^{249} + \left(7 a^{14} + 2 a^{13} + 4 a^{12} + 7 a^{11} + 16 a^{10} + a^{9} + 7 a^{8} + 4 a^{7} + 9 a^{6} + 18 a^{5} + 12 a^{4} + a^{3} + 8 a^{2} + 6 a + 17\right)\cdot 23^{250} + \left(9 a^{13} + 11 a^{12} + 12 a^{11} + 2 a^{10} + 3 a^{9} + 4 a^{8} + 10 a^{7} + 5 a^{6} + 12 a^{5} + 14 a^{4} + 9 a^{3} + 19 a^{2} + 16 a + 12\right)\cdot 23^{251} + \left(17 a^{14} + 10 a^{13} + 4 a^{12} + 17 a^{11} + 3 a^{10} + 20 a^{9} + 4 a^{8} + 6 a^{7} + 20 a^{6} + 12 a^{5} + 19 a^{4} + 20 a^{3} + 17 a^{2} + 9 a + 20\right)\cdot 23^{252} + \left(4 a^{14} + a^{13} + 13 a^{12} + 2 a^{11} + 7 a^{10} + 12 a^{9} + 11 a^{8} + 22 a^{7} + 8 a^{6} + 7 a^{5} + 4 a^{4} + 2 a^{3} + 17 a^{2} + 6 a + 3\right)\cdot 23^{253} + \left(13 a^{14} + 10 a^{13} + 19 a^{12} + 20 a^{11} + 5 a^{10} + 5 a^{9} + 12 a^{8} + 15 a^{7} + a^{6} + 8 a^{5} + 3 a^{4} + 19 a^{3} + 3 a^{2} + 22 a + 5\right)\cdot 23^{254} + \left(9 a^{14} + 2 a^{13} + 4 a^{12} + 15 a^{11} + 10 a^{10} + 4 a^{9} + 8 a^{8} + 3 a^{7} + 21 a^{6} + 17 a^{5} + 21 a^{4} + 12 a^{3} + 20 a^{2} + 20 a + 7\right)\cdot 23^{255} + \left(17 a^{14} + 11 a^{13} + 5 a^{12} + 12 a^{11} + 20 a^{10} + 7 a^{9} + 13 a^{8} + 8 a^{7} + 22 a^{6} + 6 a^{5} + 22 a^{4} + 14 a^{3} + a^{2} + 10\right)\cdot 23^{256} + \left(17 a^{13} + 20 a^{12} + 8 a^{11} + 14 a^{10} + 21 a^{9} + 6 a^{8} + 20 a^{7} + 6 a^{6} + 12 a^{5} + 3 a^{4} + a^{3} + 5 a^{2} + 2 a + 5\right)\cdot 23^{257} + \left(4 a^{14} + 12 a^{13} + 21 a^{12} + 22 a^{11} + 4 a^{10} + 10 a^{9} + 15 a^{8} + 5 a^{7} + 4 a^{6} + 18 a^{5} + 18 a^{4} + 12 a^{3} + 16 a^{2} + 17 a + 4\right)\cdot 23^{258} + \left(8 a^{14} + 7 a^{13} + 7 a^{12} + 14 a^{11} + 3 a^{9} + 3 a^{8} + 7 a^{7} + 15 a^{6} + 20 a^{5} + 2 a^{4} + a^{3} + 12 a^{2} + 6 a + 9\right)\cdot 23^{259} + \left(3 a^{14} + 19 a^{13} + 18 a^{12} + 20 a^{11} + 12 a^{10} + 22 a^{9} + 21 a^{8} + 16 a^{7} + 10 a^{6} + 20 a^{5} + 16 a^{4} + 12 a^{3} + 22 a^{2} + 3 a + 15\right)\cdot 23^{260} + \left(11 a^{13} + 15 a^{12} + 6 a^{11} + 11 a^{10} + 7 a^{9} + 21 a^{8} + 11 a^{7} + 4 a^{6} + 4 a^{5} + 9 a^{4} + a^{3} + 6 a^{2} + 4 a + 8\right)\cdot 23^{261} + \left(9 a^{14} + 5 a^{13} + 18 a^{12} + 17 a^{11} + a^{10} + 16 a^{9} + 10 a^{8} + 11 a^{7} + 2 a^{6} + 22 a^{5} + 13 a^{4} + 2 a^{3} + 16 a^{2} + 7 a + 13\right)\cdot 23^{262} + \left(10 a^{14} + 6 a^{13} + 13 a^{11} + 18 a^{10} + 15 a^{8} + 16 a^{7} + 21 a^{6} + 3 a^{5} + 20 a^{4} + 17 a^{3} + a^{2} + 20 a + 20\right)\cdot 23^{263} + \left(16 a^{14} + 20 a^{13} + 21 a^{12} + 8 a^{11} + 22 a^{10} + 8 a^{9} + 8 a^{8} + 11 a^{7} + 22 a^{6} + 15 a^{4} + 14 a^{3} + 21 a^{2} + 19 a + 11\right)\cdot 23^{264} + \left(14 a^{14} + 6 a^{13} + 2 a^{12} + 11 a^{11} + 6 a^{10} + 3 a^{9} + 16 a^{8} + 8 a^{7} + a^{6} + 13 a^{5} + a^{4} + 12 a^{3} + 16 a^{2} + 9 a + 10\right)\cdot 23^{265} + \left(3 a^{14} + 6 a^{13} + 21 a^{12} + 4 a^{10} + 10 a^{9} + 8 a^{8} + 20 a^{7} + 7 a^{6} + 10 a^{5} + 11 a^{4} + a^{3} + 10 a^{2} + 11 a + 6\right)\cdot 23^{266} + \left(6 a^{14} + 11 a^{13} + 16 a^{12} + 14 a^{11} + 20 a^{9} + 12 a^{8} + 2 a^{7} + 22 a^{6} + 19 a^{5} + 20 a^{4} + 4 a^{3} + 19 a + 21\right)\cdot 23^{267} + \left(10 a^{14} + 20 a^{13} + 14 a^{12} + 4 a^{11} + 12 a^{10} + 4 a^{9} + a^{8} + 22 a^{7} + 18 a^{6} + 20 a^{5} + 13 a^{4} + 12 a^{3} + 21 a^{2} + 2\right)\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 10 }$ $=$ $4 a^{14} + 21 a^{13} + 17 a^{12} + 19 a^{11} + 16 a^{10} + 9 a^{9} + 15 a^{8} + 2 a^{7} + 11 a^{6} + 20 a^{5} + 14 a^{4} + 7 a^{3} + 2 a^{2} + 8 a + 10 + \left(15 a^{14} + 20 a^{13} + 12 a^{12} + 20 a^{11} + 19 a^{10} + 9 a^{9} + 17 a^{8} + 19 a^{7} + 15 a^{6} + 22 a^{5} + 7 a^{4} + a^{3} + a^{2} + 12 a\right)\cdot 23 + \left(8 a^{14} + 10 a^{13} + 17 a^{12} + 15 a^{11} + 8 a^{10} + 21 a^{9} + 12 a^{8} + a^{7} + 17 a^{6} + 7 a^{5} + 10 a^{3} + 18 a^{2} + 12\right)\cdot 23^{2} + \left(12 a^{14} + 15 a^{13} + 14 a^{12} + 15 a^{11} + 7 a^{10} + 21 a^{9} + 11 a^{8} + 14 a^{7} + 20 a^{6} + 11 a^{5} + 4 a^{4} + 6 a^{3} + 12 a^{2} + 5 a + 12\right)\cdot 23^{3} + \left(9 a^{14} + 4 a^{13} + 2 a^{12} + 16 a^{11} + 13 a^{10} + 12 a^{9} + 7 a^{8} + 22 a^{7} + 8 a^{6} + a^{5} + 4 a^{4} + 6 a^{3} + 2 a^{2} + 20 a + 16\right)\cdot 23^{4} + \left(15 a^{14} + 9 a^{13} + 7 a^{12} + 14 a^{11} + 15 a^{10} + 14 a^{9} + 6 a^{8} + 21 a^{7} + a^{6} + 3 a^{5} + 6 a^{4} + 22 a^{3} + 19 a^{2} + 6 a + 13\right)\cdot 23^{5} + \left(11 a^{14} + 22 a^{13} + 14 a^{12} + 22 a^{11} + 6 a^{10} + 13 a^{9} + 11 a^{8} + 7 a^{7} + 15 a^{6} + 20 a^{5} + 12 a^{4} + 7 a^{3} + a^{2} + 8 a + 2\right)\cdot 23^{6} + \left(16 a^{14} + 5 a^{13} + 14 a^{12} + 20 a^{11} + 22 a^{10} + 22 a^{9} + 7 a^{8} + 11 a^{7} + 22 a^{6} + 21 a^{5} + 13 a^{4} + 13 a^{3} + 3 a^{2} + 7 a\right)\cdot 23^{7} + \left(16 a^{14} + 19 a^{13} + 11 a^{12} + 10 a^{11} + 10 a^{10} + 18 a^{9} + 5 a^{8} + 16 a^{7} + 4 a^{6} + 12 a^{4} + 16 a^{3} + 16 a^{2} + 13 a + 21\right)\cdot 23^{8} + \left(22 a^{14} + 7 a^{13} + 4 a^{12} + 3 a^{11} + 8 a^{10} + 2 a^{8} + 5 a^{7} + 14 a^{6} + 10 a^{5} + 6 a^{4} + 18 a^{3} + 14 a^{2} + 21 a + 13\right)\cdot 23^{9} + \left(17 a^{14} + 17 a^{13} + 4 a^{12} + 13 a^{11} + 12 a^{10} + 20 a^{9} + 13 a^{8} + 17 a^{7} + 18 a^{6} + 10 a^{5} + 17 a^{4} + 11 a^{3} + 20 a^{2} + 15 a + 1\right)\cdot 23^{10} + \left(18 a^{14} + 7 a^{13} + a^{12} + 15 a^{11} + 12 a^{10} + 22 a^{9} + 15 a^{8} + 2 a^{7} + 16 a^{6} + 20 a^{5} + 9 a^{4} + 14 a^{3} + 2 a^{2} + 12 a + 2\right)\cdot 23^{11} + \left(7 a^{14} + 14 a^{13} + 22 a^{11} + 2 a^{10} + 2 a^{9} + 13 a^{8} + 6 a^{6} + 17 a^{5} + 9 a^{4} + 4 a^{3} + 19 a^{2} + 21 a + 17\right)\cdot 23^{12} + \left(4 a^{14} + 20 a^{13} + 18 a^{12} + 15 a^{11} + 14 a^{10} + 17 a^{9} + 5 a^{8} + 11 a^{7} + 16 a^{6} + 12 a^{5} + 10 a^{4} + 5 a^{3} + 17 a^{2} + 11 a + 17\right)\cdot 23^{13} + \left(2 a^{14} + 14 a^{13} + 7 a^{12} + a^{11} + 14 a^{10} + 21 a^{9} + 16 a^{8} + 16 a^{7} + 15 a^{6} + 6 a^{5} + 22 a^{4} + 13 a^{3} + 15 a^{2} + 21 a + 15\right)\cdot 23^{14} + \left(5 a^{14} + 19 a^{13} + 18 a^{12} + 14 a^{11} + 2 a^{10} + 20 a^{9} + 3 a^{8} + 15 a^{7} + 2 a^{6} + 18 a^{5} + 2 a^{4} + 10 a^{3} + 12 a^{2} + 14 a + 11\right)\cdot 23^{15} + \left(21 a^{14} + 19 a^{13} + a^{12} + 20 a^{11} + 5 a^{10} + 13 a^{9} + 6 a^{8} + 11 a^{7} + 11 a^{6} + 4 a^{5} + 11 a^{4} + 10 a^{3} + 17 a^{2} + 2 a + 3\right)\cdot 23^{16} + \left(18 a^{14} + 8 a^{13} + 2 a^{12} + 14 a^{11} + 6 a^{10} + 19 a^{9} + 14 a^{8} + 15 a^{7} + 18 a^{6} + 7 a^{5} + 20 a^{4} + a^{3} + 22 a^{2} + 4 a + 19\right)\cdot 23^{17} + \left(6 a^{14} + 10 a^{13} + 13 a^{12} + 7 a^{11} + 2 a^{10} + a^{9} + 19 a^{8} + 2 a^{7} + 4 a^{6} + 5 a^{5} + 16 a^{4} + 19 a^{3} + 18 a^{2} + 3 a + 15\right)\cdot 23^{18} + \left(22 a^{13} + 22 a^{12} + 18 a^{11} + 18 a^{10} + 19 a^{9} + 17 a^{8} + 7 a^{7} + 20 a^{6} + 22 a^{5} + 14 a^{4} + a^{3} + 2 a^{2} + 10 a + 10\right)\cdot 23^{19} + \left(15 a^{14} + 3 a^{13} + 8 a^{12} + 10 a^{11} + 2 a^{10} + 14 a^{8} + 21 a^{7} + 19 a^{6} + 17 a^{5} + 2 a^{4} + 6 a^{3} + 19 a^{2} + 8 a + 16\right)\cdot 23^{20} + \left(4 a^{14} + 21 a^{13} + 15 a^{12} + 13 a^{11} + 3 a^{10} + 4 a^{9} + 5 a^{8} + 4 a^{7} + 3 a^{6} + 9 a^{5} + 13 a^{4} + 16 a^{3} + 10 a^{2} + 6 a + 8\right)\cdot 23^{21} + \left(19 a^{14} + 21 a^{13} + a^{12} + 11 a^{11} + 15 a^{10} + 2 a^{9} + 5 a^{8} + 10 a^{7} + 13 a^{6} + 3 a^{5} + a^{4} + 17 a^{3} + 14 a^{2} + 3 a + 5\right)\cdot 23^{22} + \left(16 a^{14} + 3 a^{13} + 22 a^{12} + 15 a^{11} + 13 a^{10} + 16 a^{9} + 22 a^{8} + 14 a^{7} + a^{6} + 6 a^{5} + 4 a^{4} + 8 a^{2} + 14 a\right)\cdot 23^{23} + \left(17 a^{14} + 18 a^{13} + 11 a^{12} + 19 a^{11} + 15 a^{10} + 22 a^{9} + 21 a^{8} + 5 a^{7} + 2 a^{6} + 7 a^{5} + 3 a^{4} + 10 a^{3} + 2 a^{2} + a + 18\right)\cdot 23^{24} + \left(12 a^{14} + 11 a^{13} + 15 a^{12} + 2 a^{11} + 3 a^{10} + 19 a^{9} + 14 a^{8} + 18 a^{6} + 4 a^{5} + 20 a^{4} + 10 a^{3} + 20 a^{2} + 12 a + 2\right)\cdot 23^{25} + \left(2 a^{14} + 16 a^{13} + 5 a^{12} + 17 a^{11} + 12 a^{10} + 6 a^{9} + 3 a^{8} + 13 a^{7} + 6 a^{6} + 3 a^{5} + a^{4} + 20 a^{3} + 22 a^{2} + 6 a + 16\right)\cdot 23^{26} + \left(5 a^{14} + 22 a^{13} + 13 a^{12} + 7 a^{11} + 22 a^{10} + 13 a^{9} + 21 a^{8} + 12 a^{7} + 15 a^{6} + a^{5} + 7 a^{4} + 8 a^{3} + 15 a^{2} + 8 a + 9\right)\cdot 23^{27} + \left(4 a^{14} + 21 a^{13} + 12 a^{12} + 19 a^{11} + 3 a^{10} + 19 a^{9} + 12 a^{8} + 6 a^{7} + 14 a^{6} + 6 a^{5} + 15 a^{4} + 5 a^{3} + 22 a^{2} + 21\right)\cdot 23^{28} + \left(21 a^{14} + 6 a^{13} + 5 a^{12} + 14 a^{11} + 15 a^{10} + 14 a^{9} + 6 a^{8} + 16 a^{7} + 6 a^{6} + a^{5} + 16 a^{4} + a^{3} + 22 a + 14\right)\cdot 23^{29} + \left(12 a^{14} + 8 a^{13} + 7 a^{12} + 18 a^{11} + 5 a^{10} + a^{9} + 14 a^{8} + 8 a^{7} + 13 a^{6} + 4 a^{5} + 16 a^{4} + 11 a^{3} + 7 a^{2} + 6 a + 2\right)\cdot 23^{30} + \left(12 a^{14} + 16 a^{13} + 22 a^{12} + a^{11} + 2 a^{10} + 21 a^{9} + 22 a^{8} + 16 a^{7} + 21 a^{6} + 3 a^{5} + a^{4} + 11 a^{3} + 2 a^{2} + 2 a + 11\right)\cdot 23^{31} + \left(3 a^{14} + a^{13} + 14 a^{12} + 6 a^{11} + 20 a^{10} + 3 a^{9} + 10 a^{8} + a^{6} + 5 a^{5} + 12 a^{4} + 21 a^{3} + 9 a^{2} + 9 a + 2\right)\cdot 23^{32} + \left(12 a^{14} + 19 a^{13} + 20 a^{12} + 12 a^{11} + 6 a^{9} + 15 a^{8} + 14 a^{7} + a^{6} + 3 a^{4} + a^{3} + 11 a^{2} + 19 a + 8\right)\cdot 23^{33} + \left(14 a^{14} + 14 a^{13} + a^{12} + 2 a^{11} + 10 a^{10} + 17 a^{8} + 2 a^{6} + 14 a^{5} + 21 a^{4} + 13 a^{3} + 7 a^{2} + 21 a + 14\right)\cdot 23^{34} + \left(19 a^{14} + 18 a^{13} + 12 a^{12} + 19 a^{11} + 20 a^{10} + 13 a^{9} + 15 a^{8} + 14 a^{7} + 10 a^{6} + 20 a^{5} + 22 a^{4} + 3 a^{3} + 4 a^{2} + a + 10\right)\cdot 23^{35} + \left(18 a^{14} + 12 a^{12} + 5 a^{11} + 7 a^{10} + 12 a^{9} + 2 a^{8} + 3 a^{7} + 2 a^{6} + 10 a^{5} + 15 a^{4} + 15 a^{3} + 9 a^{2} + 18 a\right)\cdot 23^{36} + \left(13 a^{14} + 21 a^{13} + 19 a^{12} + 17 a^{11} + 20 a^{10} + 14 a^{9} + 14 a^{8} + 5 a^{7} + 20 a^{6} + 10 a^{5} + 7 a^{4} + 15 a^{3} + 5 a^{2} + 9 a + 13\right)\cdot 23^{37} + \left(17 a^{14} + 8 a^{12} + 19 a^{11} + 11 a^{10} + 20 a^{9} + 6 a^{8} + 3 a^{7} + 17 a^{6} + 22 a^{5} + 8 a^{4} + 20 a^{3} + 6 a^{2} + 2 a + 14\right)\cdot 23^{38} + \left(7 a^{14} + 12 a^{13} + a^{12} + 19 a^{11} + 4 a^{10} + 15 a^{9} + 5 a^{8} + 5 a^{7} + 5 a^{6} + 19 a^{5} + 10 a^{4} + 14 a^{3} + 11 a^{2} + 9 a\right)\cdot 23^{39} + \left(21 a^{14} + 22 a^{12} + 13 a^{11} + 12 a^{10} + 15 a^{9} + 18 a^{8} + 15 a^{7} + 2 a^{6} + 12 a^{5} + 16 a^{4} + 4 a^{3} + 2 a^{2} + 8 a + 1\right)\cdot 23^{40} + \left(4 a^{14} + 12 a^{13} + 11 a^{12} + 8 a^{11} + 19 a^{10} + 22 a^{9} + 16 a^{8} + 16 a^{7} + 14 a^{6} + 16 a^{5} + 8 a^{4} + 12 a^{3} + 22\right)\cdot 23^{41} + \left(17 a^{14} + 21 a^{12} + 11 a^{11} + 5 a^{10} + 10 a^{9} + a^{8} + 7 a^{6} + 14 a^{5} + 12 a^{4} + 11 a^{3} + 18 a^{2} + 11 a + 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15 a^{7} + a^{6} + 10 a^{4} + 22 a^{3} + 9\right)\cdot 23^{48} + \left(10 a^{14} + 21 a^{13} + 4 a^{12} + 2 a^{11} + 21 a^{10} + a^{9} + 17 a^{8} + 3 a^{7} + a^{6} + 10 a^{5} + 15 a^{4} + 11 a^{3} + 11 a^{2} + 9 a + 4\right)\cdot 23^{49} + \left(15 a^{14} + 5 a^{13} + 4 a^{12} + 7 a^{11} + 11 a^{10} + 18 a^{9} + 9 a^{8} + 13 a^{7} + a^{6} + 18 a^{5} + 6 a^{4} + 19 a^{3} + 17 a^{2} + 11 a + 4\right)\cdot 23^{50} + \left(11 a^{14} + a^{13} + 7 a^{12} + 21 a^{11} + 5 a^{10} + 19 a^{9} + 9 a^{8} + 11 a^{7} + 4 a^{6} + 10 a^{5} + 4 a^{4} + 12 a^{3} + 20 a^{2} + 2 a + 21\right)\cdot 23^{51} + \left(11 a^{14} + 8 a^{12} + 5 a^{11} + 13 a^{10} + 16 a^{9} + 18 a^{8} + 11 a^{7} + 3 a^{6} + 18 a^{5} + 12 a^{4} + 20 a^{3} + 13 a^{2} + 2 a + 5\right)\cdot 23^{52} + \left(14 a^{14} + 10 a^{13} + 17 a^{12} + 2 a^{11} + 13 a^{10} + a^{9} + 12 a^{8} + 20 a^{7} + 7 a^{6} + 14 a^{5} + a^{4} + 6 a^{3} + 7 a^{2} + 7 a + 4\right)\cdot 23^{53} + \left(20 a^{14} + 5 a^{13} + 12 a^{12} + 22 a^{11} + 9 a^{10} + 19 a^{9} + 8 a^{8} + 5 a^{6} + 7 a^{5} + 17 a^{4} + 17 a^{3} + 2 a^{2} + 11 a + 6\right)\cdot 23^{54} + \left(5 a^{14} + 17 a^{13} + 2 a^{12} + 3 a^{11} + 17 a^{10} + 22 a^{9} + 12 a^{8} + 2 a^{6} + 17 a^{5} + 20 a^{4} + 9 a^{2} + 8 a + 12\right)\cdot 23^{55} + \left(3 a^{14} + 6 a^{13} + 10 a^{12} + 6 a^{11} + 8 a^{10} + 13 a^{9} + 15 a^{8} + 4 a^{7} + 13 a^{6} + 6 a^{5} + 7 a^{4} + a^{3} + 18 a^{2} + 11 a + 21\right)\cdot 23^{56} + \left(15 a^{14} + a^{13} + 22 a^{12} + 14 a^{11} + 3 a^{9} + 19 a^{8} + 9 a^{7} + 12 a^{6} + 16 a^{5} + 9 a^{4} + 13 a^{3} + 6 a^{2} + 4 a + 6\right)\cdot 23^{57} + \left(19 a^{14} + 2 a^{13} + a^{12} + 9 a^{11} + 14 a^{10} + 15 a^{9} + 17 a^{8} + 21 a^{7} + 7 a^{6} + 22 a^{5} + 21 a^{4} + a^{3} + 13 a^{2} + 9 a + 4\right)\cdot 23^{58} + \left(8 a^{14} + 18 a^{13} + 4 a^{12} + 16 a^{11} + 3 a^{10} + 12 a^{9} + 20 a^{8} + 8 a^{7} + 2 a^{6} + 2 a^{5} + 5 a^{4} + 2 a^{3} + 12 a^{2} + 11\right)\cdot 23^{59} + \left(17 a^{14} + 10 a^{13} + 11 a^{12} + 20 a^{11} + 10 a^{10} + 21 a^{9} + 8 a^{8} + 9 a^{7} + 7 a^{6} + 14 a^{5} + 22 a^{4} + 22 a^{3} + 19 a^{2} + 22 a + 6\right)\cdot 23^{60} + \left(5 a^{14} + 2 a^{13} + 10 a^{12} + 15 a^{11} + 20 a^{10} + 20 a^{9} + 16 a^{8} + 16 a^{7} + 17 a^{6} + a^{5} + 16 a^{4} + 13 a^{3} + 6 a^{2} + 21 a + 6\right)\cdot 23^{61} + \left(8 a^{14} + 15 a^{13} + 7 a^{12} + 3 a^{11} + 14 a^{10} + 17 a^{9} + 5 a^{8} + 16 a^{7} + 18 a^{6} + 2 a^{5} + 18 a^{4} + 2 a^{3} + 20 a^{2} + 7 a + 17\right)\cdot 23^{62} + \left(17 a^{14} + 20 a^{13} + 19 a^{12} + 18 a^{11} + 8 a^{10} + 2 a^{9} + 11 a^{8} + 21 a^{7} + 19 a^{6} + a^{5} + 5 a^{4} + 17 a^{3} + 22 a^{2} + 19 a + 1\right)\cdot 23^{63} + \left(21 a^{13} + 7 a^{12} + 9 a^{11} + 21 a^{10} + 2 a^{9} + a^{8} + 11 a^{7} + 18 a^{6} + 11 a^{5} + 6 a^{4} + 21 a^{3} + 14 a^{2} + 16 a + 2\right)\cdot 23^{64} + \left(21 a^{14} + 21 a^{13} + 10 a^{12} + 21 a^{11} + 3 a^{10} + 2 a^{9} + 11 a^{8} + 17 a^{7} + 9 a^{5} + 14 a^{4} + 19 a^{3} + 6 a^{2} + 4 a + 17\right)\cdot 23^{65} 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11\right)\cdot 23^{71} + \left(5 a^{14} + 21 a^{13} + 20 a^{12} + 3 a^{11} + 15 a^{10} + 15 a^{9} + 9 a^{8} + 8 a^{7} + 18 a^{6} + 2 a^{5} + 22 a^{4} + 10 a^{3} + 9 a^{2} + 17 a + 6\right)\cdot 23^{72} + \left(15 a^{14} + 9 a^{13} + 16 a^{12} + 4 a^{11} + 16 a^{10} + 5 a^{9} + 21 a^{8} + 8 a^{6} + 21 a^{5} + 14 a^{3} + 21 a^{2} + 22 a + 22\right)\cdot 23^{73} + \left(2 a^{14} + 8 a^{13} + 22 a^{12} + 9 a^{11} + 8 a^{10} + 20 a^{9} + 19 a^{8} + 16 a^{7} + 18 a^{6} + 18 a^{5} + 22 a^{4} + 9 a^{3} + 22 a^{2} + 17 a + 6\right)\cdot 23^{74} + \left(22 a^{14} + 9 a^{13} + 3 a^{12} + 16 a^{11} + 15 a^{10} + 6 a^{9} + a^{8} + 19 a^{7} + 22 a^{6} + 14 a^{5} + 22 a^{4} + 17 a^{3} + 21 a^{2} + 5 a + 20\right)\cdot 23^{75} + \left(15 a^{14} + 20 a^{13} + 18 a^{12} + 6 a^{11} + 12 a^{10} + 6 a^{9} + 7 a^{8} + 7 a^{7} + 13 a^{6} + 12 a^{5} + a^{3} + 18 a^{2} + 22 a + 18\right)\cdot 23^{76} + \left(18 a^{14} + 5 a^{13} + 21 a^{12} + 8 a^{11} + 7 a^{10} + a^{9} + 12 a^{8} + 18 a^{7} + 2 a^{6} + 21 a^{5} + 16 a^{4} + 4 a^{3} + 11 a^{2} + 11 a + 9\right)\cdot 23^{77} + \left(5 a^{14} + 21 a^{13} + 10 a^{12} + 17 a^{11} + 18 a^{10} + 6 a^{9} + 15 a^{7} + 2 a^{5} + 20 a^{4} + 20 a^{3} + 19 a^{2} + 21 a + 8\right)\cdot 23^{78} + \left(19 a^{14} + 9 a^{13} + 18 a^{11} + 14 a^{9} + 11 a^{8} + 3 a^{7} + 13 a^{6} + 21 a^{5} + 5 a^{4} + 15 a^{3} + 10 a^{2} + 17 a + 6\right)\cdot 23^{79} + \left(3 a^{14} + 13 a^{13} + 11 a^{12} + 2 a^{11} + 20 a^{10} + 6 a^{8} + 12 a^{7} + 18 a^{6} + 6 a^{5} + 2 a^{4} + 3 a^{3} + 18 a^{2} + 19 a + 14\right)\cdot 23^{80} + \left(10 a^{14} + 9 a^{13} + 7 a^{12} + 22 a^{11} + 17 a^{10} + 20 a^{9} + 20 a^{8} + 7 a^{7} + 14 a^{6} + 15 a^{5} + 4 a^{4} + 4 a^{3} + 10 a^{2} + 2\right)\cdot 23^{81} + \left(22 a^{14} + 19 a^{13} + 20 a^{12} + 12 a^{11} + 17 a^{10} + 16 a^{9} + 9 a^{8} + 3 a^{7} + 12 a^{6} + 2 a^{5} + 12 a^{4} + 9 a^{3} + 13 a^{2} + 5 a + 8\right)\cdot 23^{82} + \left(14 a^{14} + 17 a^{13} + 2 a^{12} + 18 a^{11} + 13 a^{10} + 13 a^{9} + 6 a^{8} + 10 a^{7} + a^{6} + 3 a^{5} + 19 a^{4} + 5 a^{3} + 22 a + 13\right)\cdot 23^{83} + \left(6 a^{14} + 20 a^{13} + 7 a^{12} + 11 a^{11} + 19 a^{10} + 4 a^{9} + 21 a^{8} + a^{7} + 11 a^{6} + a^{5} + a^{4} + 11 a^{3} + 19 a^{2} + 13 a + 10\right)\cdot 23^{84} + \left(a^{14} + 13 a^{13} + a^{12} + 7 a^{11} + 4 a^{10} + 16 a^{9} + 3 a^{8} + 15 a^{6} + 11 a^{5} + 6 a^{4} + 17 a^{3} + 4 a^{2} + 10 a + 8\right)\cdot 23^{85} + \left(10 a^{14} + 8 a^{13} + 9 a^{12} + 12 a^{11} + 9 a^{10} + 7 a^{9} + 2 a^{8} + 19 a^{7} + 9 a^{6} + 18 a^{5} + 21 a^{4} + 11 a^{3} + 22 a^{2} + a + 11\right)\cdot 23^{86} + \left(22 a^{14} + 5 a^{13} + 14 a^{12} + 22 a^{11} + 2 a^{10} + 8 a^{9} + 7 a^{8} + 11 a^{7} + 14 a^{6} + 6 a^{5} + 6 a^{4} + 2 a^{3} + 6 a + 21\right)\cdot 23^{87} + \left(16 a^{14} + 17 a^{13} + 6 a^{12} + 14 a^{11} + 7 a^{10} + 13 a^{9} + 4 a^{8} + a^{7} + 22 a^{6} + 9 a^{5} + 7 a^{4} + 10 a^{3} + 15 a^{2} + 9 a + 18\right)\cdot 23^{88} + \left(8 a^{14} + 5 a^{13} + 18 a^{12} + 19 a^{11} + 19 a^{10} + 10 a^{9} + 3 a^{8} + 2 a^{7} + 13 a^{6} + 16 a^{5} + 2 a^{3} + 8 a^{2} + 13\right)\cdot 23^{89} + \left(19 a^{14} + 3 a^{12} + 16 a^{11} + 8 a^{10} + 20 a^{9} + 4 a^{8} + 6 a^{7} + 13 a^{6} + 2 a^{5} + 13 a^{4} + 19 a^{3} + 8 a + 20\right)\cdot 23^{90} + \left(14 a^{14} + 10 a^{13} + 15 a^{12} + 4 a^{11} + 20 a^{10} + 6 a^{9} + 10 a^{8} + 13 a^{7} + 8 a^{5} + 17 a^{4} + 14 a^{3} + 15 a^{2} + 20 a + 9\right)\cdot 23^{91} + \left(a^{14} + a^{13} + 3 a^{12} + 21 a^{11} + 6 a^{10} + 19 a^{9} + 13 a^{8} + 20 a^{7} + 11 a^{6} + 7 a^{5} + 14 a^{4} + a^{3} + 17 a^{2} + 20 a + 17\right)\cdot 23^{92} + \left(14 a^{14} + 10 a^{13} + 10 a^{11} + 8 a^{10} + 6 a^{9} + 5 a^{8} + 17 a^{6} + 15 a^{5} + 6 a^{4} + 21 a^{3} + 9 a^{2} + 14 a + 18\right)\cdot 23^{93} + \left(19 a^{14} + 4 a^{13} + 21 a^{12} + 2 a^{11} + 10 a^{10} + 18 a^{9} + 4 a^{8} + 13 a^{6} + 14 a^{5} + 11 a^{4} + 12 a^{3} + 21 a^{2} + 8 a + 19\right)\cdot 23^{94} + \left(2 a^{13} + 15 a^{12} + 5 a^{11} + 16 a^{10} + 16 a^{9} + 5 a^{8} + a^{7} + 22 a^{6} + 6 a^{5} + 15 a^{4} + 3 a^{3} + 18 a^{2} + 22 a + 17\right)\cdot 23^{95} + \left(12 a^{14} + 2 a^{13} + 10 a^{12} + 14 a^{11} + 15 a^{10} + 2 a^{9} + 12 a^{8} + 17 a^{7} + 20 a^{6} + 9 a^{5} + 7 a^{4} + 21 a^{3} + 12 a^{2} + 22 a + 2\right)\cdot 23^{96} + \left(7 a^{14} + 13 a^{13} + 11 a^{12} + 4 a^{11} + 20 a^{10} + 12 a^{9} + 22 a^{8} + 16 a^{7} + 7 a^{6} + 14 a^{5} + 16 a^{4} + 17 a^{3} + 19 a^{2} + 12 a + 12\right)\cdot 23^{97} + \left(9 a^{14} + 6 a^{13} + 2 a^{12} + 21 a^{11} + 21 a^{10} + 21 a^{9} + 17 a^{8} + 20 a^{7} + 11 a^{6} + 17 a^{5} + 6 a^{4} + 16 a^{3} + 22 a^{2} + 16 a + 16\right)\cdot 23^{98} + \left(8 a^{14} + 15 a^{13} + 15 a^{12} + 7 a^{11} + 19 a^{10} + 18 a^{9} + 5 a^{8} + 4 a^{7} + 18 a^{6} + 6 a^{5} + 20 a^{4} + 8 a^{3} + 6 a^{2} + 15\right)\cdot 23^{99} + \left(2 a^{14} + 13 a^{13} + 9 a^{12} + 18 a^{11} + 22 a^{10} + 20 a^{9} + 17 a^{8} + 9 a^{7} + 3 a^{6} + 14 a^{5} + 5 a^{4} + 8 a^{3} + 9 a^{2} + 4 a + 13\right)\cdot 23^{100} + \left(8 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19\right)\cdot 23^{106} + \left(20 a^{14} + 21 a^{13} + 14 a^{12} + 17 a^{11} + 21 a^{10} + 7 a^{9} + 3 a^{8} + 22 a^{7} + 19 a^{6} + 4 a^{5} + a^{4} + 13 a^{3} + 22 a^{2} + 9 a + 19\right)\cdot 23^{107} + \left(22 a^{13} + 4 a^{12} + 2 a^{11} + 16 a^{10} + 16 a^{9} + 2 a^{8} + 7 a^{6} + 4 a^{5} + 20 a^{4} + 9 a^{3} + 22 a^{2} + 17 a + 22\right)\cdot 23^{108} + \left(15 a^{14} + 2 a^{13} + 19 a^{12} + 7 a^{11} + 2 a^{10} + 7 a^{9} + 22 a^{8} + 20 a^{7} + 21 a^{6} + 19 a^{5} + 6 a^{4} + 9 a^{3} + 13 a^{2} + 20 a + 9\right)\cdot 23^{109} + \left(5 a^{14} + 6 a^{13} + 21 a^{12} + 14 a^{11} + 3 a^{10} + 17 a^{9} + 14 a^{8} + 7 a^{7} + 2 a^{6} + 9 a^{5} + 18 a^{4} + 12 a^{3} + 6 a^{2} + 3 a + 14\right)\cdot 23^{110} + \left(6 a^{14} + 14 a^{13} + 14 a^{12} + 9 a^{11} + 19 a^{10} + 22 a^{9} + 18 a^{8} + 6 a^{7} + 19 a^{6} + 16 a^{5} + 19 a^{4} + 8 a^{3} + 6 a^{2} + 2 a + 16\right)\cdot 23^{111} + \left(13 a^{14} + 7 a^{13} + 14 a^{12} + 12 a^{11} + 15 a^{10} + 4 a^{9} + 8 a^{8} + 5 a^{6} + 6 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8 a^{7} + 21 a^{6} + 2 a^{5} + 10 a^{4} + 10 a^{3} + 21 a^{2} + 11 a + 22\right)\cdot 23^{129} + \left(14 a^{14} + 12 a^{12} + 19 a^{11} + 14 a^{10} + 17 a^{9} + 17 a^{8} + 5 a^{7} + 2 a^{6} + 20 a^{5} + 6 a^{4} + 19 a^{3} + 3 a^{2} + 7 a + 4\right)\cdot 23^{130} + \left(12 a^{14} + 8 a^{13} + 18 a^{12} + 13 a^{11} + 9 a^{10} + 14 a^{9} + 11 a^{8} + 12 a^{7} + 22 a^{6} + 8 a^{5} + 10 a^{4} + 22 a^{3} + 2 a^{2} + 12 a + 12\right)\cdot 23^{131} + \left(22 a^{14} + 16 a^{13} + 20 a^{12} + a^{11} + 9 a^{10} + a^{9} + 14 a^{8} + 21 a^{7} + 5 a^{6} + 19 a^{5} + 16 a^{4} + 9 a^{2} + 17 a + 21\right)\cdot 23^{132} + \left(11 a^{14} + 16 a^{13} + 9 a^{12} + 20 a^{11} + 11 a^{10} + 16 a^{9} + 16 a^{8} + 12 a^{7} + 18 a^{6} + 13 a^{5} + 12 a^{4} + 9 a^{3} + 2 a^{2} + 21 a + 6\right)\cdot 23^{133} + \left(4 a^{14} + 19 a^{13} + 15 a^{12} + 14 a^{11} + 18 a^{10} + 8 a^{9} + 5 a^{8} + 5 a^{7} + 15 a^{5} + 11 a^{4} + 14 a^{3} + 18 a^{2} + 3 a + 19\right)\cdot 23^{134} + \left(19 a^{14} + 3 a^{13} + 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a + 10\right)\cdot 23^{152} + \left(15 a^{14} + 2 a^{13} + 8 a^{11} + 13 a^{10} + 10 a^{9} + a^{8} + 19 a^{7} + 11 a^{6} + a^{5} + 14 a^{4} + 8 a^{3} + 17 a^{2} + 20\right)\cdot 23^{153} + \left(8 a^{14} + 21 a^{13} + 18 a^{12} + 19 a^{11} + 10 a^{10} + 16 a^{9} + 3 a^{8} + 17 a^{7} + 20 a^{6} + 3 a^{5} + 10 a^{4} + 22 a^{3} + 8 a^{2} + 7 a + 5\right)\cdot 23^{154} + \left(21 a^{14} + 5 a^{12} + 12 a^{11} + 14 a^{10} + 21 a^{9} + 14 a^{8} + 8 a^{7} + 11 a^{6} + 17 a^{5} + 10 a^{4} + 17 a^{2} + 21 a + 14\right)\cdot 23^{155} + \left(7 a^{14} + 18 a^{13} + 19 a^{12} + 4 a^{11} + 12 a^{10} + 3 a^{9} + 3 a^{8} + 18 a^{7} + 11 a^{6} + 20 a^{5} + 10 a^{4} + 18 a^{3} + 11 a\right)\cdot 23^{156} + \left(20 a^{14} + 17 a^{13} + 9 a^{12} + 7 a^{11} + 17 a^{10} + 8 a^{8} + 15 a^{7} + 13 a^{6} + 10 a^{5} + 14 a^{4} + 7 a^{2} + 11\right)\cdot 23^{157} + \left(21 a^{14} + 18 a^{13} + 15 a^{12} + 4 a^{11} + 5 a^{10} + 11 a^{9} + 9 a^{8} + 8 a^{7} + 15 a^{6} + 17 a^{5} + 14 a^{4} + 5 a^{3} + 22 a^{2} + 21 a + 4\right)\cdot 23^{158} + \left(6 a^{14} + 10 a^{13} + 17 a^{12} + 2 a^{10} + 8 a^{9} + 19 a^{8} + 2 a^{7} + 15 a^{6} + 18 a^{5} + 12 a^{4} + 7 a^{3} + 16 a^{2} + 14 a + 14\right)\cdot 23^{159} + \left(22 a^{13} + 15 a^{12} + 6 a^{11} + 7 a^{10} + 18 a^{8} + 14 a^{7} + 20 a^{6} + 10 a^{5} + 8 a^{4} + 16 a^{3} + 22 a^{2} + 17 a + 4\right)\cdot 23^{160} + \left(10 a^{14} + 22 a^{12} + 15 a^{11} + 22 a^{10} + 19 a^{9} + 4 a^{8} + 10 a^{7} + a^{6} + 12 a^{5} + 16 a^{4} + 21 a^{3} + 3 a^{2} + 21 a + 22\right)\cdot 23^{161} + \left(13 a^{14} + 2 a^{13} + 22 a^{12} + 4 a^{11} + 14 a^{10} + 15 a^{9} + 21 a^{7} + 5 a^{6} + 14 a^{5} + 17 a^{4} + 17 a^{3} + 11 a^{2} + 5 a + 19\right)\cdot 23^{162} + \left(4 a^{14} + 21 a^{13} + 20 a^{12} + 16 a^{11} + 5 a^{10} + 8 a^{9} + 4 a^{7} + 2 a^{6} + 21 a^{5} + 3 a^{4} + 2 a^{3} + 8 a^{2} + 8 a + 4\right)\cdot 23^{163} + \left(18 a^{14} + 12 a^{13} + 3 a^{12} + 11 a^{11} + 6 a^{10} + 14 a^{9} + 11 a^{8} + 14 a^{7} + 7 a^{6} + 9 a^{5} + 11 a^{4} + 18 a^{3} + 3 a^{2} + 3 a + 14\right)\cdot 23^{164} + \left(16 a^{14} + 19 a^{13} + 22 a^{12} + 6 a^{11} + 18 a^{10} + 8 a^{9} + 8 a^{8} + 22 a^{7} + 11 a^{6} + 14 a^{5} + 15 a^{4} + 18 a^{3} + 18 a^{2} + 4 a + 7\right)\cdot 23^{165} + \left(19 a^{14} + 9 a^{13} + 3 a^{12} + 15 a^{11} + 22 a^{10} + 16 a^{9} + 12 a^{8} + 18 a^{7} + 14 a^{6} + 20 a^{5} + a^{4} + 11 a^{3} + 13 a^{2} + 8 a + 1\right)\cdot 23^{166} + \left(19 a^{14} + 20 a^{13} + a^{12} + 21 a^{11} + 7 a^{10} + 14 a^{9} + 17 a^{8} + 17 a^{7} + 5 a^{6} + 2 a^{5} + 13 a^{4} + 2 a^{3} + 15 a^{2} + 7 a + 11\right)\cdot 23^{167} + \left(12 a^{13} + 14 a^{12} + 20 a^{11} + 21 a^{9} + 8 a^{8} + 15 a^{7} + 20 a^{6} + 4 a^{5} + 7 a^{4} + 8 a^{3} + 12 a^{2} + 22 a + 12\right)\cdot 23^{168} + \left(18 a^{14} + 2 a^{13} + 6 a^{12} + 12 a^{11} + 4 a^{10} + 2 a^{9} + 2 a^{8} + 15 a^{7} + 15 a^{6} + 9 a^{5} + 16 a^{3} + 15 a^{2} + 18 a + 18\right)\cdot 23^{169} + \left(12 a^{14} + 15 a^{13} + 17 a^{12} + 8 a^{11} + 14 a^{10} + 21 a^{9} + 21 a^{8} + 19 a^{6} + 15 a^{5} + 12 a^{4} + 18 a^{3} + 12 a^{2} + 4 a + 19\right)\cdot 23^{170} + \left(22 a^{14} + 8 a^{13} + 5 a^{12} + 2 a^{11} + 18 a^{10} + 9 a^{9} + 20 a^{7} + 12 a^{6} + 16 a^{5} + 20 a^{4} + 15 a^{3} + 18 a^{2} + 7 a + 9\right)\cdot 23^{171} + \left(3 a^{14} + a^{13} + 4 a^{12} + 17 a^{11} + 17 a^{10} + 20 a^{9} + 20 a^{8} + 13 a^{7} + 18 a^{6} + 19 a^{5} + 10 a^{4} + 7 a^{3} + 21 a^{2} + 20\right)\cdot 23^{172} + \left(13 a^{14} + 20 a^{13} + 7 a^{12} + 19 a^{11} + 17 a^{10} + 22 a^{9} + 7 a^{8} + 17 a^{7} + 6 a^{6} + 17 a^{5} + 7 a^{4} + 15 a^{3} + 15 a^{2} + 14\right)\cdot 23^{173} + \left(12 a^{14} + 7 a^{13} + 14 a^{12} + 13 a^{11} + 14 a^{10} + 16 a^{9} + 11 a^{8} + 7 a^{7} + 13 a^{6} + 16 a^{5} + 10 a^{4} + 10 a^{3} + 7 a^{2} + 9 a + 13\right)\cdot 23^{174} + \left(17 a^{14} + 17 a^{13} + 10 a^{12} + 14 a^{11} + 16 a^{10} + 15 a^{9} + 18 a^{8} + 17 a^{7} + 17 a^{6} + 2 a^{5} + 5 a^{4} + 16 a^{3} + 15 a^{2} + 19 a + 18\right)\cdot 23^{175} + \left(20 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a^{3} + 11 a^{2} + 12 a + 14\right)\cdot 23^{181} + \left(22 a^{14} + 18 a^{13} + 2 a^{12} + 5 a^{11} + 9 a^{10} + 18 a^{9} + a^{8} + 12 a^{7} + 21 a^{6} + 8 a^{5} + 21 a^{4} + 2 a^{3} + 8 a^{2} + a + 18\right)\cdot 23^{182} + \left(2 a^{14} + 20 a^{13} + 13 a^{12} + 20 a^{11} + 21 a^{10} + 22 a^{9} + 12 a^{8} + 9 a^{7} + 7 a^{6} + 12 a^{5} + 12 a^{4} + 22 a^{3} + 21 a^{2} + 11 a + 10\right)\cdot 23^{183} + \left(5 a^{14} + a^{12} + 3 a^{10} + 9 a^{9} + 21 a^{8} + 20 a^{7} + 2 a^{6} + 14 a^{5} + 22 a^{4} + 19 a^{3} + 22 a^{2} + 18 a + 17\right)\cdot 23^{184} + \left(4 a^{14} + 3 a^{13} + 9 a^{11} + 20 a^{10} + 21 a^{9} + 18 a^{8} + 8 a^{7} + 15 a^{6} + 15 a^{5} + a^{4} + 9 a^{3} + 14 a^{2} + 15 a + 18\right)\cdot 23^{185} + \left(20 a^{14} + 12 a^{12} + 20 a^{11} + 3 a^{10} + 22 a^{9} + 8 a^{8} + 6 a^{7} + 18 a^{6} + 10 a^{5} + 8 a^{4} + 10 a^{3} + 6 a^{2} + 19 a + 21\right)\cdot 23^{186} + \left(14 a^{14} + 20 a^{13} + 11 a^{12} + 18 a^{11} + 12 a^{10} + 8 a^{9} + a^{8} + 15 a^{7} + 17 a^{6} + 10 a^{5} + 22 a^{4} + a^{3} + 8 a^{2} + 19 a + 20\right)\cdot 23^{187} + \left(19 a^{14} + 22 a^{13} + 21 a^{12} + 11 a^{11} + 4 a^{10} + 15 a^{9} + 19 a^{8} + 10 a^{7} + 13 a^{6} + 20 a^{5} + 12 a^{4} + 14 a^{3} + 5 a + 5\right)\cdot 23^{188} + \left(a^{14} + 7 a^{13} + a^{12} + 8 a^{11} + 8 a^{10} + 12 a^{9} + 10 a^{8} + 17 a^{7} + 8 a^{6} + 7 a^{5} + 20 a^{4} + 7 a^{3} + 14 a^{2} + 11 a + 12\right)\cdot 23^{189} + \left(20 a^{14} + 21 a^{13} + 15 a^{12} + 22 a^{11} + 12 a^{10} + 10 a^{9} + 13 a^{8} + 6 a^{7} + 17 a^{6} + 14 a^{5} + 21 a^{4} + 14 a^{3} + 10 a^{2} + 20 a + 20\right)\cdot 23^{190} + \left(20 a^{14} + 21 a^{13} + 18 a^{12} + 21 a^{11} + 13 a^{10} + 21 a^{9} + 21 a^{8} + 18 a^{7} + 12 a^{6} + 21 a^{5} + 15 a^{4} + 5 a^{3} + 19 a^{2} + 18 a + 9\right)\cdot 23^{191} + \left(10 a^{14} + 14 a^{13} + 16 a^{12} + 22 a^{11} + 20 a^{10} + 8 a^{9} + 15 a^{8} + 9 a^{7} + 2 a^{6} + 5 a^{5} + 19 a^{4} + 8 a^{3} + 13 a^{2} + 8 a + 12\right)\cdot 23^{192} + \left(2 a^{14} + 5 a^{13} + 7 a^{12} + 15 a^{11} + 10 a^{10} + 5 a^{9} + 20 a^{8} + 21 a^{7} + 2 a^{6} + 17 a^{5} + 4 a^{3} + 21 a^{2} + 7 a + 21\right)\cdot 23^{193} + \left(7 a^{14} + a^{13} + 3 a^{12} + 15 a^{11} + 16 a^{10} + 21 a^{9} + a^{8} + 7 a^{7} + 6 a^{6} + 16 a^{5} + 11 a^{4} + 22 a^{3} + 10 a^{2} + 19 a + 6\right)\cdot 23^{194} + \left(7 a^{13} + 3 a^{12} + 6 a^{11} + 15 a^{10} + 19 a^{9} + 20 a^{8} + 4 a^{7} + a^{6} + 9 a^{5} + 22 a^{4} + 18 a^{3} + 20 a^{2} + 21 a + 18\right)\cdot 23^{195} + \left(5 a^{14} + 16 a^{13} + 12 a^{12} + 6 a^{11} + 6 a^{9} + 3 a^{8} + 14 a^{7} + 15 a^{6} + 8 a^{5} + 12 a^{4} + 10 a^{3} + 10 a^{2} + 4 a + 6\right)\cdot 23^{196} + \left(5 a^{14} + 8 a^{13} + 20 a^{12} + 10 a^{11} + 19 a^{10} + 21 a^{9} + 7 a^{8} + 20 a^{7} + a^{6} + 11 a^{5} + 10 a^{3} + 22 a^{2} + 3 a + 10\right)\cdot 23^{197} + \left(12 a^{14} + 18 a^{13} + 9 a^{12} + 20 a^{11} + 17 a^{10} + 20 a^{9} + 11 a^{8} + 15 a^{7} + 4 a^{6} + 16 a^{5} + 19 a^{4} + 13 a^{3} + 12 a^{2} + 18 a + 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a^{5} + 6 a^{4} + 5 a^{3} + 13 a^{2} + 21 a + 13\right)\cdot 23^{204} + \left(19 a^{14} + 15 a^{13} + 13 a^{12} + 16 a^{11} + 16 a^{10} + 10 a^{7} + 19 a^{6} + 14 a^{5} + 3 a^{4} + 6 a^{3} + 15 a^{2} + 11\right)\cdot 23^{205} + \left(10 a^{14} + 10 a^{13} + 17 a^{12} + 6 a^{11} + 7 a^{10} + 6 a^{9} + 2 a^{8} + 7 a^{7} + 3 a^{6} + 20 a^{5} + 7 a^{4} + 20 a^{3} + 4 a^{2} + a + 10\right)\cdot 23^{206} + \left(3 a^{14} + 6 a^{13} + 6 a^{12} + 11 a^{11} + 10 a^{10} + 6 a^{9} + 10 a^{8} + 5 a^{7} + 20 a^{6} + 9 a^{5} + 6 a^{4} + 9 a^{3} + 12 a^{2} + 10\right)\cdot 23^{207} + \left(8 a^{14} + 8 a^{13} + 12 a^{12} + 15 a^{11} + 7 a^{10} + a^{9} + 3 a^{8} + 4 a^{7} + 9 a^{6} + 16 a^{5} + 11 a^{4} + 4 a^{3} + 14 a^{2} + 8 a + 15\right)\cdot 23^{208} + \left(22 a^{14} + 9 a^{13} + 18 a^{12} + 8 a^{11} + 6 a^{10} + 18 a^{9} + 16 a^{8} + 2 a^{7} + 11 a^{6} + 7 a^{5} + 22 a^{4} + 6 a^{3} + 7 a^{2} + 7 a + 12\right)\cdot 23^{209} + \left(10 a^{14} + 6 a^{13} + 14 a^{12} + 14 a^{11} + 18 a^{10} + 22 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a^{4} + 3 a^{3} + 17 a^{2} + 22 a + 17\right)\cdot 23^{221} + \left(5 a^{14} + 22 a^{13} + 17 a^{12} + 16 a^{11} + 14 a^{10} + 16 a^{9} + 8 a^{8} + 21 a^{7} + 13 a^{6} + 2 a^{5} + 7 a^{4} + a^{3} + 7 a^{2} + a + 6\right)\cdot 23^{222} + \left(16 a^{14} + 2 a^{13} + 10 a^{12} + 5 a^{11} + 19 a^{10} + 15 a^{9} + 17 a^{8} + a^{7} + 12 a^{6} + 6 a^{5} + 12 a^{4} + 5 a^{3} + 11 a^{2} + 21 a + 11\right)\cdot 23^{223} + \left(18 a^{14} + 2 a^{13} + 22 a^{12} + 5 a^{11} + 17 a^{10} + 7 a^{9} + 15 a^{8} + 9 a^{7} + 14 a^{6} + 13 a^{5} + 8 a^{4} + 6 a^{3} + 9 a + 15\right)\cdot 23^{224} + \left(15 a^{14} + 15 a^{13} + 8 a^{12} + 15 a^{11} + 15 a^{10} + 8 a^{9} + 16 a^{8} + 10 a^{7} + 16 a^{6} + 3 a^{4} + 21 a^{3} + 11 a^{2} + 17 a + 13\right)\cdot 23^{225} + \left(20 a^{14} + 7 a^{13} + 3 a^{12} + 2 a^{11} + 22 a^{10} + 13 a^{9} + 18 a^{8} + 6 a^{7} + 19 a^{6} + 20 a^{5} + 21 a^{4} + 6 a^{3} + 20 a^{2} + 9 a + 21\right)\cdot 23^{226} + \left(3 a^{14} + 14 a^{13} + 8 a^{12} + 18 a^{11} + 8 a^{10} + 22 a^{9} + 2 a^{8} + 10 a^{7} + 4 a^{6} + 8 a^{5} + 15 a^{4} + 4 a^{3} + a^{2} + 17 a + 1\right)\cdot 23^{227} + \left(21 a^{14} + a^{13} + 9 a^{12} + 16 a^{11} + 14 a^{10} + 4 a^{9} + 7 a^{8} + 7 a^{7} + 8 a^{6} + 7 a^{5} + 7 a^{4} + 10 a^{3} + 4 a^{2} + 2 a + 20\right)\cdot 23^{228} + \left(16 a^{13} + 16 a^{12} + 20 a^{11} + 4 a^{10} + 4 a^{9} + 3 a^{8} + 15 a^{7} + 4 a^{6} + 16 a^{5} + 2 a^{4} + 10 a^{3} + 14 a^{2} + 10 a + 20\right)\cdot 23^{229} + \left(17 a^{14} + 18 a^{13} + 6 a^{12} + 6 a^{11} + 11 a^{10} + 12 a^{9} + a^{8} + 8 a^{7} + 11 a^{6} + 18 a^{5} + a^{4} + 22 a^{3} + 17 a^{2} + 3 a + 19\right)\cdot 23^{230} + \left(19 a^{14} + 18 a^{13} + 8 a^{12} + 7 a^{11} + 3 a^{10} + 10 a^{9} + a^{8} + 17 a^{7} + 15 a^{6} + 6 a^{5} + 13 a^{4} + 15 a^{3} + 3 a^{2} + 11 a + 1\right)\cdot 23^{231} + \left(22 a^{14} + 22 a^{13} + 9 a^{12} + 12 a^{11} + 11 a^{10} + 18 a^{9} + 13 a^{8} + 18 a^{7} + 21 a^{6} + 15 a^{5} + 19 a^{4} + 9 a^{3} + 6 a^{2} + 18 a + 10\right)\cdot 23^{232} + \left(19 a^{14} + 5 a^{12} + 16 a^{11} + 14 a^{10} + 6 a^{9} + 4 a^{8} + 19 a^{7} + 2 a^{6} + 10 a^{4} + 12 a^{3} + 12 a^{2} + 5 a + 19\right)\cdot 23^{233} + \left(3 a^{14} + a^{13} + 21 a^{12} + 18 a^{11} + 6 a^{10} + 16 a^{9} + 20 a^{8} + 2 a^{7} + 6 a^{6} + 2 a^{5} + 13 a^{4} + 6 a^{3} + 11 a^{2} + 7 a + 14\right)\cdot 23^{234} + \left(10 a^{14} + 10 a^{13} + 22 a^{11} + 20 a^{10} + a^{9} + a^{8} + 20 a^{7} + 5 a^{6} + 20 a^{5} + 20 a^{4} + 9 a^{3} + 8 a^{2} + 21 a + 7\right)\cdot 23^{235} + \left(3 a^{14} + 13 a^{13} + 2 a^{12} + 13 a^{11} + 3 a^{10} + 7 a^{9} + 7 a^{8} + 4 a^{6} + 19 a^{5} + 5 a^{4} + 19 a^{3} + 15 a^{2} + 12 a + 21\right)\cdot 23^{236} + \left(14 a^{14} + 18 a^{13} + 7 a^{12} + 20 a^{11} + 11 a^{10} + 8 a^{9} + 18 a^{8} + 22 a^{7} + 14 a^{6} + 16 a^{5} + 5 a^{4} + 13 a^{3} + 5 a^{2} + 16 a + 3\right)\cdot 23^{237} + \left(17 a^{13} + 17 a^{12} + 21 a^{11} + 22 a^{10} + 15 a^{9} + 14 a^{8} + 9 a^{7} + 11 a^{6} + a^{5} + 16 a^{4} + 18 a^{3} + 13 a^{2} + 19 a + 20\right)\cdot 23^{238} + \left(10 a^{14} + 11 a^{13} + 11 a^{12} + 12 a^{11} + 7 a^{10} + 21 a^{9} + 19 a^{8} + 3 a^{7} + 4 a^{6} + 18 a^{5} + 22 a^{4} + 10 a^{3} + 4 a^{2} + 19 a + 22\right)\cdot 23^{239} + \left(7 a^{14} + 2 a^{13} + 7 a^{12} + 16 a^{11} + 20 a^{10} + 9 a^{9} + 12 a^{8} + 15 a^{7} + 13 a^{6} + 6 a^{5} + 16 a^{4} + 21 a^{3} + 5 a^{2} + 9 a + 10\right)\cdot 23^{240} + \left(20 a^{14} + 3 a^{13} + 6 a^{12} + 10 a^{10} + 16 a^{9} + 4 a^{8} + 11 a^{7} + 22 a^{6} + a^{5} + 10 a^{4} + 13 a^{3} + 17 a^{2} + 16 a + 2\right)\cdot 23^{241} + \left(6 a^{14} + 6 a^{13} + 11 a^{12} + 4 a^{11} + 5 a^{10} + 20 a^{9} + 10 a^{8} + 9 a^{7} + 6 a^{6} + 3 a^{5} + 11 a^{4} + 15 a^{3} + 7 a^{2} + 2 a\right)\cdot 23^{242} + \left(17 a^{14} + 21 a^{13} + 22 a^{12} + 5 a^{11} + 8 a^{10} + 17 a^{9} + 15 a^{8} + 12 a^{7} + 6 a^{6} + 7 a^{5} + 19 a^{4} + 12 a^{3} + 20 a^{2} + 20 a + 15\right)\cdot 23^{243} + \left(13 a^{14} + 13 a^{13} + 17 a^{12} + a^{11} + 7 a^{10} + 22 a^{9} + 16 a^{8} + 19 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13 a^{11} + 10 a^{10} + 5 a^{9} + 15 a^{8} + 11 a^{7} + 8 a^{6} + 18 a^{5} + 11 a^{4} + 21 a^{3} + 19 a^{2} + 15 a + 9\right)\cdot 23^{250} + \left(6 a^{14} + 6 a^{13} + 4 a^{12} + 18 a^{11} + 8 a^{10} + 19 a^{9} + 5 a^{8} + 4 a^{7} + 10 a^{6} + 17 a^{5} + 22 a^{4} + 13 a^{3} + 6 a^{2} + 15 a + 4\right)\cdot 23^{251} + \left(8 a^{14} + 12 a^{13} + 16 a^{12} + 13 a^{11} + 3 a^{10} + 10 a^{9} + 6 a^{8} + 20 a^{7} + 7 a^{6} + 5 a^{5} + 16 a^{4} + 17 a^{3} + 13 a^{2} + 21 a + 15\right)\cdot 23^{252} + \left(20 a^{14} + 2 a^{13} + 8 a^{12} + 5 a^{11} + 6 a^{10} + 6 a^{9} + 10 a^{8} + 15 a^{7} + 6 a^{6} + 7 a^{5} + 21 a^{4} + 4 a^{3} + 17 a^{2} + 13 a + 22\right)\cdot 23^{253} + \left(6 a^{14} + a^{13} + 20 a^{12} + 10 a^{11} + 17 a^{10} + 14 a^{9} + 13 a^{8} + 4 a^{7} + 5 a^{6} + 14 a^{5} + 15 a^{4} + 6 a^{3} + 14 a^{2} + 7 a + 19\right)\cdot 23^{254} + \left(19 a^{14} + 9 a^{13} + 15 a^{12} + 20 a^{11} + 18 a^{10} + 20 a^{9} + a^{8} + 15 a^{7} + 4 a^{6} + 10 a^{5} + a^{4} + 14 a^{3} + 14 a^{2} + 14 a + 6\right)\cdot 23^{255} + \left(21 a^{14} + 8 a^{13} + 17 a^{12} + 4 a^{11} + 6 a^{10} + 4 a^{9} + 2 a^{8} + 21 a^{7} + 22 a^{6} + 2 a^{5} + 12 a^{4} + 17 a^{3} + 16 a^{2} + 16 a + 21\right)\cdot 23^{256} + \left(12 a^{14} + 20 a^{13} + 13 a^{12} + 2 a^{11} + 12 a^{10} + 15 a^{9} + 16 a^{8} + 2 a^{7} + 11 a^{6} + 14 a^{5} + 5 a^{3} + 6 a^{2} + 22\right)\cdot 23^{257} + \left(3 a^{13} + 8 a^{12} + a^{11} + 12 a^{10} + 20 a^{9} + 2 a^{8} + 19 a^{7} + 12 a^{6} + 11 a^{5} + 15 a^{4} + 13 a^{3} + 3 a^{2} + 17 a + 20\right)\cdot 23^{258} + \left(4 a^{14} + 22 a^{13} + 5 a^{12} + 4 a^{11} + 10 a^{10} + 9 a^{9} + 8 a^{8} + 16 a^{7} + 22 a^{6} + 16 a^{5} + 8 a^{4} + 9 a^{3} + 9 a^{2} + 7 a + 5\right)\cdot 23^{259} + \left(6 a^{14} + 2 a^{13} + 3 a^{12} + a^{11} + 15 a^{10} + 22 a^{9} + 18 a^{8} + 3 a^{7} + 6 a^{6} + 17 a^{5} + 5 a^{4} + 20 a^{3} + 16 a^{2} + 19 a + 7\right)\cdot 23^{260} + \left(2 a^{14} + 14 a^{13} + 22 a^{12} + 13 a^{11} + 7 a^{10} + 17 a^{9} + 7 a^{8} + 18 a^{6} + 15 a^{5} + 22 a^{4} + 6 a^{3} + a^{2} + 13 a + 14\right)\cdot 23^{261} + \left(5 a^{13} + 2 a^{12} + 6 a^{11} + 8 a^{10} + 11 a^{9} + 15 a^{8} + 13 a^{7} + 13 a^{6} + 2 a^{5} + 13 a^{4} + 12 a^{3} + 17 a^{2} + 5 a + 13\right)\cdot 23^{262} + \left(21 a^{14} + 10 a^{13} + 5 a^{12} + 4 a^{10} + 21 a^{9} + 17 a^{8} + 21 a^{7} + 21 a^{4} + 12 a^{3} + 13 a^{2} + 20 a + 16\right)\cdot 23^{263} + \left(a^{14} + 4 a^{12} + 9 a^{11} + 19 a^{10} + 10 a^{9} + 12 a^{8} + 8 a^{7} + 12 a^{6} + 13 a^{5} + 9 a^{4} + 3 a^{3} + 12 a^{2} + 14 a + 22\right)\cdot 23^{264} + \left(20 a^{14} + 3 a^{13} + 17 a^{12} + 18 a^{11} + 11 a^{10} + 20 a^{9} + a^{8} + 16 a^{7} + 6 a^{6} + 2 a^{5} + 7 a^{4} + 15 a^{3} + 21 a^{2} + 12 a + 22\right)\cdot 23^{265} + \left(22 a^{14} + 13 a^{13} + 5 a^{12} + 5 a^{11} + 17 a^{10} + 19 a^{9} + 11 a^{8} + 16 a^{7} + 9 a^{6} + 3 a^{5} + 22 a^{4} + 9 a^{3} + 11 a^{2} + 18\right)\cdot 23^{266} + \left(20 a^{14} + 17 a^{13} + 11 a^{12} + 3 a^{11} + 10 a^{10} + 2 a^{9} + 17 a^{8} + 4 a^{7} + 2 a^{6} + 18 a^{5} + a^{4} + 19 a^{3} + 15 a^{2} + a + 3\right)\cdot 23^{267} + \left(6 a^{14} + 7 a^{13} + 16 a^{12} + 2 a^{11} + 8 a^{10} + 15 a^{9} + 14 a^{8} + 11 a^{7} + 4 a^{6} + 8 a^{5} + 3 a^{4} + 11 a^{3} + 19 a^{2} + 17 a + 6\right)\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 11 }$ $=$ $13 a^{14} + 16 a^{12} + 20 a^{11} + 6 a^{10} + 15 a^{9} + 13 a^{8} + 18 a^{7} + 3 a^{6} + 7 a^{5} + 11 a^{4} + 6 a^{3} + 6 a^{2} + 13 a + 16 + \left(6 a^{14} + 7 a^{13} + 5 a^{12} + 22 a^{11} + 12 a^{9} + 3 a^{8} + 20 a^{7} + 19 a^{6} + 8 a^{5} + 11 a^{4} + 12 a^{3} + 17 a^{2} + 7 a\right)\cdot 23 + \left(8 a^{14} + 11 a^{12} + 4 a^{11} + 16 a^{10} + 3 a^{9} + 12 a^{8} + 16 a^{7} + 16 a^{5} + 5 a^{4} + 8 a^{3} + 22 a + 15\right)\cdot 23^{2} + \left(14 a^{14} + 21 a^{13} + 21 a^{12} + 3 a^{11} + a^{10} + 6 a^{9} + 15 a^{8} + 18 a^{7} + 2 a^{6} + 20 a^{5} + a^{4} + 16 a^{3} + 14 a^{2} + 22\right)\cdot 23^{3} + \left(20 a^{14} + 11 a^{13} + 2 a^{12} + 21 a^{11} + 19 a^{10} + 4 a^{9} + 4 a^{8} + 22 a^{7} + a^{6} + 20 a^{5} + 20 a^{4} + 22 a^{3} + 9 a^{2} + 12 a + 12\right)\cdot 23^{4} + \left(2 a^{14} + 18 a^{13} + 9 a^{12} + 20 a^{11} + 5 a^{10} + 21 a^{9} + 21 a^{8} + 14 a^{7} + 18 a^{6} + 13 a^{5} + 11 a^{4} + 14 a^{3} + 6 a^{2} + 20 a + 16\right)\cdot 23^{5} + \left(15 a^{14} + 2 a^{13} + 13 a^{12} + 9 a^{11} + 14 a^{10} + 8 a^{9} + 4 a^{8} + 7 a^{7} + 7 a^{6} + 3 a^{5} + 11 a^{4} + 8 a^{2} + 4 a + 4\right)\cdot 23^{6} + \left(10 a^{14} + 12 a^{13} + 2 a^{12} + 3 a^{11} + 18 a^{10} + 20 a^{9} + 7 a^{8} + 20 a^{6} + 13 a^{5} + 11 a^{4} + 21 a^{3} + 5 a + 8\right)\cdot 23^{7} + \left(8 a^{14} + 18 a^{13} + 6 a^{12} + 9 a^{11} + 21 a^{10} + 16 a^{9} + 13 a^{8} + 18 a^{7} + 13 a^{6} + 19 a^{5} + 9 a^{4} + 8 a^{3} + a^{2} + 13 a + 4\right)\cdot 23^{8} + \left(15 a^{14} + 17 a^{13} + 5 a^{12} + 22 a^{11} + 8 a^{10} + 15 a^{9} + 4 a^{7} + 15 a^{6} + 19 a^{5} + 7 a^{4} + 5 a^{3} + 20 a^{2} + 15 a + 19\right)\cdot 23^{9} + \left(21 a^{14} + 13 a^{13} + 21 a^{12} + 10 a^{11} + 16 a^{10} + 7 a^{9} + 7 a^{8} + 4 a^{7} + 15 a^{6} + 15 a^{5} + 15 a^{4} + 6 a^{3} + 6 a^{2} + 3 a + 19\right)\cdot 23^{10} + \left(5 a^{13} + 15 a^{12} + 9 a^{11} + 19 a^{10} + 2 a^{9} + 8 a^{7} + 2 a^{6} + 12 a^{5} + 16 a^{4} + 8 a^{3} + 13 a^{2} + 5 a + 11\right)\cdot 23^{11} + \left(9 a^{14} + 15 a^{13} + 22 a^{12} + 19 a^{11} + 8 a^{10} + 10 a^{9} + 18 a^{8} + 3 a^{7} + 8 a^{6} + 20 a^{5} + 12 a^{4} + 3 a^{3} + 7 a^{2} + 19 a + 8\right)\cdot 23^{12} + \left(2 a^{14} + 21 a^{13} + 11 a^{12} + 2 a^{11} + 22 a^{10} + 6 a^{9} + 20 a^{8} + 20 a^{7} + 7 a^{6} + a^{5} + 4 a^{4} + 2 a^{3} + 14 a^{2}\right)\cdot 23^{13} + \left(9 a^{14} + 10 a^{13} + 9 a^{12} + 17 a^{11} + 4 a^{10} + 14 a^{9} + 19 a^{8} + 13 a^{7} + 18 a^{6} + 9 a^{5} + a^{4} + 2 a^{3} + 2 a^{2} + 15 a + 9\right)\cdot 23^{14} + \left(9 a^{14} + 2 a^{13} + 7 a^{12} + 14 a^{11} + 12 a^{10} + 20 a^{9} + 17 a^{7} + 12 a^{6} + 18 a^{5} + a^{4} + 16 a^{3} + 5 a^{2} + 5 a + 14\right)\cdot 23^{15} + \left(4 a^{14} + 17 a^{13} + a^{12} + 8 a^{11} + 4 a^{10} + 9 a^{9} + 9 a^{8} + 3 a^{7} + 9 a^{6} + 9 a^{5} + 6 a^{4} + 20 a^{3} + a^{2} + 12 a + 10\right)\cdot 23^{16} + \left(a^{14} + 15 a^{13} + 9 a^{12} + 7 a^{11} + 6 a^{10} + 3 a^{9} + 11 a^{8} + 21 a^{7} + 11 a^{6} + 22 a^{5} + 22 a^{4} + 22 a^{3} + 4 a^{2} + 7 a + 8\right)\cdot 23^{17} + \left(10 a^{14} + 3 a^{13} + 11 a^{12} + 3 a^{11} + 5 a^{10} + 20 a^{9} + 17 a^{8} + 14 a^{7} + 22 a^{6} + 22 a^{5} + 8 a^{4} + 9 a^{3} + 18 a^{2} + 7 a + 21\right)\cdot 23^{18} + \left(21 a^{14} + 7 a^{13} + 13 a^{12} + 11 a^{11} + 9 a^{10} + 18 a^{9} + 12 a^{7} + 17 a^{6} + 15 a^{5} + 22 a^{4} + 4 a^{3} + 14 a^{2} + 19 a + 2\right)\cdot 23^{19} + \left(12 a^{14} + 17 a^{13} + 2 a^{12} + 18 a^{11} + a^{10} + a^{9} + 5 a^{8} + 6 a^{7} + 21 a^{6} + 14 a^{5} + 16 a^{4} + 2 a^{3} + 15 a^{2} + 11 a + 21\right)\cdot 23^{20} + \left(18 a^{14} + 13 a^{13} + 21 a^{12} + 9 a^{11} + 22 a^{10} + 16 a^{9} + 14 a^{8} + a^{7} + 2 a^{6} + 12 a^{5} + 4 a^{4} + 3 a^{3} + 19 a^{2} + 14 a\right)\cdot 23^{21} + \left(4 a^{14} + 3 a^{13} + 15 a^{11} + 5 a^{10} + 15 a^{9} + 13 a^{8} + 16 a^{7} + a^{6} + 12 a^{5} + 3 a^{4} + 3 a^{3} + 14 a^{2} + 18 a + 3\right)\cdot 23^{22} + \left(14 a^{14} + 15 a^{13} + 21 a^{12} + 16 a^{11} + 10 a^{10} + a^{9} + 15 a^{8} + 3 a^{6} + 20 a^{5} + 3 a^{4} + 4 a^{3} + 22 a^{2} + 19 a + 20\right)\cdot 23^{23} + \left(20 a^{14} + 10 a^{12} + 13 a^{11} + 14 a^{10} + 19 a^{9} + 10 a^{8} + 8 a^{7} + 10 a^{6} + 5 a^{5} + 8 a^{4} + 2 a^{3} + 16 a^{2} + 5 a + 17\right)\cdot 23^{24} + \left(9 a^{14} + 4 a^{13} + 19 a^{12} + 2 a^{11} + 3 a^{10} + 13 a^{9} + 15 a^{8} + 22 a^{7} + 8 a^{6} + 15 a^{5} + 19 a^{4} + 16 a^{3} + 9 a + 12\right)\cdot 23^{25} + \left(19 a^{14} + 15 a^{12} + 17 a^{11} + 10 a^{9} + 4 a^{8} + 15 a^{7} + 12 a^{6} + 3 a^{5} + 12 a^{4} + 8 a^{3} + 12 a^{2} + 6 a + 5\right)\cdot 23^{26} + \left(12 a^{14} + 5 a^{13} + 2 a^{12} + 5 a^{11} + 3 a^{10} + 21 a^{9} + 21 a^{8} + 9 a^{7} + 13 a^{6} + 12 a^{5} + 8 a^{4} + 6 a^{3} + 16 a^{2} + 19\right)\cdot 23^{27} + \left(9 a^{14} + 8 a^{13} + a^{12} + 16 a^{11} + 8 a^{10} + 5 a^{9} + 11 a^{8} + 2 a^{7} + 6 a^{6} + 19 a^{5} + 17 a^{3} + 4 a^{2} + 22 a + 8\right)\cdot 23^{28} + \left(a^{14} + 3 a^{13} + 7 a^{12} + 18 a^{11} + 10 a^{10} + 9 a^{9} + 19 a^{8} + 20 a^{7} + 11 a^{6} + 14 a^{5} + 6 a^{4} + 19 a^{3} + 12 a^{2} + 5 a + 2\right)\cdot 23^{29} + \left(21 a^{14} + 5 a^{13} + 21 a^{12} + 8 a^{11} + 20 a^{10} + 16 a^{9} + 16 a^{8} + 7 a^{7} + 9 a^{5} + 12 a^{4} + 21 a^{3} + 21 a^{2} + 11 a + 9\right)\cdot 23^{30} + \left(22 a^{14} + 17 a^{13} + 11 a^{12} + 17 a^{11} + 17 a^{10} + 3 a^{9} + a^{8} + 2 a^{7} + 19 a^{6} + 11 a^{4} + a^{3} + 11 a^{2} + a + 4\right)\cdot 23^{31} + \left(a^{14} + 15 a^{13} + 4 a^{12} + a^{11} + 12 a^{10} + 3 a^{9} + 20 a^{8} + 16 a^{7} + 22 a^{6} + 8 a^{5} + a^{4} + 7 a^{3} + 21 a^{2} + 4 a + 6\right)\cdot 23^{32} + \left(8 a^{14} + 19 a^{13} + 15 a^{12} + 10 a^{11} + 3 a^{10} + 7 a^{9} + 3 a^{8} + 3 a^{7} + 6 a^{6} + 18 a^{5} + 4 a^{4} + 13 a^{3} + 16 a^{2} + 22 a + 18\right)\cdot 23^{33} + \left(17 a^{14} + 22 a^{13} + 17 a^{12} + 6 a^{11} + 12 a^{10} + 15 a^{9} + 22 a^{8} + 15 a^{7} + 6 a^{6} + 9 a^{5} + 17 a^{4} + 3 a^{3} + 6 a^{2} + 21 a + 21\right)\cdot 23^{34} + \left(22 a^{14} + 21 a^{13} + 12 a^{12} + 5 a^{11} + 17 a^{10} + 21 a^{9} + 14 a^{8} + 3 a^{7} + 3 a^{6} + 10 a^{5} + 14 a^{4} + 4 a^{3} + 9 a^{2} + 2 a + 16\right)\cdot 23^{35} + \left(7 a^{14} + 22 a^{13} + 15 a^{12} + a^{11} + a^{10} + 16 a^{9} + 20 a^{8} + 18 a^{7} + 7 a^{6} + 3 a^{5} + 21 a^{4} + 17 a^{3} + 7 a^{2} + 21 a + 18\right)\cdot 23^{36} + \left(15 a^{14} + 13 a^{13} + 22 a^{12} + 11 a^{11} + 2 a^{10} + 6 a^{9} + 18 a^{7} + 11 a^{6} + 2 a^{5} + 8 a^{4} + 16 a^{3} + 3 a^{2} + 15 a + 16\right)\cdot 23^{37} + \left(12 a^{14} + 16 a^{13} + 5 a^{11} + 7 a^{10} + 13 a^{9} + 4 a^{8} + a^{7} + 21 a^{5} + 6 a^{4} + 19 a^{3} + 7 a^{2} + 17 a + 22\right)\cdot 23^{38} + \left(7 a^{14} + 2 a^{13} + 6 a^{12} + 4 a^{11} + 3 a^{10} + 9 a^{9} + 2 a^{8} + 8 a^{7} + 14 a^{6} + 19 a^{5} + 9 a^{4} + 4 a^{3} + 6 a^{2} + 5 a + 8\right)\cdot 23^{39} + \left(20 a^{14} + 20 a^{13} + 18 a^{12} + 18 a^{11} + a^{10} + 14 a^{9} + 11 a^{8} + 4 a^{7} + 6 a^{6} + 12 a^{5} + 18 a^{4} + 19 a^{3} + 9 a^{2} + 12 a + 4\right)\cdot 23^{40} + \left(12 a^{14} + 12 a^{13} + 10 a^{12} + 15 a^{11} + 10 a^{10} + 13 a^{9} + 8 a^{8} + 15 a^{7} + 11 a^{6} + 20 a^{5} + 7 a^{4} + 19 a^{3} + 14 a^{2} + a + 12\right)\cdot 23^{41} + \left(3 a^{14} + 4 a^{13} + 19 a^{12} + 17 a^{11} + 19 a^{10} + 5 a^{9} + 4 a^{8} + 9 a^{7} + 22 a^{6} + 18 a^{5} + 17 a^{4} + 12 a^{3} + 22 a^{2} + 11 a + 21\right)\cdot 23^{42} + \left(12 a^{14} + 18 a^{13} + 2 a^{12} + 22 a^{11} + 2 a^{10} + 15 a^{9} + 13 a^{8} + 15 a^{7} + 3 a^{6} + 7 a^{5} + 22 a^{4} + 12 a^{3} + 11 a^{2} + 8 a + 7\right)\cdot 23^{43} + \left(10 a^{14} + 3 a^{13} + 2 a^{12} + 13 a^{11} + 3 a^{10} + 19 a^{9} + 2 a^{8} + 19 a^{7} + 18 a^{6} + a^{5} + 18 a^{4} + 10 a^{3} + 10 a^{2} + 21 a + 8\right)\cdot 23^{44} + \left(18 a^{13} + 7 a^{12} + 10 a^{11} + 6 a^{10} + 2 a^{9} + 11 a^{8} + 5 a^{7} + 7 a^{6} + 16 a^{5} + a^{4} + 16 a^{3} + 7 a^{2} + a + 13\right)\cdot 23^{45} + \left(7 a^{14} + 7 a^{12} + 17 a^{11} + 16 a^{10} + 14 a^{9} + 2 a^{8} + 16 a^{7} + 3 a^{6} + 16 a^{5} + 7 a^{4} + 6 a^{2} + 9\right)\cdot 23^{46} + \left(20 a^{14} + 20 a^{13} + 11 a^{12} + 15 a^{11} + 4 a^{10} + 14 a^{9} + 13 a^{8} + 5 a^{7} + 15 a^{6} + 11 a^{5} + 16 a^{4} + 3 a^{3} + 22 a^{2} + 20 a + 19\right)\cdot 23^{47} + \left(4 a^{14} + 4 a^{13} + 9 a^{12} + 21 a^{11} + 18 a^{10} + 12 a^{9} + 6 a^{8} + 8 a^{7} + 5 a^{6} + 11 a^{5} + 5 a^{4} + 14 a^{3} + 21 a^{2} + a + 12\right)\cdot 23^{48} + \left(8 a^{14} + 21 a^{13} + 17 a^{12} + 18 a^{11} + 16 a^{10} + 20 a^{9} + a^{8} + 2 a^{7} + 10 a^{6} + 17 a^{4} + 2 a^{3} + 2 a^{2} + 13 a + 16\right)\cdot 23^{49} + \left(13 a^{14} + 17 a^{13} + 17 a^{12} + 2 a^{11} + 4 a^{10} + 16 a^{9} + 4 a^{8} + 15 a^{7} + 6 a^{6} + 20 a^{5} + 19 a^{4} + 15 a^{3} + 2 a^{2} + 22 a + 4\right)\cdot 23^{50} + \left(16 a^{14} + 10 a^{13} + 12 a^{12} + 22 a^{11} + 18 a^{10} + 17 a^{9} + 9 a^{8} + 7 a^{7} + 5 a^{6} + 17 a^{5} + 14 a^{4} + 21 a^{3} + 16 a^{2} + 18 a + 18\right)\cdot 23^{51} + \left(17 a^{14} + 16 a^{13} + 14 a^{12} + 4 a^{11} + 17 a^{10} + 12 a^{9} + 14 a^{8} + 8 a^{7} + 15 a^{5} + 14 a^{4} + 21 a^{2} + 8 a + 19\right)\cdot 23^{52} + \left(13 a^{14} + 4 a^{13} + 18 a^{12} + 13 a^{11} + 14 a^{10} + 5 a^{9} + 19 a^{8} + 5 a^{7} + 5 a^{6} + a^{5} + 16 a^{4} + 22 a^{3} + 10 a^{2} + 9 a + 20\right)\cdot 23^{53} + \left(15 a^{14} + 13 a^{13} + 6 a^{12} + 6 a^{11} + 9 a^{10} + 9 a^{9} + 2 a^{8} + 13 a^{7} + 19 a^{6} + 20 a^{5} + 11 a^{4} + 6 a^{3} + 9 a^{2} + 15 a + 5\right)\cdot 23^{54} + \left(15 a^{14} + 5 a^{13} + a^{12} + 4 a^{11} + 22 a^{10} + 18 a^{9} + 18 a^{8} + 15 a^{7} + 11 a^{6} + 20 a^{5} + 12 a^{4} + 6 a^{3} + 4 a^{2} + 22 a + 8\right)\cdot 23^{55} + \left(13 a^{14} + 3 a^{13} + 10 a^{12} + 16 a^{11} + 6 a^{10} + 10 a^{9} + 20 a^{8} + 13 a^{7} + 16 a^{6} + 22 a^{5} + a^{4} + 7 a^{3} + 11 a^{2} + 13 a + 20\right)\cdot 23^{56} + \left(2 a^{14} + 13 a^{13} + 19 a^{12} + 19 a^{11} + 4 a^{10} + 16 a^{9} + 8 a^{8} + 13 a^{7} + 4 a^{6} + 18 a^{5} + 19 a^{4} + 12 a^{3} + 5 a^{2} + 22 a + 3\right)\cdot 23^{57} + \left(2 a^{14} + 16 a^{13} + 6 a^{12} + 5 a^{11} + 2 a^{10} + 8 a^{9} + 4 a^{8} + 7 a^{7} + 22 a^{6} + 6 a^{5} + 21 a^{4} + 11 a^{2} + 9 a + 7\right)\cdot 23^{58} + \left(a^{14} + 10 a^{13} + 2 a^{12} + 19 a^{11} + 12 a^{10} + 2 a^{9} + 2 a^{8} + 5 a^{7} + 15 a^{6} + 9 a^{5} + 5 a^{4} + 22 a^{3} + 14 a^{2} + 13 a + 2\right)\cdot 23^{59} + \left(21 a^{14} + 22 a^{13} + 17 a^{12} + 21 a^{11} + 10 a^{10} + 16 a^{9} + a^{8} + 20 a^{7} + 18 a^{6} + 7 a^{4} + 10 a^{3} + 21 a^{2} + 21 a + 19\right)\cdot 23^{60} + \left(6 a^{14} + 22 a^{13} + 6 a^{12} + 3 a^{11} + 10 a^{10} + 7 a^{9} + 21 a^{8} + 21 a^{7} + 18 a^{6} + 9 a^{5} + 22 a^{4} + 11 a^{3} + 10 a^{2} + 2 a\right)\cdot 23^{61} + \left(2 a^{14} + 14 a^{13} + 22 a^{12} + 19 a^{11} + 22 a^{10} + 3 a^{9} + 13 a^{8} + 9 a^{7} + 4 a^{6} + 8 a^{5} + 7 a^{4} + 21 a^{3} + 20 a^{2} + 10 a + 5\right)\cdot 23^{62} + \left(3 a^{14} + 10 a^{13} + 11 a^{12} + 7 a^{11} + 15 a^{10} + 17 a^{9} + 4 a^{8} + 18 a^{7} + 8 a^{6} + 4 a^{5} + 3 a^{4} + 3 a^{3} + 22 a^{2} + 17 a + 2\right)\cdot 23^{63} + \left(14 a^{14} + 17 a^{13} + 18 a^{12} + 15 a^{11} + 18 a^{10} + 7 a^{9} + 10 a^{8} + 9 a^{7} + 22 a^{6} + 7 a^{5} + 22 a^{4} + 11 a^{3} + 22 a^{2} + 6 a + 19\right)\cdot 23^{64} + \left(18 a^{14} + 9 a^{13} + 10 a^{12} + 5 a^{11} + 19 a^{10} + 10 a^{9} + 6 a^{7} + 14 a^{6} + 17 a^{5} + 13 a^{3} + 6 a^{2} + 13 a + 21\right)\cdot 23^{65} + \left(21 a^{13} + 14 a^{12} + 21 a^{11} + 20 a^{10} + 18 a^{9} + 4 a^{8} + 16 a^{7} + 20 a^{6} + 17 a^{5} + 22 a^{4} + 13 a^{3} + 7 a^{2} + 15 a + 20\right)\cdot 23^{66} + \left(10 a^{14} + 13 a^{13} + 10 a^{11} + 9 a^{10} + 6 a^{9} + a^{8} + 7 a^{7} + 13 a^{6} + 17 a^{5} + 5 a^{4} + 10 a^{2} + 5 a + 19\right)\cdot 23^{67} + \left(2 a^{14} + 5 a^{13} + 13 a^{12} + 15 a^{11} + 18 a^{10} + 19 a^{9} + 3 a^{8} + 21 a^{7} + 8 a^{6} + 14 a^{5} + 21 a^{4} + 2 a^{3} + 17 a^{2} + 16 a + 10\right)\cdot 23^{68} + \left(20 a^{14} + 2 a^{13} + 4 a^{12} + 2 a^{11} + 13 a^{10} + 10 a^{9} + 6 a^{8} + 5 a^{7} + 16 a^{6} + 10 a^{5} + 9 a^{4} + 6 a^{3} + 6 a^{2} + 8 a + 15\right)\cdot 23^{69} + \left(21 a^{14} + 4 a^{13} + 10 a^{12} + 2 a^{11} + 2 a^{10} + 21 a^{9} + 5 a^{8} + 21 a^{7} + 17 a^{6} + 2 a^{5} + 21 a^{4} + 14 a^{2} + 16 a + 3\right)\cdot 23^{70} + \left(a^{14} + 19 a^{13} + 6 a^{12} + 17 a^{11} + a^{10} + 11 a^{9} + 22 a^{8} + 4 a^{7} + 8 a^{6} + 9 a^{5} + 6 a^{4} + 13 a^{3} + 19 a^{2} + 21 a + 16\right)\cdot 23^{71} + \left(21 a^{14} + a^{13} + 17 a^{12} + 3 a^{10} + 15 a^{9} + 7 a^{8} + 19 a^{7} + a^{6} + 20 a^{5} + 2 a^{4} + 22 a^{3} + 18 a^{2} + 3 a + 2\right)\cdot 23^{72} + \left(8 a^{14} + 12 a^{13} + 5 a^{12} + 12 a^{11} + 8 a^{10} + 10 a^{9} + 22 a^{8} + 5 a^{7} + 14 a^{6} + 12 a^{5} + 3 a^{4} + 3 a^{3} + 14 a^{2} + 17 a + 5\right)\cdot 23^{73} + \left(14 a^{14} + 9 a^{13} + 19 a^{12} + 6 a^{11} + 22 a^{10} + 17 a^{9} + 18 a^{8} + 7 a^{7} + 2 a^{6} + 20 a^{5} + 16 a^{4} + 5 a^{3} + 3 a^{2} + a\right)\cdot 23^{74} + \left(6 a^{14} + 16 a^{12} + 16 a^{11} + 10 a^{10} + 20 a^{9} + 14 a^{8} + 12 a^{7} + 13 a^{6} + 3 a^{5} + a^{4} + 15 a^{3} + 9 a^{2} + 14\right)\cdot 23^{75} + \left(18 a^{14} + 17 a^{13} + 9 a^{12} + 8 a^{11} + 4 a^{10} + a^{8} + 12 a^{7} + 20 a^{5} + a^{4} + 12 a^{3} + 6 a + 3\right)\cdot 23^{76} + \left(11 a^{14} + 3 a^{13} + 4 a^{12} + 6 a^{11} + a^{10} + 8 a^{9} + 9 a^{8} + 22 a^{7} + 3 a^{6} + a^{5} + 11 a^{4} + a^{3} + 15 a^{2} + 22 a + 14\right)\cdot 23^{77} + \left(21 a^{14} + 21 a^{13} + 12 a^{12} + 3 a^{11} + 19 a^{10} + 13 a^{9} + 17 a^{8} + 9 a^{7} + 21 a^{6} + 22 a^{5} + 13 a^{4} + 6 a^{3} + 22 a^{2} + 8 a + 15\right)\cdot 23^{78} + \left(15 a^{14} + 12 a^{12} + 20 a^{11} + 14 a^{10} + 8 a^{9} + 15 a^{8} + 15 a^{7} + 2 a^{6} + a^{5} + 8 a^{4} + 8 a^{3} + 22 a^{2} + 20 a + 1\right)\cdot 23^{79} + \left(9 a^{14} + 21 a^{13} + a^{12} + 5 a^{11} + 4 a^{10} + 6 a^{9} + 2 a^{7} + 10 a^{5} + 3 a^{4} + 14 a^{3} + 20 a^{2} + 3 a + 3\right)\cdot 23^{80} + \left(14 a^{14} + 3 a^{13} + 8 a^{12} + 21 a^{11} + 10 a^{10} + 3 a^{9} + 6 a^{8} + 6 a^{7} + a^{6} + 10 a^{5} + 21 a^{4} + 21 a^{3} + 5 a^{2} + 5 a + 7\right)\cdot 23^{81} + \left(10 a^{14} + 5 a^{13} + 7 a^{12} + 19 a^{11} + 5 a^{10} + 6 a^{9} + 7 a^{8} + 16 a^{7} + 17 a^{6} + 4 a^{5} + 5 a^{4} + 15 a^{3} + 11 a^{2} + 4 a\right)\cdot 23^{82} + \left(22 a^{14} + 20 a^{13} + 4 a^{12} + 7 a^{11} + a^{10} + 22 a^{9} + 20 a^{8} + 6 a^{7} + 17 a^{6} + 16 a^{5} + 16 a^{4} + 19 a^{3} + 20 a^{2} + a + 8\right)\cdot 23^{83} + \left(20 a^{14} + 14 a^{12} + 2 a^{11} + 2 a^{10} + 5 a^{9} + 13 a^{8} + 7 a^{7} + 2 a^{6} + 18 a^{5} + 12 a^{4} + 20 a^{3} + a^{2} + 8\right)\cdot 23^{84} + \left(16 a^{14} + 21 a^{13} + 3 a^{12} + 21 a^{11} + 16 a^{10} + 11 a^{9} + 17 a^{8} + 15 a^{7} + 13 a^{6} + 9 a^{5} + 4 a^{4} + 14 a^{3} + 6 a^{2} + 15 a 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a^{3} + 12 a^{2} + 11 a + 1\right)\cdot 23^{97} + \left(13 a^{14} + 3 a^{13} + 13 a^{12} + 20 a^{11} + 7 a^{10} + 9 a^{9} + 14 a^{8} + 20 a^{7} + a^{6} + 3 a^{5} + 8 a^{4} + 11 a^{3} + 14 a^{2} + 10 a + 4\right)\cdot 23^{98} + \left(14 a^{14} + 15 a^{13} + 17 a^{12} + 2 a^{11} + 8 a^{10} + 8 a^{9} + 21 a^{8} + 8 a^{7} + a^{6} + 6 a^{5} + 15 a^{4} + 9 a^{3} + 20 a + 2\right)\cdot 23^{99} + \left(13 a^{13} + 5 a^{12} + 18 a^{11} + 12 a^{10} + 12 a^{9} + 18 a^{8} + 10 a^{7} + 10 a^{6} + 11 a^{5} + 4 a^{4} + 13 a^{3} + 14 a^{2} + 12 a + 9\right)\cdot 23^{100} + \left(22 a^{14} + 16 a^{13} + 2 a^{12} + 21 a^{11} + 21 a^{10} + 14 a^{8} + 6 a^{7} + 8 a^{6} + 12 a^{5} + 4 a^{4} + 18 a^{3} + 2 a^{2} + 21 a + 18\right)\cdot 23^{101} + \left(11 a^{14} + 4 a^{13} + 15 a^{12} + 7 a^{11} + 19 a^{10} + 5 a^{9} + 6 a^{8} + 16 a^{7} + 15 a^{6} + 2 a^{5} + 4 a^{4} + 3 a^{3} + 7 a^{2} + 9 a + 2\right)\cdot 23^{102} + \left(14 a^{14} + 15 a^{13} + 11 a^{10} + 15 a^{9} + 19 a^{8} + 18 a^{7} + 12 a^{6} + 15 a^{5} + 22 a^{4} + 16 a^{3} + 4 a^{2} + 4 a + 10\right)\cdot 23^{103} + \left(4 a^{14} + 6 a^{13} + 6 a^{12} + 6 a^{11} + 5 a^{10} + 8 a^{9} + 7 a^{8} + 4 a^{7} + 14 a^{6} + 21 a^{5} + 18 a^{4} + 19 a^{3} + 17 a^{2} + 22 a + 10\right)\cdot 23^{104} + \left(19 a^{14} + 7 a^{13} + 11 a^{12} + 9 a^{11} + 6 a^{10} + a^{9} + 4 a^{8} + 22 a^{7} + 13 a^{6} + 6 a^{5} + 4 a^{4} + 3 a^{2} + 9 a + 14\right)\cdot 23^{105} + \left(21 a^{13} + 5 a^{12} + 9 a^{11} + 9 a^{10} + 18 a^{9} + 16 a^{8} + 12 a^{7} + a^{6} + 15 a^{4} + 7 a^{3} + 8 a^{2} + 15 a + 12\right)\cdot 23^{106} + \left(18 a^{14} + 5 a^{13} + 16 a^{12} + 3 a^{11} + 11 a^{10} + 12 a^{9} + 20 a^{8} + 16 a^{6} + 19 a^{5} + 11 a^{4} + 6 a^{3} + a^{2} + 14 a + 22\right)\cdot 23^{107} + \left(17 a^{14} + 22 a^{13} + 10 a^{12} + 3 a^{11} + 21 a^{10} + 7 a^{9} + 10 a^{8} + 20 a^{7} + 19 a^{6} + 15 a^{5} + 18 a^{4} + 5 a^{3} + 19 a^{2} + 8\right)\cdot 23^{108} + \left(18 a^{14} + 8 a^{13} + 5 a^{12} + 13 a^{11} + 12 a^{10} + 18 a^{9} + 5 a^{8} + 22 a^{7} + a^{6} + 22 a^{4} + 6 a^{3} + a + 5\right)\cdot 23^{109} + \left(21 a^{13} + 6 a^{11} + 16 a^{10} + 22 a^{9} + 5 a^{8} + 12 a^{7} + 18 a^{6} + 6 a^{5} + 7 a^{4} + 8 a^{3} + 16 a^{2} + 19 a + 18\right)\cdot 23^{110} + \left(10 a^{14} + 16 a^{13} + 18 a^{12} + 17 a^{11} + 17 a^{10} + 13 a^{9} + 13 a^{8} + 4 a^{7} + 5 a^{6} + 19 a^{5} + 10 a^{4} + 15 a^{3} + 12 a^{2} + 11 a + 3\right)\cdot 23^{111} + \left(3 a^{14} + 10 a^{13} + 9 a^{12} + 20 a^{10} + 8 a^{9} + 11 a^{8} + 8 a^{7} + 18 a^{6} + 10 a^{5} + 3 a^{4} + 19 a^{3} + 3 a^{2} + 2 a + 14\right)\cdot 23^{112} + \left(8 a^{14} + 18 a^{13} + 16 a^{12} + 20 a^{11} + 16 a^{10} + 19 a^{9} + 14 a^{8} + 7 a^{7} + 13 a^{6} + 2 a^{5} + 9 a^{4} + 19 a^{3} + 4 a^{2} + 12 a + 8\right)\cdot 23^{113} + \left(13 a^{14} + 18 a^{13} + 12 a^{12} + 17 a^{11} + 12 a^{9} + 15 a^{8} + 9 a^{7} + 19 a^{6} + 11 a^{5} + 7 a^{4} + 11 a^{3} + 4 a^{2} + a + 9\right)\cdot 23^{114} + \left(9 a^{14} + 13 a^{13} + 21 a^{12} + 4 a^{11} + 7 a^{10} + 11 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a^{5} + 5 a^{4} + 13 a^{3} + 12 a^{2} + 16 a + 1\right)\cdot 23^{126} + \left(11 a^{14} + 13 a^{13} + 16 a^{12} + 8 a^{11} + 20 a^{10} + 8 a^{9} + 4 a^{8} + 2 a^{7} + 3 a^{6} + 15 a^{5} + 15 a^{4} + 21 a^{3} + 3 a^{2} + 21\right)\cdot 23^{127} + \left(15 a^{14} + 6 a^{13} + 18 a^{12} + 12 a^{11} + 2 a^{10} + a^{9} + 4 a^{8} + 7 a^{6} + a^{5} + 15 a^{4} + 20 a^{3} + 17 a^{2} + 3 a + 15\right)\cdot 23^{128} + \left(12 a^{14} + 22 a^{13} + 22 a^{12} + 9 a^{11} + 19 a^{10} + 4 a^{9} + 15 a^{8} + 15 a^{7} + 5 a^{6} + 19 a^{5} + 17 a^{4} + 12 a^{3} + 13 a^{2} + 9 a + 20\right)\cdot 23^{129} + \left(3 a^{14} + 15 a^{13} + 13 a^{12} + 13 a^{11} + 21 a^{10} + 16 a^{9} + 20 a^{8} + 10 a^{7} + 21 a^{6} + 16 a^{4} + 12 a^{3} + 9 a^{2} + 15 a + 6\right)\cdot 23^{130} + \left(4 a^{14} + 10 a^{13} + 4 a^{12} + 3 a^{11} + 3 a^{10} + 20 a^{9} + 15 a^{7} + 10 a^{6} + 14 a^{5} + 17 a^{4} + 5 a^{3} + 12 a^{2} + 11 a\right)\cdot 23^{131} + \left(a^{14} + 12 a^{13} + a^{12} + a^{11} + a^{10} + 15 a^{9} + a^{8} + 8 a^{7} + 16 a^{6} + 6 a^{5} + 21 a^{4} + 8 a^{3} + 16 a^{2} + 22 a + 18\right)\cdot 23^{132} + \left(13 a^{14} + 22 a^{13} + 22 a^{12} + 2 a^{11} + 7 a^{10} + 7 a^{9} + 15 a^{8} + 18 a^{7} + a^{6} + 22 a^{5} + 9 a^{3} + 16 a^{2} + 15 a + 9\right)\cdot 23^{133} + \left(21 a^{14} + 4 a^{13} + 10 a^{12} + 14 a^{11} + 2 a^{10} + 21 a^{9} + 10 a^{8} + 12 a^{7} + 16 a^{6} + 11 a^{5} + 10 a^{4} + a^{3} + 19 a^{2} + 7 a + 21\right)\cdot 23^{134} + \left(4 a^{14} + 9 a^{13} + 16 a^{11} + 17 a^{10} + 19 a^{9} + 19 a^{8} + 5 a^{7} + 9 a^{6} + 2 a^{5} + 22 a^{3} + 3 a^{2} + 7\right)\cdot 23^{135} + \left(16 a^{14} + 16 a^{13} + 13 a^{12} + 10 a^{11} + 2 a^{10} + 7 a^{9} + 18 a^{8} + 7 a^{7} + 11 a^{6} + 22 a^{5} + a^{4} + 13 a^{3} + 14 a^{2} + 5 a + 7\right)\cdot 23^{136} + \left(15 a^{14} + a^{13} + a^{12} + 12 a^{11} + 5 a^{10} + 7 a^{9} + 8 a^{8} + 18 a^{7} + 17 a^{6} + 13 a^{5} + 11 a^{4} + 2 a^{3} + 15 a^{2} + 22 a + 20\right)\cdot 23^{137} + \left(17 a^{14} + 9 a^{13} + 21 a^{12} + 4 a^{11} + 4 a^{10} + 6 a^{9} + 22 a^{8} + 19 a^{7} + 14 a^{6} + a^{5} + 6 a^{4} + a^{3} + 6 a^{2} + 21 a\right)\cdot 23^{138} + \left(4 a^{14} + a^{13} + 14 a^{12} + 3 a^{11} + 2 a^{10} + 11 a^{9} + 18 a^{8} + 12 a^{7} + 13 a^{6} + 21 a^{5} + 19 a^{4} + 4 a^{3} + 6 a^{2} + 14 a + 22\right)\cdot 23^{139} + \left(4 a^{14} + 17 a^{13} + 16 a^{11} + 7 a^{10} + 4 a^{9} + 15 a^{8} + 2 a^{7} + 21 a^{6} + 4 a^{5} + 6 a^{4} + 4 a^{3} + 5 a^{2} + 7\right)\cdot 23^{140} + \left(10 a^{14} + 12 a^{13} + 16 a^{12} + 8 a^{11} + a^{10} + 16 a^{9} + 21 a^{8} + 4 a^{7} + 13 a^{5} + 5 a^{4} + 12 a^{3} + 12 a^{2} + 12 a + 16\right)\cdot 23^{141} + \left(20 a^{14} + 19 a^{13} + 5 a^{12} + 13 a^{11} + 19 a^{10} + 20 a^{9} + 2 a^{8} + 8 a^{7} + 11 a^{6} + 8 a^{5} + 13 a^{4} + 6 a^{3} + 2 a^{2} + 18 a + 5\right)\cdot 23^{142} + \left(15 a^{14} + 18 a^{13} + 7 a^{12} + 17 a^{11} + 7 a^{10} + 14 a^{9} + a^{8} + 14 a^{7} + 9 a^{6} + 12 a^{5} + a^{4} + 4 a^{3} + 3 a^{2} + 20 a + 12\right)\cdot 23^{143} + \left(22 a^{14} + 16 a^{13} + 22 a^{12} + 21 a^{11} + 13 a^{10} + 20 a^{9} + 22 a^{8} + 3 a^{7} + 10 a^{6} + 8 a^{5} + 2 a^{4} + 13 a^{3} + 11 a^{2} + 12 a + 12\right)\cdot 23^{144} + \left(6 a^{14} + 10 a^{13} + 13 a^{12} + 22 a^{11} + 10 a^{10} + 11 a^{9} + 17 a^{8} + 3 a^{7} + 3 a^{6} + a^{5} + 15 a^{4} + 2 a^{3} + 16 a^{2} + 8 a + 20\right)\cdot 23^{145} + \left(7 a^{14} + 10 a^{13} + 17 a^{12} + 14 a^{11} + 18 a^{10} + a^{9} + 8 a^{8} + 21 a^{7} + 11 a^{6} + 14 a^{5} + 3 a^{4} + 3 a^{3} + 21 a^{2} + 11 a + 18\right)\cdot 23^{146} + \left(15 a^{14} + 2 a^{13} + 11 a^{12} + 21 a^{11} + 14 a^{10} + 12 a^{9} + 10 a^{8} + 5 a^{7} + 21 a^{6} + 5 a^{5} + 5 a^{4} + 8 a^{3} + 6 a^{2} + 11 a + 19\right)\cdot 23^{147} + \left(22 a^{14} + 22 a^{13} + 11 a^{12} + 7 a^{11} + 20 a^{10} + 2 a^{9} + 16 a^{8} + 7 a^{7} + 21 a^{6} + 15 a^{5} + 13 a^{4} + 13 a^{3} + 10 a^{2} + 10 a + 15\right)\cdot 23^{148} + \left(a^{14} + a^{13} + 19 a^{11} + 12 a^{10} + 13 a^{9} + 8 a^{8} + 18 a^{7} + 18 a^{6} + 10 a^{4} + 6 a^{3} + 14 a^{2} + 16 a + 5\right)\cdot 23^{149} + \left(a^{14} + 13 a^{13} + 2 a^{12} + 17 a^{11} + 18 a^{10} + 8 a^{9} + 19 a^{8} + 7 a^{7} + 17 a^{6} + 16 a^{5} + 19 a^{4} + 22 a^{3} + 21 a^{2} + 16 a + 4\right)\cdot 23^{150} + \left(10 a^{14} + 16 a^{13} + 22 a^{12} + 13 a^{11} + 22 a^{10} + 5 a^{9} + 20 a^{8} + 17 a^{7} + 6 a^{6} + 12 a^{5} + 10 a^{4} + 16 a^{2} + 11 a + 18\right)\cdot 23^{151} + \left(11 a^{14} + 22 a^{13} + 3 a^{12} + 4 a^{11} + 5 a^{10} + 6 a^{9} + 21 a^{8} + 10 a^{7} + 17 a^{6} + 20 a^{5} + 21 a^{4} + 13 a^{3} + 20 a^{2} + 9 a + 19\right)\cdot 23^{152} + \left(15 a^{14} + 7 a^{13} + 12 a^{12} + 9 a^{11} + a^{10} + 19 a^{9} + 10 a^{8} + 12 a^{7} + 17 a^{6} + 7 a^{5} + 17 a^{4} + 11 a^{3} + 18 a^{2} + 8 a + 14\right)\cdot 23^{153} + \left(9 a^{14} + 7 a^{13} + 21 a^{12} + 20 a^{11} + 11 a^{10} + 22 a^{9} + a^{8} + 14 a^{7} + 18 a^{6} + 17 a^{5} + 12 a^{4} + 10 a^{3} + 20 a^{2} + 8 a + 14\right)\cdot 23^{154} + \left(7 a^{14} + 18 a^{13} + 9 a^{12} + 14 a^{11} + 10 a^{10} + 15 a^{9} + 5 a^{8} + 10 a^{7} + 22 a^{6} + 19 a^{5} + 18 a^{4} + 18 a^{3} + a^{2} + 4 a + 1\right)\cdot 23^{155} + \left(19 a^{14} + 2 a^{13} + a^{12} + 7 a^{11} + 4 a^{10} + 5 a^{9} + 5 a^{8} + 9 a^{7} + 10 a^{6} + 22 a^{5} + 14 a^{4} + 14 a^{3} + 16 a^{2} + 19 a + 2\right)\cdot 23^{156} + \left(7 a^{14} + 18 a^{13} + 14 a^{12} + 2 a^{11} + 4 a^{10} + 14 a^{9} + 5 a^{8} + a^{7} + 5 a^{6} + 13 a^{5} + 11 a^{4} + 12 a^{3} + 22 a^{2} + 16 a + 3\right)\cdot 23^{157} + \left(7 a^{14} + 17 a^{13} + 13 a^{12} + 6 a^{11} + 9 a^{10} + 19 a^{9} + 8 a^{8} + 15 a^{7} + 19 a^{6} + 17 a^{4} + 12 a^{3} + 5 a^{2} + 3 a + 10\right)\cdot 23^{158} + \left(5 a^{14} + 18 a^{13} + 11 a^{12} + 8 a^{10} + 9 a^{9} + 13 a^{8} + 4 a^{7} + 15 a^{6} + 18 a^{5} + 11 a^{4} + 5 a^{3} + a + 20\right)\cdot 23^{159} + \left(3 a^{14} + a^{13} + 7 a^{12} + 9 a^{11} + 11 a^{10} + 6 a^{9} + 5 a^{8} + 5 a^{7} + 2 a^{6} + 9 a^{4} + 4 a^{3} + 2 a^{2} + 20 a + 14\right)\cdot 23^{160} + \left(2 a^{14} + 22 a^{12} + 6 a^{11} + 10 a^{10} + 4 a^{9} + 17 a^{8} + 16 a^{7} + 14 a^{6} + 10 a^{5} + 13 a^{4} + 19 a^{3} + 6 a^{2} + 12 a + 2\right)\cdot 23^{161} + \left(12 a^{14} + 5 a^{13} + 22 a^{12} + 18 a^{11} + 17 a^{10} + 10 a^{9} + 19 a^{8} + 13 a^{7} + 6 a^{6} + 22 a^{5} + 5 a^{4} + 14 a^{3} + 20 a^{2} + 14 a + 15\right)\cdot 23^{162} + \left(7 a^{14} + 6 a^{13} + 19 a^{12} + 13 a^{11} + 13 a^{10} + 20 a^{9} + 7 a^{8} + 11 a^{7} + 3 a^{6} + a^{5} + 6 a^{4} + 8 a^{3} + 21 a^{2} + 16 a + 7\right)\cdot 23^{163} + \left(4 a^{14} + 11 a^{13} + 16 a^{12} + a^{11} + 10 a^{10} + 7 a^{9} + a^{8} + a^{7} + 5 a^{5} + 11 a^{4} + 17 a^{3} + 16 a^{2} + a + 3\right)\cdot 23^{164} + \left(19 a^{14} + 19 a^{13} + a^{12} + 17 a^{11} + 2 a^{10} + 5 a^{9} + 17 a^{8} + 22 a^{7} + 6 a^{6} + 16 a^{5} + 15 a^{4} + 14 a^{2} + 14 a + 17\right)\cdot 23^{165} + \left(20 a^{14} + 9 a^{13} + 3 a^{12} + 18 a^{11} + 6 a^{10} + 22 a^{9} + 21 a^{8} + 7 a^{7} + 12 a^{6} + 17 a^{5} + 13 a^{4} + 14 a^{3} + 3 a^{2} + 4 a + 1\right)\cdot 23^{166} + \left(15 a^{14} + 20 a^{12} + 21 a^{11} + 18 a^{10} + 21 a^{9} + 20 a^{8} + 2 a^{7} + 19 a^{6} + 20 a^{5} + 3 a^{4} + 17 a^{3} + 3 a^{2} + 22 a + 5\right)\cdot 23^{167} + \left(18 a^{14} + a^{12} + 11 a^{11} + 15 a^{10} + 15 a^{9} + a^{8} + 13 a^{7} + 13 a^{6} + 3 a^{5} + 15 a^{4} + 21 a^{3} + 21 a^{2} + 2 a\right)\cdot 23^{168} + \left(7 a^{14} + 21 a^{13} + 16 a^{12} + 17 a^{11} + 9 a^{10} + 21 a^{9} + 4 a^{8} + 9 a^{7} + 2 a^{6} + 8 a^{5} + 11 a^{4} + 21 a^{3} + a^{2} + 22 a + 3\right)\cdot 23^{169} + \left(12 a^{14} + 7 a^{13} + a^{12} + 18 a^{11} + 19 a^{10} + 9 a^{9} + 11 a^{8} + 4 a^{7} + 20 a^{6} + 7 a^{5} + 5 a^{4} + 22 a^{3} + 6 a^{2} + 5 a + 3\right)\cdot 23^{170} + \left(6 a^{14} + 6 a^{13} + 10 a^{12} + 5 a^{11} + 8 a^{10} + 22 a^{9} + 15 a^{8} + 20 a^{7} + 11 a^{6} + 9 a^{5} + 5 a^{4} + 18 a^{3} + 12 a^{2} + 7 a + 13\right)\cdot 23^{171} + \left(19 a^{14} + 14 a^{13} + 9 a^{12} + 16 a^{11} + 19 a^{10} + a^{9} + 14 a^{8} + 11 a^{7} + 20 a^{6} + 11 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+ 8 a^{9} + a^{7} + 14 a^{6} + 4 a^{5} + 7 a^{4} + 20 a^{3} + 14 a^{2} + 3\right)\cdot 23^{178} + \left(2 a^{14} + 15 a^{13} + 11 a^{12} + 22 a^{11} + 3 a^{10} + 19 a^{9} + 4 a^{8} + 5 a^{7} + 7 a^{6} + 9 a^{5} + 22 a^{4} + a^{3} + 9 a^{2} + 4 a + 9\right)\cdot 23^{179} + \left(8 a^{14} + 20 a^{13} + 20 a^{12} + 7 a^{11} + 3 a^{10} + 4 a^{9} + 3 a^{8} + 16 a^{7} + 22 a^{6} + 13 a^{5} + 5 a^{4} + 19 a^{3} + 11 a^{2} + 3 a + 14\right)\cdot 23^{180} + \left(a^{14} + 22 a^{13} + 21 a^{12} + 13 a^{11} + 4 a^{10} + 19 a^{9} + 2 a^{8} + 11 a^{7} + 8 a^{6} + 6 a^{5} + 6 a^{4} + 7 a^{3} + 3 a^{2} + 8\right)\cdot 23^{181} + \left(16 a^{14} + 2 a^{13} + 9 a^{12} + 9 a^{11} + 18 a^{10} + 21 a^{9} + 7 a^{8} + 5 a^{7} + 19 a^{6} + 8 a^{5} + 19 a^{4} + 12 a^{3} + a^{2} + 10 a\right)\cdot 23^{182} + \left(17 a^{14} + 9 a^{13} + 14 a^{12} + 5 a^{11} + 20 a^{10} + 3 a^{9} + 10 a^{8} + 18 a^{7} + 5 a^{6} + 9 a^{5} + 12 a^{4} + 21 a^{3} + 7 a^{2} + 16 a\right)\cdot 23^{183} + \left(5 a^{14} + 4 a^{13} + 10 a^{12} + 14 a^{11} + 16 a^{9} + 11 a^{8} + a^{7} + 6 a^{5} + 19 a^{4} + 8 a^{3} + 8 a^{2} + 18 a + 9\right)\cdot 23^{184} + \left(10 a^{14} + 10 a^{13} + 12 a^{11} + a^{10} + 13 a^{9} + 20 a^{8} + 7 a^{7} + 3 a^{6} + 9 a^{5} + 17 a^{4} + 8 a^{3} + 16 a^{2} + 19 a + 5\right)\cdot 23^{185} + \left(8 a^{14} + 8 a^{13} + 19 a^{12} + 12 a^{11} + 11 a^{10} + 5 a^{9} + a^{8} + a^{7} + 13 a^{6} + 14 a^{5} + 17 a^{4} + 21 a^{3} + 7 a^{2} + 21 a + 19\right)\cdot 23^{186} + \left(11 a^{14} + 10 a^{13} + 12 a^{12} + 12 a^{11} + 11 a^{10} + 10 a^{9} + 5 a^{8} + a^{7} + 13 a^{6} + 21 a^{5} + 5 a^{4} + 11 a^{3} + 20 a^{2} + 5 a + 18\right)\cdot 23^{187} + \left(21 a^{14} + 11 a^{13} + 7 a^{12} + 4 a^{11} + 2 a^{10} + 12 a^{9} + 14 a^{8} + 8 a^{7} + 12 a^{6} + 15 a^{5} + 6 a^{4} + 5 a^{3} + 7 a^{2} + 3 a + 22\right)\cdot 23^{188} + \left(15 a^{14} + 5 a^{13} + 11 a^{12} + 3 a^{11} + 9 a^{10} + 15 a^{8} + 16 a^{7} + 14 a^{6} + 5 a^{5} + 14 a^{4} + 9 a^{3} + 16 a^{2} + 12 a + 19\right)\cdot 23^{189} + \left(2 a^{14} + 4 a^{13} + 15 a^{12} + 19 a^{11} + 18 a^{10} + 9 a^{9} + 20 a^{8} + 21 a^{7} + 10 a^{6} + 6 a^{5} + 16 a^{4} + 20 a^{3} + 13 a^{2} + 14 a + 11\right)\cdot 23^{190} + \left(8 a^{14} + 6 a^{13} + 5 a^{12} + 4 a^{10} + 10 a^{9} + 11 a^{7} + 5 a^{6} + 18 a^{5} + 2 a^{4} + 6 a^{3} + 7 a^{2} + 15 a + 19\right)\cdot 23^{191} + \left(17 a^{14} + 15 a^{13} + a^{12} + 19 a^{11} + 7 a^{10} + 17 a^{9} + 12 a^{8} + 22 a^{7} + 18 a^{6} + 4 a^{5} + 2 a^{4} + 14 a^{3} + 14 a^{2} + 18 a + 22\right)\cdot 23^{192} + \left(3 a^{14} + 10 a^{13} + 18 a^{12} + 16 a^{11} + 2 a^{10} + 17 a^{9} + 3 a^{8} + 8 a^{7} + 17 a^{6} + 17 a^{5} + 21 a^{4} + 7 a^{3} + a^{2} + 11 a + 10\right)\cdot 23^{193} + \left(5 a^{14} + 7 a^{13} + 9 a^{12} + 6 a^{10} + 6 a^{9} + 5 a^{8} + 10 a^{6} + 14 a^{5} + 12 a^{4} + 12 a^{3} + a^{2} + 14 a + 20\right)\cdot 23^{194} + \left(12 a^{14} + 18 a^{12} + 4 a^{11} + 14 a^{10} + 7 a^{9} + 12 a^{8} + a^{7} + 22 a^{6} + 7 a^{5} + 10 a^{4} + 20 a^{3} + 22 a^{2} + 15 a + 17\right)\cdot 23^{195} + \left(7 a^{14} + 4 a^{13} + 20 a^{12} + 12 a^{11} + 3 a^{10} + a^{8} + 9 a^{7} + 4 a^{6} + 10 a^{5} + 14 a^{4} + 3 a^{3} + 22 a^{2} + 3 a + 9\right)\cdot 23^{196} + \left(15 a^{14} + 17 a^{13} + 17 a^{12} + 16 a^{11} + 9 a^{10} + 3 a^{9} + 14 a^{8} + 19 a^{7} + 4 a^{6} + 6 a^{5} + 9 a^{4} + 9 a^{3} + 13 a^{2} + 22 a + 4\right)\cdot 23^{197} + \left(5 a^{14} + 16 a^{13} + a^{12} + 6 a^{11} + 8 a^{10} + 20 a^{9} + 22 a^{8} + 4 a^{7} + 5 a^{6} + 12 a^{5} + 12 a^{4} + 20 a^{3} + 8 a^{2} + 18 a + 19\right)\cdot 23^{198} + \left(20 a^{13} + 14 a^{12} + 19 a^{11} + 21 a^{10} + 16 a^{9} + 19 a^{8} + 18 a^{7} + 21 a^{6} + 6 a^{5} + 18 a^{4} + 21 a^{3} + 3 a^{2} + 22 a + 20\right)\cdot 23^{199} + \left(13 a^{14} + 3 a^{13} + 5 a^{12} + 7 a^{11} + 13 a^{10} + 5 a^{9} + 22 a^{8} + 17 a^{7} + 22 a^{6} + 9 a^{5} + 10 a^{4} + 19 a^{2} + 8 a + 7\right)\cdot 23^{200} + \left(22 a^{14} + a^{13} + 15 a^{11} + 22 a^{10} + 19 a^{9} + 16 a^{8} + 8 a^{7} + 22 a^{6} + 11 a^{5} + 13 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+ 15 a^{5} + 12 a^{4} + 7 a^{3} + 20 a^{2} + 6 a + 22\right)\cdot 23^{207} + \left(18 a^{14} + 16 a^{13} + 6 a^{12} + 17 a^{11} + 10 a^{10} + 3 a^{9} + 6 a^{8} + 12 a^{7} + a^{6} + 12 a^{5} + 22 a^{4} + 2 a^{3} + 5 a^{2} + 12 a + 17\right)\cdot 23^{208} + \left(9 a^{14} + 16 a^{13} + 18 a^{12} + 10 a^{11} + 22 a^{10} + 5 a^{9} + 2 a^{8} + a^{7} + 9 a^{5} + 16 a^{4} + 13 a^{2} + 14 a + 19\right)\cdot 23^{209} + \left(15 a^{13} + 21 a^{12} + 11 a^{11} + 10 a^{10} + 20 a^{9} + 16 a^{8} + 18 a^{7} + 22 a^{6} + 14 a^{5} + 15 a^{4} + 13 a^{3} + 8 a^{2} + 4 a + 3\right)\cdot 23^{210} + \left(8 a^{14} + a^{13} + 12 a^{12} + 12 a^{11} + 19 a^{10} + a^{9} + 22 a^{8} + 15 a^{6} + 21 a^{5} + 14 a^{4} + 4 a^{3} + 16 a^{2} + 9 a + 12\right)\cdot 23^{211} + \left(9 a^{14} + 20 a^{13} + 18 a^{12} + 18 a^{11} + 7 a^{10} + 14 a^{9} + 9 a^{8} + 2 a^{7} + 14 a^{6} + 7 a^{5} + 15 a^{4} + 13 a^{3} + 12 a^{2} + 20 a + 18\right)\cdot 23^{212} + \left(2 a^{14} + 10 a^{13} + 11 a^{12} + 16 a^{11} + 2 a^{9} + 5 a^{8} + 7 a^{7} + 9 a^{6} + 8 a^{5} + 13 a^{4} + 21 a^{3} + 10 a^{2} + 7 a + 2\right)\cdot 23^{213} + \left(13 a^{14} + 2 a^{13} + 3 a^{12} + 13 a^{11} + 7 a^{10} + 10 a^{9} + 19 a^{8} + 6 a^{7} + 3 a^{6} + 11 a^{4} + 15 a^{3} + 12 a^{2} + 11 a + 21\right)\cdot 23^{214} + \left(4 a^{14} + 15 a^{13} + 20 a^{12} + 18 a^{11} + 21 a^{10} + 18 a^{9} + 17 a^{8} + 14 a^{7} + 11 a^{6} + 8 a^{5} + 20 a^{4} + 3 a^{3} + 21 a^{2} + 15 a + 22\right)\cdot 23^{215} + \left(8 a^{14} + 18 a^{13} + 22 a^{12} + 22 a^{11} + 16 a^{10} + 11 a^{9} + 11 a^{8} + 22 a^{7} + 9 a^{6} + 17 a^{5} + 13 a^{4} + 10 a^{3} + 11 a^{2} + 8 a + 3\right)\cdot 23^{216} + \left(17 a^{14} + 3 a^{13} + a^{12} + 18 a^{11} + 10 a^{10} + 22 a^{9} + 14 a^{8} + 2 a^{7} + 5 a^{6} + 13 a^{5} + 4 a^{4} + 2 a^{3} + 6 a^{2} + 21 a + 3\right)\cdot 23^{217} + \left(21 a^{14} + a^{13} + 22 a^{12} + 16 a^{11} + 16 a^{10} + 14 a^{9} + 6 a^{8} + 16 a^{7} + a^{6} + 19 a^{5} + 4 a^{4} + 22 a^{3} + 14 a^{2} + a + 11\right)\cdot 23^{218} + \left(11 a^{14} + 14 a^{13} + 22 a^{12} + 22 a^{11} + a^{10} + 7 a^{9} + 8 a^{8} + 17 a^{7} + 7 a^{6} + 10 a^{5} + 6 a^{4} + 9 a^{3} + 20 a^{2} + 21 a + 4\right)\cdot 23^{219} + \left(21 a^{14} + 12 a^{13} + 2 a^{12} + 17 a^{11} + 9 a^{10} + 15 a^{9} + 2 a^{8} + 22 a^{7} + 7 a^{6} + 5 a^{5} + 22 a^{4} + 14 a^{3} + 14 a^{2} + 10 a + 7\right)\cdot 23^{220} + \left(5 a^{14} + 17 a^{13} + 21 a^{12} + 21 a^{11} + 16 a^{10} + 13 a^{9} + 2 a^{8} + 9 a^{7} + 8 a^{6} + 4 a^{5} + 9 a^{4} + 3 a^{3} + 3 a^{2} + 17 a + 1\right)\cdot 23^{221} + \left(15 a^{14} + 10 a^{13} + 11 a^{12} + 7 a^{10} + 19 a^{9} + 13 a^{8} + 15 a^{7} + 17 a^{6} + 10 a^{5} + 20 a^{4} + 8 a^{3} + 17 a^{2} + 22 a + 14\right)\cdot 23^{222} + \left(20 a^{14} + 12 a^{13} + a^{11} + 13 a^{10} + 3 a^{8} + 11 a^{7} + 2 a^{6} + 3 a^{5} + 7 a^{4} + 3 a^{3} + 9 a^{2} + 4 a\right)\cdot 23^{223} + \left(5 a^{14} + a^{13} + 16 a^{12} + 3 a^{11} + 18 a^{10} + 2 a^{9} + 10 a^{8} + a^{7} + 8 a^{6} + 6 a^{5} + 17 a^{4} + 19 a^{3} + 11 a^{2} + 15 a + 13\right)\cdot 23^{224} + \left(4 a^{14} + 2 a^{13} + 16 a^{12} + 3 a^{11} + 11 a^{9} + 15 a^{8} + 10 a^{6} + 13 a^{5} + 13 a^{4} + 2 a^{3} + 13 a^{2} + 7 a + 3\right)\cdot 23^{225} + \left(15 a^{14} + 20 a^{13} + 16 a^{12} + 10 a^{11} + 20 a^{10} + 11 a^{9} + 15 a^{8} + 10 a^{7} + 13 a^{6} + 19 a^{5} + 12 a^{4} + 12 a^{3} + 7 a^{2} + 11 a + 16\right)\cdot 23^{226} + \left(22 a^{14} + 7 a^{12} + 18 a^{11} + 9 a^{10} + 17 a^{9} + 9 a^{8} + 10 a^{7} + 13 a^{6} + 6 a^{5} + 9 a^{4} + 2 a^{3} + 21 a^{2} + 17 a + 7\right)\cdot 23^{227} + \left(8 a^{14} + 3 a^{12} + 10 a^{11} + 11 a^{10} + 6 a^{9} + 14 a^{8} + 10 a^{7} + 11 a^{6} + 8 a^{5} + 17 a^{4} + 21 a^{3} + 15 a^{2} + 21 a\right)\cdot 23^{228} + \left(10 a^{14} + 20 a^{13} + 5 a^{12} + 12 a^{11} + 15 a^{10} + 10 a^{9} + 8 a^{8} + 8 a^{7} + 4 a^{6} + 12 a^{5} + 8 a^{4} + 14 a^{3} + 6 a^{2} + 19 a + 22\right)\cdot 23^{229} + \left(20 a^{14} + 4 a^{13} + 3 a^{12} + 18 a^{11} + 13 a^{10} + 8 a^{9} + 17 a^{8} + 12 a^{7} + 16 a^{6} + a^{5} + 14 a^{4} + 21 a^{3} + 12 a^{2} + 6 a + 12\right)\cdot 23^{230} + \left(12 a^{14} + 22 a^{13} + 7 a^{12} + 15 a^{11} + 2 a^{10} + 19 a^{9} + 14 a^{8} + 15 a^{7} + 9 a^{6} + 6 a^{5} + 8 a^{4} + 17 a^{3} + 10 a^{2} + 12 a + 1\right)\cdot 23^{231} + \left(16 a^{14} + 16 a^{13} + 4 a^{12} + 6 a^{11} + 10 a^{10} + 5 a^{8} + 16 a^{7} + 6 a^{6} + 7 a^{5} + 8 a^{3} + 21 a^{2} + 13 a + 9\right)\cdot 23^{232} + \left(15 a^{14} + 2 a^{13} + 8 a^{12} + 8 a^{11} + 6 a^{10} + a^{9} + 8 a^{8} + a^{6} + 6 a^{5} + 18 a^{4} + 19 a^{3} + 7 a^{2} + 6 a + 6\right)\cdot 23^{233} + \left(20 a^{14} + 17 a^{13} + 6 a^{12} + 4 a^{11} + 17 a^{10} + a^{9} + 22 a^{8} + 17 a^{7} + 17 a^{6} + 16 a^{5} + 12 a^{3} + 12 a^{2} + 4 a + 7\right)\cdot 23^{234} + \left(6 a^{14} + 21 a^{13} + 14 a^{12} + 22 a^{11} + 14 a^{10} + 19 a^{9} + 17 a^{8} + 22 a^{7} + 12 a^{6} + 16 a^{5} + 13 a^{4} + a^{3} + 2 a^{2} + a + 15\right)\cdot 23^{235} + \left(13 a^{14} + 13 a^{13} + 18 a^{12} + 21 a^{11} + 7 a^{10} + 7 a^{9} + 14 a^{8} + 13 a^{7} + 3 a^{6} + 10 a^{5} + 22 a^{4} + 8 a^{2} + 11 a + 8\right)\cdot 23^{236} + \left(16 a^{14} + 18 a^{13} + 3 a^{12} + 17 a^{11} + 20 a^{10} + 9 a^{9} + 6 a^{8} + 20 a^{7} + 6 a^{6} + 18 a^{5} + 6 a^{4} + 9 a^{3} + 7 a^{2} + 5 a + 5\right)\cdot 23^{237} + \left(16 a^{14} + 4 a^{13} + 5 a^{12} + 10 a^{11} + 21 a^{10} + 11 a^{9} + 13 a^{8} + 16 a^{7} + 14 a^{6} + 4 a^{5} + 18 a^{4} + 19 a^{3} + 11 a + 17\right)\cdot 23^{238} + \left(13 a^{14} + 17 a^{13} + 11 a^{12} + 4 a^{11} + 22 a^{10} + a^{9} + 13 a^{8} + 6 a^{7} + 13 a^{6} + 4 a^{5} + 15 a^{3} + 4 a^{2} + 11 a + 16\right)\cdot 23^{239} + \left(3 a^{14} + 3 a^{13} + 13 a^{12} + 5 a^{11} + 13 a^{10} + 22 a^{9} + 21 a^{8} + 20 a^{7} + 19 a^{6} + 14 a^{5} + 15 a^{4} + 13 a^{3} + 5 a^{2} + 9 a + 16\right)\cdot 23^{240} + \left(4 a^{14} + 9 a^{13} + 19 a^{12} + 21 a^{11} + 19 a^{10} + 17 a^{9} + 12 a^{8} + 16 a^{7} + 6 a^{6} + 11 a^{5} + 19 a^{4} + 3 a^{3} + 11 a^{2} + 18 a + 9\right)\cdot 23^{241} + \left(5 a^{14} + 18 a^{13} + 10 a^{12} + 3 a^{11} + a^{10} + 11 a^{9} + 8 a^{8} + 8 a^{7} + 3 a^{6} + 12 a^{5} + 19 a^{4} + 5 a^{3} + 4 a^{2} + 7 a + 2\right)\cdot 23^{242} + \left(5 a^{14} + 9 a^{13} + 5 a^{12} + 13 a^{11} + 2 a^{10} + 12 a^{9} + 7 a^{8} + 6 a^{7} + 10 a^{6} + 20 a^{5} + 10 a^{4} + 3 a^{3} + 15 a^{2} + 20 a + 9\right)\cdot 23^{243} + \left(21 a^{14} + 18 a^{13} + 7 a^{12} + 10 a^{11} + 13 a^{10} + 19 a^{9} + 6 a^{8} + 9 a^{7} + 21 a^{6} + 3 a^{5} + 9 a^{4} + 2 a^{3} + 2 a^{2} + a + 13\right)\cdot 23^{244} + \left(16 a^{13} + 9 a^{12} + 13 a^{11} + 18 a^{10} + 4 a^{9} + 21 a^{8} + 17 a^{7} + 16 a^{6} + 15 a^{5} + 5 a^{4} + 16 a^{3} + 22 a^{2} + 19 a + 17\right)\cdot 23^{245} + \left(19 a^{14} + 3 a^{13} + 14 a^{12} + 11 a^{11} + 18 a^{10} + 11 a^{9} + a^{8} + 7 a^{7} + 3 a^{6} + 22 a^{5} + 4 a^{4} + 13 a^{3} + 6 a^{2} + 17 a + 5\right)\cdot 23^{246} + \left(10 a^{14} + 11 a^{12} + 5 a^{11} + 20 a^{9} + 20 a^{8} + 7 a^{7} + 3 a^{6} + 15 a^{5} + 21 a^{4} + 3 a^{3} + 4 a^{2} + 18 a + 14\right)\cdot 23^{247} + \left(a^{14} + 6 a^{13} + 16 a^{12} + 8 a^{11} + 18 a^{10} + 16 a^{9} + 18 a^{8} + 10 a^{7} + 11 a^{6} + 4 a^{5} + 20 a^{4} + 2 a^{2} + 21 a\right)\cdot 23^{248} + \left(18 a^{14} + 9 a^{13} + 20 a^{12} + 9 a^{11} + 14 a^{10} + 7 a^{9} + 5 a^{8} + 12 a^{7} + 4 a^{6} + 5 a^{5} + a^{4} + 15 a^{3} + 12 a^{2} + 16 a + 11\right)\cdot 23^{249} + \left(22 a^{14} + 17 a^{13} + 15 a^{12} + 18 a^{11} + 10 a^{10} + 4 a^{9} + 8 a^{8} + 2 a^{7} + 20 a^{6} + 4 a^{5} + 13 a^{4} + 7 a^{3} + 2 a^{2} + 6 a + 21\right)\cdot 23^{250} + \left(5 a^{14} + 17 a^{13} + 10 a^{12} + 11 a^{11} + 16 a^{10} + 9 a^{8} + 4 a^{7} + 21 a^{6} + 2 a^{5} + 16 a^{4} + 11 a^{3} + 15 a^{2} + 2 a + 18\right)\cdot 23^{251} + \left(4 a^{14} + 6 a^{13} + 8 a^{12} + 16 a^{11} + 2 a^{10} + 9 a^{9} + 2 a^{8} + 21 a^{6} + 20 a^{5} + 9 a^{4} + 7 a^{3} + 3 a^{2} + 8 a + 20\right)\cdot 23^{252} + \left(3 a^{14} + 19 a^{12} + 8 a^{11} + 4 a^{10} + 8 a^{9} + 4 a^{8} + 19 a^{7} + 9 a^{5} + 17 a^{4} + 4 a^{3} + 21 a^{2} + 2 a + 20\right)\cdot 23^{253} + \left(14 a^{14} + 5 a^{13} + 10 a^{12} + 19 a^{11} + 16 a^{10} + 20 a^{9} + 13 a^{8} + 8 a^{7} + 21 a^{6} + 15 a^{5} + a^{4} + 12 a^{3} + 4 a^{2} + 17 a + 17\right)\cdot 23^{254} + \left(18 a^{14} + 13 a^{13} + 10 a^{12} + 6 a^{11} + 19 a^{10} + 11 a^{9} + 17 a^{8} + 7 a^{7} + 22 a^{6} + 19 a^{5} + 14 a^{4} + 3 a^{3} + 5 a^{2} + 22 a + 16\right)\cdot 23^{255} + \left(10 a^{14} + 8 a^{13} + 17 a^{12} + 20 a^{11} + 20 a^{10} + 5 a^{9} + 6 a^{8} + 21 a^{7} + 19 a^{6} + 9 a^{5} + 19 a^{4} + 15 a^{3} + 20 a^{2} + 10 a + 20\right)\cdot 23^{256} + \left(9 a^{14} + 8 a^{13} + 13 a^{12} + 20 a^{11} + 2 a^{10} + 2 a^{9} + 15 a^{8} + 13 a^{7} + 9 a^{6} + 7 a^{5} + 7 a^{4} + 14 a^{3} + 17 a^{2} + 14 a + 9\right)\cdot 23^{257} + \left(2 a^{14} + 22 a^{13} + 14 a^{12} + 4 a^{11} + 22 a^{10} + 13 a^{9} + 13 a^{8} + 12 a^{7} + 7 a^{6} + 5 a^{5} + 13 a^{4} + 9 a^{3} + 14 a^{2} + 16 a + 18\right)\cdot 23^{258} + \left(21 a^{14} + 5 a^{13} + 12 a^{12} + 7 a^{11} + 13 a^{10} + 21 a^{9} + 22 a^{8} + 11 a^{7} + 4 a^{6} + a^{4} + 6 a^{3} + 19 a^{2} + a + 8\right)\cdot 23^{259} + \left(17 a^{14} + 17 a^{13} + 19 a^{12} + 5 a^{11} + 8 a^{10} + 7 a^{8} + 20 a^{7} + 2 a^{6} + 3 a^{5} + 22 a^{4} + 10 a^{3} + 3 a^{2} + 19 a + 22\right)\cdot 23^{260} + \left(3 a^{14} + 9 a^{13} + a^{12} + 17 a^{11} + 17 a^{10} + 9 a^{9} + a^{7} + 6 a^{6} + 18 a^{5} + 2 a^{4} + 9 a^{3} + 3 a^{2} + 19 a + 19\right)\cdot 23^{261} + \left(9 a^{14} + 21 a^{13} + 7 a^{12} + 16 a^{11} + 4 a^{10} + 21 a^{9} + 2 a^{8} + 8 a^{7} + 3 a^{6} + 8 a^{5} + 13 a^{4} + 22 a^{3} + 17 a^{2} + 5 a + 1\right)\cdot 23^{262} + \left(21 a^{14} + 20 a^{13} + 10 a^{12} + 7 a^{11} + a^{10} + 20 a^{9} + 14 a^{8} + 13 a^{7} + 3 a^{6} + 11 a^{5} + 10 a^{4} + 11 a^{3} + 4 a^{2} + 3 a + 1\right)\cdot 23^{263} + \left(12 a^{14} + 4 a^{13} + 9 a^{12} + 16 a^{11} + 9 a^{10} + 8 a^{8} + 15 a^{7} + 11 a^{6} + 5 a^{4} + 18 a^{3} + 9 a^{2} + 13 a + 14\right)\cdot 23^{264} + \left(a^{14} + 8 a^{13} + 9 a^{12} + 15 a^{10} + 20 a^{9} + 15 a^{8} + 12 a^{7} + 7 a^{6} + 5 a^{5} + 16 a^{4} + 13 a^{3} + 22 a^{2} + 14 a + 6\right)\cdot 23^{265} + \left(15 a^{14} + 10 a^{13} + 16 a^{12} + 5 a^{11} + 21 a^{10} + a^{9} + 14 a^{8} + 12 a^{7} + 15 a^{6} + 16 a^{5} + 15 a^{4} + 13 a^{3} + 19 a^{2} + 3 a + 4\right)\cdot 23^{266} + \left(16 a^{14} + 15 a^{13} + 20 a^{12} + 12 a^{11} + 21 a^{10} + 22 a^{9} + 19 a^{8} + 13 a^{7} + 18 a^{6} + 3 a^{5} + 9 a^{4} + 8 a^{3} + 11 a + 14\right)\cdot 23^{267} + \left(17 a^{14} + 13 a^{12} + 11 a^{11} + 16 a^{10} + 7 a^{9} + 21 a^{8} + 14 a^{7} + 21 a^{6} + 3 a^{5} + 22 a^{4} + 2 a^{3} + 10 a + 14\right)\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 12 }$ $=$ $9 a^{13} + 7 a^{12} + 7 a^{11} + 22 a^{10} + 8 a^{8} + 15 a^{7} + 12 a^{6} + 18 a^{5} + 15 a^{4} + 19 a^{3} + 19 a^{2} + 20 a + \left(7 a^{14} + 20 a^{11} + 7 a^{10} + 14 a^{9} + 18 a^{8} + 22 a^{7} + 6 a^{6} + 10 a^{5} + 9 a^{4} + 15 a^{3} + 7 a^{2} + 13 a + 11\right)\cdot 23 + \left(17 a^{14} + 18 a^{13} + 2 a^{12} + 3 a^{11} + a^{10} + 9 a^{9} + 3 a^{8} + 13 a^{7} + 20 a^{6} + 19 a^{5} + 17 a^{4} + 17 a^{3} + 10 a^{2} + 17 a + 15\right)\cdot 23^{2} + \left(21 a^{13} + 17 a^{11} + 11 a^{10} + 12 a^{9} + 8 a^{8} + 20 a^{7} + 21 a^{6} + 16 a^{5} + 22 a^{4} + a^{3} + 5 a^{2} + 9 a + 22\right)\cdot 23^{3} + \left(12 a^{14} + 20 a^{13} + 6 a^{12} + a^{11} + 4 a^{10} + 14 a^{9} + 12 a^{8} + a^{7} + 12 a^{6} + 22 a^{5} + a^{4} + 10 a^{3} + 2 a^{2} + 18 a\right)\cdot 23^{4} + \left(3 a^{14} + 18 a^{13} + 17 a^{12} + 17 a^{11} + 14 a^{10} + 13 a^{9} + 9 a^{8} + 11 a^{7} + 16 a^{6} + 22 a^{5} + 2 a^{4} + 16 a^{3} + a^{2} + 7 a\right)\cdot 23^{5} + \left(19 a^{14} + 14 a^{13} + 22 a^{12} + 4 a^{11} + 10 a^{10} + 13 a^{9} + 13 a^{7} + 6 a^{6} + 14 a^{5} + 18 a^{4} + 17 a^{3} + 2 a^{2} + 18 a + 4\right)\cdot 23^{6} + \left(17 a^{14} + 19 a^{13} + 9 a^{12} + 8 a^{11} + 14 a^{10} + 7 a^{9} + 14 a^{8} + 6 a^{7} + 19 a^{6} + 8 a^{5} + 7 a^{4} + 12 a^{3} + 13 a^{2} + 12 a + 11\right)\cdot 23^{7} + \left(13 a^{14} + 19 a^{13} + 6 a^{12} + 14 a^{11} + 5 a^{10} + 20 a^{9} + 11 a^{8} + 20 a^{7} + 15 a^{6} + 2 a^{5} + 16 a^{4} + 15 a^{3} + 22 a^{2} + 5\right)\cdot 23^{8} + \left(12 a^{14} + 4 a^{13} + 3 a^{12} + 16 a^{11} + 16 a^{10} + 10 a^{9} + 9 a^{8} + 20 a^{7} + 3 a^{6} + 5 a^{5} + 17 a^{4} + 22 a^{3} + 22 a^{2} + 20 a + 1\right)\cdot 23^{9} + \left(15 a^{14} + 17 a^{13} + 18 a^{12} + 3 a^{11} + a^{10} + 22 a^{9} + 13 a^{8} + 14 a^{7} + a^{6} + 19 a^{5} + 15 a^{4} + 21 a^{3} + 7 a^{2} + 13 a + 16\right)\cdot 23^{10} + \left(a^{14} + 11 a^{12} + 12 a^{11} + 19 a^{10} + 15 a^{9} + 14 a^{8} + 17 a^{7} + 4 a^{6} + 6 a^{5} + 2 a^{4} + 16 a^{3} + 19 a + 16\right)\cdot 23^{11} + \left(15 a^{14} + 17 a^{13} + 9 a^{12} + 5 a^{11} + 9 a^{10} + 8 a^{9} + 17 a^{8} + a^{7} + a^{6} + 14 a^{5} + 14 a^{4} + 22 a^{3} + 7 a^{2} + 6 a + 20\right)\cdot 23^{12} + \left(21 a^{14} + 14 a^{13} + 4 a^{12} + 16 a^{11} + 15 a^{10} + 19 a^{9} + 19 a^{8} + 19 a^{7} + 14 a^{6} + 6 a^{5} + 5 a^{4} + 16 a^{3} + 14 a^{2} + 5 a + 15\right)\cdot 23^{13} + \left(3 a^{14} + 7 a^{13} + 11 a^{12} + 15 a^{11} + 14 a^{10} + 15 a^{9} + 22 a^{8} + 18 a^{7} + 18 a^{6} + 3 a^{5} + 7 a^{4} + 8 a^{3} + 14 a^{2} + 11 a + 16\right)\cdot 23^{14} + \left(21 a^{14} + 11 a^{13} + 3 a^{12} + 7 a^{11} + 10 a^{10} + 16 a^{9} + 14 a^{8} + 16 a^{7} + 9 a^{6} + 17 a^{5} + 20 a^{4} + 3 a^{3} + 16 a^{2} + 10 a + 2\right)\cdot 23^{15} + \left(16 a^{14} + 12 a^{13} + 15 a^{11} + 20 a^{10} + 4 a^{9} + 9 a^{8} + 8 a^{7} + 12 a^{6} + 19 a^{5} + 11 a^{4} + 12 a^{3} + 18 a^{2} + 6 a + 7\right)\cdot 23^{16} + \left(a^{14} + 21 a^{13} + 7 a^{12} + 8 a^{11} + 21 a^{10} + 18 a^{9} + 20 a^{8} + 12 a^{7} + 13 a^{6} + 20 a^{5} + 17 a^{4} + 10 a^{3} + 15 a^{2} + 18 a\right)\cdot 23^{17} + \left(11 a^{14} + 15 a^{13} + 17 a^{12} + 18 a^{11} + 18 a^{10} + a^{9} + 13 a^{8} + 13 a^{7} + 5 a^{6} + 18 a^{5} + 9 a^{4} + 6 a^{3} + 8 a^{2} + 16 a + 4\right)\cdot 23^{18} + \left(6 a^{14} + 4 a^{13} + 7 a^{12} + 13 a^{11} + 22 a^{10} + 13 a^{9} + 10 a^{8} + 22 a^{7} + 22 a^{6} + 11 a^{5} + 15 a^{4} + 8 a^{3} + 10 a^{2} + 11 a\right)\cdot 23^{19} + \left(14 a^{14} + 20 a^{13} + 3 a^{12} + 3 a^{11} + 3 a^{10} + 13 a^{9} + a^{8} + 2 a^{7} + 7 a^{6} + 8 a^{4} + 13 a^{3} + 19 a^{2} + 6 a + 3\right)\cdot 23^{20} + \left(12 a^{14} + 6 a^{13} + 14 a^{12} + 11 a^{11} + 10 a^{10} + 11 a^{9} + 9 a^{8} + 5 a^{7} + 20 a^{6} + 9 a^{5} + 2 a^{4} + a^{3} + 8 a^{2} + a + 22\right)\cdot 23^{21} + \left(16 a^{14} + 11 a^{13} + 21 a^{12} + 17 a^{11} + 13 a^{10} + 9 a^{9} + 12 a^{8} + 14 a^{7} + 6 a^{6} + 7 a^{4} + 16 a^{3} + 7 a^{2} + 12 a + 4\right)\cdot 23^{22} + \left(22 a^{14} + 7 a^{13} + 8 a^{12} + 16 a^{11} + 10 a^{10} + 16 a^{9} + 14 a^{8} + 9 a^{7} + 16 a^{6} + 7 a^{5} + 18 a^{3} + 12 a^{2} + 19\right)\cdot 23^{23} + \left(9 a^{14} + 9 a^{13} + 6 a^{12} + 7 a^{11} + 11 a^{10} + 18 a^{9} + 8 a^{8} + 5 a^{7} + 16 a^{6} + 16 a^{5} + a^{4} + 11 a^{3} + 20 a^{2} + 17 a\right)\cdot 23^{24} + \left(8 a^{14} + 18 a^{13} + 11 a^{12} + a^{11} + 9 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\left(a^{14} + 5 a^{13} + 21 a^{12} + 13 a^{11} + 6 a^{10} + 16 a^{9} + 22 a^{8} + 21 a^{7} + 22 a^{6} + 18 a^{5} + 5 a^{4} + 17 a^{3} + 8 a^{2} + 5 a + 8\right)\cdot 23^{31} + \left(17 a^{14} + 3 a^{13} + 8 a^{12} + a^{11} + 11 a^{10} + 18 a^{8} + 14 a^{7} + 5 a^{6} + 18 a^{4} + 22 a^{3} + 20 a^{2} + 17 a + 18\right)\cdot 23^{32} + \left(13 a^{14} + 2 a^{13} + 22 a^{12} + 7 a^{11} + 9 a^{10} + 17 a^{9} + 4 a^{7} + 20 a^{6} + 16 a^{5} + 16 a^{4} + 12 a^{3} + 22 a^{2} + 15 a + 10\right)\cdot 23^{33} + \left(17 a^{14} + 20 a^{12} + 17 a^{11} + 7 a^{10} + 14 a^{9} + 12 a^{8} + 22 a^{7} + 4 a^{6} + 5 a^{5} + 14 a^{4} + 8 a^{3} + 19 a^{2} + 16 a + 14\right)\cdot 23^{34} + \left(4 a^{14} + 17 a^{13} + 14 a^{11} + 8 a^{10} + 3 a^{9} + 21 a^{8} + 9 a^{7} + 6 a^{6} + 2 a^{5} + 13 a^{4} + 19 a^{3} + 21 a^{2} + 17 a + 3\right)\cdot 23^{35} + \left(5 a^{14} + 12 a^{13} + a^{12} + 21 a^{11} + 6 a^{10} + 16 a^{9} + 17 a^{8} + 3 a^{7} + 4 a^{6} + 7 a^{5} + 4 a^{4} + 15 a^{3} + 13 a + 9\right)\cdot 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20\right)\cdot 23^{42} + \left(6 a^{13} + 3 a^{12} + 5 a^{11} + 4 a^{10} + 7 a^{9} + 22 a^{8} + 20 a^{7} + 3 a^{6} + 11 a^{5} + 5 a^{4} + a^{3} + 21 a^{2} + 8 a\right)\cdot 23^{43} + \left(2 a^{14} + 5 a^{13} + 22 a^{12} + 4 a^{10} + 6 a^{9} + 21 a^{8} + 17 a^{7} + 11 a^{6} + 10 a^{5} + 21 a^{4} + 6 a^{3} + 11 a^{2} + 9 a\right)\cdot 23^{44} + \left(20 a^{14} + 6 a^{13} + 17 a^{12} + 3 a^{11} + a^{10} + 15 a^{9} + 21 a^{8} + 13 a^{7} + 8 a^{6} + 11 a^{5} + 8 a^{4} + a^{3} + 15 a^{2} + 15 a + 4\right)\cdot 23^{45} + \left(21 a^{14} + 12 a^{13} + 12 a^{12} + 16 a^{11} + 15 a^{10} + 7 a^{9} + 15 a^{8} + 21 a^{7} + 9 a^{6} + 10 a^{5} + 9 a^{4} + 17 a^{3} + 10 a^{2} + 16 a + 11\right)\cdot 23^{46} + \left(a^{14} + 12 a^{13} + 18 a^{11} + 12 a^{10} + 8 a^{9} + 21 a^{8} + 20 a^{7} + 16 a^{6} + 22 a^{5} + 10 a^{4} + 15 a^{3} + 11 a^{2} + 20 a + 9\right)\cdot 23^{47} + \left(7 a^{14} + 13 a^{13} + 4 a^{12} + 16 a^{11} + 9 a^{10} + 5 a^{9} + 6 a^{8} + 15 a^{7} + 4 a^{6} + 4 a^{5} + 19 a^{4} + 6 a^{3} + 3 a^{2} + 4 a + 14\right)\cdot 23^{48} + \left(2 a^{14} + 7 a^{13} + 3 a^{12} + 19 a^{11} + 7 a^{10} + 8 a^{9} + 5 a^{8} + 21 a^{7} + 19 a^{6} + 3 a^{5} + 18 a^{4} + 15 a^{3} + 15 a^{2} + 6 a + 22\right)\cdot 23^{49} + \left(13 a^{14} + 21 a^{13} + 13 a^{12} + 2 a^{11} + 6 a^{10} + 14 a^{9} + 17 a^{8} + 16 a^{7} + 21 a^{6} + 15 a^{5} + 17 a^{4} + 3 a^{3} + 5 a^{2} + 9 a + 8\right)\cdot 23^{50} + \left(6 a^{14} + 6 a^{13} + a^{12} + 5 a^{11} + 18 a^{10} + 2 a^{9} + 4 a^{8} + 4 a^{7} + 4 a^{6} + 4 a^{5} + 12 a^{4} + 22 a^{3} + 3 a^{2} + 14 a + 7\right)\cdot 23^{51} + \left(7 a^{14} + 14 a^{13} + 6 a^{12} + 13 a^{11} + 16 a^{10} + 19 a^{9} + a^{8} + 19 a^{7} + 19 a^{6} + 13 a^{5} + 4 a^{4} + 18 a^{2} + 12 a + 21\right)\cdot 23^{52} + \left(20 a^{14} + 12 a^{13} + 20 a^{12} + 13 a^{11} + 18 a^{10} + 13 a^{9} + 18 a^{7} + 15 a^{6} + 4 a^{5} + 20 a^{3} + 11 a^{2} + 19 a + 13\right)\cdot 23^{53} + \left(5 a^{14} + 18 a^{13} + 10 a^{12} + 4 a^{11} + 20 a^{10} + 17 a^{9} + 10 a^{8} + 12 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+ 8 a^{9} + 12 a^{8} + 5 a^{7} + 14 a^{6} + 7 a^{5} + a^{4} + 12 a^{3} + 11 a^{2} + 12 a + 12\right)\cdot 23^{60} + \left(6 a^{14} + 19 a^{13} + 19 a^{12} + 15 a^{11} + 20 a^{10} + 18 a^{9} + 14 a^{8} + 4 a^{7} + 12 a^{6} + 14 a^{5} + 2 a^{4} + 5 a^{2} + 22 a + 8\right)\cdot 23^{61} + \left(12 a^{14} + 9 a^{13} + 15 a^{12} + 12 a^{11} + 7 a^{10} + 5 a^{9} + 17 a^{8} + 12 a^{7} + 19 a^{6} + 17 a^{5} + 6 a^{4} + 20 a^{3} + 21 a^{2} + 21 a + 1\right)\cdot 23^{62} + \left(3 a^{14} + 22 a^{13} + 8 a^{12} + 11 a^{11} + 6 a^{10} + 22 a^{9} + 20 a^{8} + 2 a^{7} + 22 a^{6} + 11 a^{5} + 9 a^{4} + 15 a^{3} + 5 a^{2} + 4 a + 10\right)\cdot 23^{63} + \left(a^{14} + 17 a^{13} + 7 a^{12} + 3 a^{11} + 22 a^{10} + 18 a^{9} + 18 a^{8} + 4 a^{7} + 5 a^{6} + 6 a^{5} + a^{4} + 8 a^{3} + 16 a^{2} + 22 a + 3\right)\cdot 23^{64} + \left(9 a^{14} + 8 a^{13} + 20 a^{12} + 12 a^{11} + 16 a^{10} + a^{9} + 14 a^{8} + 7 a^{7} + 12 a^{6} + 2 a^{5} + 2 a^{4} + 9 a^{3} + 9 a^{2} + 13 a + 14\right)\cdot 23^{65} + \left(11 a^{14} + 15 a^{13} + 17 a^{12} + 5 a^{11} + 7 a^{10} + 15 a^{9} + 18 a^{8} + 20 a^{7} + 10 a^{6} + 19 a^{5} + 17 a^{4} + 13 a^{3} + 20 a^{2} + 5 a + 2\right)\cdot 23^{66} + \left(16 a^{14} + 16 a^{13} + 8 a^{12} + a^{11} + 16 a^{10} + 7 a^{9} + 7 a^{7} + 19 a^{6} + 15 a^{5} + 15 a^{4} + 8 a^{3} + 21 a^{2} + 19 a + 22\right)\cdot 23^{67} + \left(16 a^{14} + 10 a^{13} + 18 a^{12} + 16 a^{11} + 18 a^{9} + 7 a^{8} + 2 a^{7} + 20 a^{6} + 6 a^{5} + 20 a^{4} + 15 a^{3} + 14 a^{2} + 17 a + 17\right)\cdot 23^{68} + \left(20 a^{14} + 22 a^{13} + 17 a^{12} + 18 a^{11} + 19 a^{10} + a^{9} + 21 a^{8} + 20 a^{7} + 3 a^{6} + 17 a^{5} + 12 a^{4} + 7 a^{3} + 10 a^{2} + 14 a + 4\right)\cdot 23^{69} + \left(21 a^{14} + 15 a^{13} + 9 a^{12} + 5 a^{11} + 8 a^{10} + 19 a^{9} + a^{8} + 2 a^{7} + 10 a^{6} + 2 a^{4} + 14 a^{3} + 2 a^{2} + 11 a + 9\right)\cdot 23^{70} + \left(10 a^{14} + 10 a^{13} + 19 a^{12} + 5 a^{11} + 3 a^{10} + 6 a^{9} + 19 a^{8} + 8 a^{7} + 16 a^{6} + 3 a^{5} + 17 a^{4} + 12 a^{3} + 11 a^{2} + 5 a + 5\right)\cdot 23^{71} + \left(14 a^{14} + 3 a^{13} + 14 a^{12} + 6 a^{11} + 2 a^{10} + a^{9} + 8 a^{8} + 22 a^{7} + 12 a^{6} + 16 a^{5} + 18 a^{4} + 15 a^{3} + 12 a^{2} + 5 a + 2\right)\cdot 23^{72} + \left(7 a^{14} + 2 a^{13} + 16 a^{12} + 16 a^{11} + 3 a^{10} + 15 a^{9} + 9 a^{8} + 21 a^{7} + a^{6} + 7 a^{5} + 6 a^{4} + 10 a^{3} + 8 a^{2} + 14 a + 22\right)\cdot 23^{73} + \left(6 a^{14} + 3 a^{13} + 5 a^{12} + 10 a^{11} + 2 a^{10} + 20 a^{9} + 12 a^{8} + 21 a^{7} + 8 a^{6} + 6 a^{5} + 5 a^{4} + 21 a^{3} + 22 a^{2} + 12 a + 13\right)\cdot 23^{74} + \left(16 a^{14} + 3 a^{13} + 13 a^{12} + 2 a^{11} + 5 a^{10} + 11 a^{9} + 22 a^{8} + 13 a^{7} + 18 a^{6} + 14 a^{5} + 21 a^{4} + 8 a^{3} + 14 a^{2} + a + 9\right)\cdot 23^{75} + \left(18 a^{14} + 9 a^{12} + 9 a^{11} + 17 a^{10} + 11 a^{9} + 9 a^{8} + 19 a^{7} + 15 a^{6} + 7 a^{5} + 6 a^{4} + 2 a^{3} + a^{2} + 22 a + 1\right)\cdot 23^{76} + \left(3 a^{14} + 15 a^{13} + 2 a^{12} + 11 a^{10} + 22 a^{9} + 17 a^{8} + 11 a^{7} + 18 a^{6} + 2 a^{5} + 4 a^{4} + 3 a^{3} + 6 a^{2} + 20 a\right)\cdot 23^{77} + \left(21 a^{14} + 11 a^{13} + 10 a^{12} + 8 a^{11} + 4 a^{10} + 19 a^{9} + 4 a^{8} + a^{7} + 22 a^{6} + 11 a^{5} + a^{4} + 8 a^{3} + 16 a + 7\right)\cdot 23^{78} + \left(a^{14} + 11 a^{13} + 4 a^{12} + 11 a^{11} + 15 a^{10} + 3 a^{9} + 15 a^{8} + 11 a^{7} + 8 a^{6} + 19 a^{5} + 3 a^{4} + a^{3} + 5 a^{2} + 6 a + 18\right)\cdot 23^{79} + \left(9 a^{14} + 9 a^{13} + 9 a^{12} + 17 a^{11} + 17 a^{9} + 17 a^{8} + 9 a^{7} + 22 a^{6} + 2 a^{5} + 3 a^{4} + 11 a^{3} + 8 a^{2} + 6 a\right)\cdot 23^{80} + \left(16 a^{14} + 5 a^{13} + 5 a^{12} + 12 a^{11} + 3 a^{10} + 16 a^{9} + 5 a^{8} + 21 a^{7} + 15 a^{6} + 11 a^{5} + 19 a^{4} + 21 a^{3} + 14 a^{2} + 18 a + 2\right)\cdot 23^{81} + \left(20 a^{14} + 9 a^{13} + 14 a^{12} + 21 a^{11} + 7 a^{10} + 8 a^{9} + 5 a^{8} + 12 a^{7} + 5 a^{6} + 13 a^{5} + 16 a^{4} + 13 a^{3} + 19 a^{2} + 9 a + 1\right)\cdot 23^{82} + \left(10 a^{14} + 8 a^{13} + 18 a^{12} + 14 a^{11} + 5 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+ 20 a^{11} + 11 a^{10} + 21 a^{9} + 9 a^{8} + 17 a^{7} + 3 a^{6} + a^{5} + 8 a^{4} + 15 a^{3} + a^{2} + 3 a + 15\right)\cdot 23^{89} + \left(7 a^{14} + 4 a^{12} + 11 a^{11} + 19 a^{10} + 5 a^{9} + 14 a^{8} + 18 a^{7} + 8 a^{6} + 13 a^{5} + 21 a^{4} + 18 a^{3} + 11 a^{2} + 16 a + 20\right)\cdot 23^{90} + \left(19 a^{14} + 14 a^{13} + 5 a^{12} + 15 a^{11} + 20 a^{10} + 8 a^{8} + 8 a^{7} + 19 a^{6} + 15 a^{5} + 8 a^{4} + 15 a^{3} + 21 a^{2} + 4 a + 18\right)\cdot 23^{91} + \left(14 a^{14} + 21 a^{13} + 21 a^{12} + 12 a^{11} + 6 a^{10} + 14 a^{9} + 15 a^{8} + 13 a^{6} + 4 a^{5} + 13 a^{4} + 3 a^{3} + 20 a^{2} + 21 a + 16\right)\cdot 23^{92} + \left(a^{14} + a^{13} + 7 a^{12} + 11 a^{11} + 10 a^{10} + 4 a^{9} + 18 a^{8} + 17 a^{7} + 20 a^{6} + 2 a^{5} + 20 a^{4} + 2 a^{3} + 18 a^{2} + 14 a + 1\right)\cdot 23^{93} + \left(11 a^{13} + 3 a^{12} + 10 a^{11} + a^{10} + 12 a^{9} + 10 a^{8} + 17 a^{7} + 10 a^{6} + 8 a^{5} + 21 a^{4} + 11 a^{3} + 22 a^{2} + 18 a + 3\right)\cdot 23^{94} + \left(6 a^{14} + 16 a^{13} + 22 a^{12} + 5 a^{11} + 6 a^{10} + 7 a^{9} + 6 a^{8} + 18 a^{7} + 7 a^{6} + 22 a^{5} + 17 a^{4} + 6 a^{3} + 16 a + 6\right)\cdot 23^{95} + \left(21 a^{14} + 21 a^{13} + 2 a^{12} + 8 a^{11} + 8 a^{10} + 12 a^{9} + 20 a^{8} + 5 a^{7} + 9 a^{6} + 12 a^{5} + 22 a^{4} + 22 a^{3} + 16 a^{2} + 7 a + 21\right)\cdot 23^{96} + \left(15 a^{14} + 17 a^{13} + 16 a^{12} + 13 a^{11} + 16 a^{10} + 14 a^{9} + 17 a^{8} + 21 a^{7} + 8 a^{6} + 22 a^{5} + a^{4} + 13 a^{2} + 16 a\right)\cdot 23^{97} + \left(14 a^{14} + 22 a^{13} + 8 a^{11} + 3 a^{10} + 17 a^{9} + 14 a^{8} + 3 a^{7} + 22 a^{6} + 4 a^{5} + 4 a^{4} + 4 a^{3} + 14 a^{2} + 15 a + 19\right)\cdot 23^{98} + \left(20 a^{14} + 16 a^{13} + 13 a^{12} + a^{11} + 8 a^{10} + 7 a^{9} + 19 a^{8} + 4 a^{7} + 9 a^{6} + 4 a^{5} + a^{4} + 5 a^{3} + 10 a^{2} + 21 a\right)\cdot 23^{99} + \left(7 a^{14} + 22 a^{13} + 6 a^{12} + 2 a^{11} + 21 a^{10} + 22 a^{9} + 12 a^{8} + 22 a^{7} + 20 a^{6} + 19 a^{5} + 9 a^{4} + 7 a^{3} + 21 a^{2} + 6 a\right)\cdot 23^{100} + \left(7 a^{14} + 2 a^{13} + 19 a^{12} + 6 a^{11} + 6 a^{10} + 14 a^{9} + 17 a^{7} + 18 a^{6} + 22 a^{5} + 2 a^{4} + a^{3} + 2 a^{2} + 11 a + 14\right)\cdot 23^{101} + \left(3 a^{14} + 19 a^{13} + 22 a^{12} + 14 a^{11} + 18 a^{10} + a^{9} + 12 a^{8} + 6 a^{7} + 8 a^{6} + 7 a^{5} + 20 a^{4} + 10 a^{3} + 17 a^{2} + 14\right)\cdot 23^{102} + \left(16 a^{14} + 9 a^{13} + 12 a^{12} + 5 a^{11} + a^{10} + 16 a^{9} + 21 a^{8} + 5 a^{7} + 22 a^{6} + 19 a^{5} + 10 a^{4} + 2 a^{3} + 3 a^{2} + 16 a + 14\right)\cdot 23^{103} + \left(13 a^{14} + 13 a^{13} + 19 a^{12} + 18 a^{11} + 9 a^{10} + 9 a^{9} + 3 a^{8} + 22 a^{7} + 20 a^{6} + 4 a^{5} + 15 a^{4} + 13 a^{3} + 13 a^{2} + 10 a + 20\right)\cdot 23^{104} + \left(2 a^{14} + 4 a^{13} + 10 a^{12} + 14 a^{11} + 12 a^{10} + 18 a^{9} + 4 a^{8} + 17 a^{7} + 12 a^{6} + 12 a^{5} + 3 a^{4} + 22 a^{3} + 13 a^{2} + 5 a + 21\right)\cdot 23^{105} + \left(20 a^{14} + 15 a^{13} + 2 a^{12} + 19 a^{11} + 5 a^{10} + 4 a^{9} + 15 a^{8} + 22 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11 a^{2} + 10 a + 18\right)\cdot 23^{129} + \left(17 a^{14} + 15 a^{13} + 12 a^{12} + 14 a^{11} + 6 a^{10} + 18 a^{9} + 17 a^{8} + 19 a^{7} + 19 a^{6} + 7 a^{5} + 11 a^{4} + 4 a^{3} + 21 a^{2} + 2 a + 14\right)\cdot 23^{130} + \left(13 a^{14} + 7 a^{13} + 6 a^{12} + 10 a^{11} + 6 a^{10} + 19 a^{9} + 6 a^{8} + 11 a^{7} + 8 a^{6} + 7 a^{5} + 21 a^{4} + 7 a^{3} + 6 a^{2} + 20 a + 9\right)\cdot 23^{131} + \left(12 a^{14} + 15 a^{13} + 15 a^{11} + 14 a^{10} + 16 a^{9} + 14 a^{8} + 17 a^{7} + 5 a^{6} + 4 a^{5} + 5 a^{4} + 9 a^{2} + 20 a + 7\right)\cdot 23^{132} + \left(18 a^{14} + 2 a^{13} + 8 a^{12} + a^{11} + 10 a^{10} + 5 a^{9} + 10 a^{8} + 15 a^{7} + 6 a^{6} + 13 a^{5} + 10 a^{4} + 13 a^{3} + 13 a^{2} + 20 a\right)\cdot 23^{133} + \left(15 a^{14} + a^{13} + 5 a^{12} + 7 a^{11} + a^{10} + 12 a^{9} + 21 a^{8} + 18 a^{7} + 6 a^{6} + 10 a^{5} + 2 a^{4} + 13 a^{3} + 10 a^{2} + 21 a + 12\right)\cdot 23^{134} + \left(13 a^{14} + 12 a^{13} + 10 a^{12} + 9 a^{11} + 6 a^{10} + 2 a^{9} + 15 a^{8} + 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\left(9 a^{14} + 6 a^{13} + 11 a^{12} + 16 a^{11} + 20 a^{10} + 2 a^{9} + 8 a^{8} + 10 a^{7} + 3 a^{6} + 8 a^{5} + 13 a^{4} + a^{3} + 11 a^{2} + 20 a + 20\right)\cdot 23^{141} + \left(12 a^{14} + 4 a^{13} + 18 a^{12} + 14 a^{11} + 17 a^{10} + 7 a^{9} + 5 a^{8} + 15 a^{7} + 14 a^{6} + 11 a^{5} + 22 a^{4} + 8 a^{3} + 22 a^{2} + 18 a + 12\right)\cdot 23^{142} + \left(9 a^{14} + 19 a^{13} + 21 a^{12} + 18 a^{11} + 2 a^{10} + 6 a^{9} + 6 a^{8} + 10 a^{7} + 5 a^{6} + 3 a^{5} + 8 a^{4} + 9 a^{2} + 5 a + 20\right)\cdot 23^{143} + \left(10 a^{14} + 5 a^{13} + 13 a^{12} + 21 a^{11} + 20 a^{10} + 20 a^{9} + 5 a^{8} + 17 a^{7} + 22 a^{6} + 22 a^{5} + 16 a^{4} + 20 a^{3} + 11 a^{2} + 4 a + 3\right)\cdot 23^{144} + \left(3 a^{14} + 4 a^{13} + 8 a^{11} + 14 a^{10} + 18 a^{9} + 7 a^{8} + 17 a^{7} + 9 a^{6} + 20 a^{5} + 11 a^{4} + 2 a^{3} + 22 a^{2} + a + 11\right)\cdot 23^{145} + \left(3 a^{14} + 13 a^{13} + 13 a^{12} + 10 a^{11} + 6 a^{10} + 3 a^{9} + 22 a^{8} + 7 a^{6} + 11 a^{5} + 8 a^{4} + 15 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a^{3} + 15 a^{2} + 13 a + 16\right)\cdot 23^{152} + \left(22 a^{14} + 10 a^{13} + 3 a^{11} + 11 a^{10} + 12 a^{9} + 17 a^{8} + 3 a^{7} + 9 a^{6} + 11 a^{5} + 6 a^{4} + a^{3} + a^{2} + 8 a\right)\cdot 23^{153} + \left(18 a^{14} + 17 a^{13} + 7 a^{12} + 7 a^{11} + 15 a^{10} + 3 a^{9} + 6 a^{8} + 14 a^{7} + 15 a^{6} + 15 a^{5} + 6 a^{4} + 10 a^{3} + 12 a^{2} + a + 12\right)\cdot 23^{154} + \left(22 a^{13} + 16 a^{12} + 13 a^{11} + 15 a^{9} + 21 a^{8} + 7 a^{7} + 11 a^{6} + 4 a^{5} + 13 a^{3} + 9 a^{2} + 12 a + 9\right)\cdot 23^{155} + \left(14 a^{14} + 3 a^{13} + 2 a^{12} + 6 a^{11} + 3 a^{10} + 5 a^{9} + 20 a^{8} + 13 a^{7} + 5 a^{6} + 3 a^{5} + 16 a^{4} + 22 a^{3} + 20 a^{2} + 5 a + 8\right)\cdot 23^{156} + \left(7 a^{14} + 9 a^{13} + 13 a^{12} + 9 a^{11} + 13 a^{10} + 11 a^{9} + a^{8} + 18 a^{7} + 22 a^{6} + 9 a^{5} + 10 a^{4} + 16 a^{3} + 12 a^{2} + 10 a + 20\right)\cdot 23^{157} + \left(13 a^{14} + 7 a^{13} + 8 a^{12} + 12 a^{11} + 20 a^{10} + 4 a^{9} + 17 a^{8} + 8 a^{7} + 7 a^{6} + 21 a^{5} + 21 a^{4} + 12 a^{3} + 9 a^{2} + 11 a + 9\right)\cdot 23^{158} + \left(a^{14} + 18 a^{13} + 18 a^{12} + a^{11} + 10 a^{10} + 20 a^{9} + 15 a^{8} + 21 a^{7} + 14 a^{6} + 7 a^{5} + 18 a^{4} + 3 a^{3} + 20 a^{2} + 14 a + 20\right)\cdot 23^{159} + \left(5 a^{14} + 19 a^{13} + 6 a^{12} + 9 a^{11} + 8 a^{10} + 22 a^{9} + 5 a^{8} + 14 a^{7} + 2 a^{6} + 3 a^{5} + 4 a^{4} + 22 a^{3} + 20 a^{2} + 3 a + 2\right)\cdot 23^{160} + \left(a^{14} + 8 a^{13} + 22 a^{12} + 2 a^{11} + 2 a^{10} + 20 a^{9} + 15 a^{8} + 20 a^{7} + a^{6} + 10 a^{5} + 16 a^{4} + 18 a^{3} + 12 a^{2} + 18 a + 2\right)\cdot 23^{161} + \left(5 a^{14} + 15 a^{13} + 19 a^{12} + 9 a^{11} + 11 a^{10} + 15 a^{9} + 22 a^{8} + 10 a^{7} + 4 a^{6} + 2 a^{5} + 15 a^{4} + 6 a^{3} + 17 a^{2} + 21 a + 8\right)\cdot 23^{162} + \left(a^{14} + 9 a^{13} + 8 a^{12} + 22 a^{11} + 5 a^{10} + 2 a^{9} + 7 a^{7} + 14 a^{5} + 22 a^{4} + 16 a^{3} + 8 a^{2} + 9 a + 17\right)\cdot 23^{163} + \left(11 a^{14} + 8 a^{13} + 6 a^{12} + 7 a^{11} + 13 a^{9} + 18 a^{8} + 8 a^{7} + 2 a^{6} + 17 a^{5} + 15 a^{4} + 8 a^{3} + 21 a^{2} + 5 a + 1\right)\cdot 23^{164} + \left(10 a^{14} + 12 a^{13} + 11 a^{12} + 7 a^{11} + 7 a^{10} + 19 a^{9} + 14 a^{8} + 7 a^{7} + 15 a^{6} + 21 a^{5} + 18 a^{4} + 20 a^{3} + 2 a^{2} + 21 a + 7\right)\cdot 23^{165} + \left(7 a^{14} + 6 a^{13} + 9 a^{11} + 4 a^{10} + 4 a^{9} + 20 a^{8} + 12 a^{7} + 14 a^{6} + 7 a^{5} + 20 a^{4} + 3 a^{3} + 4 a^{2} + 13 a + 1\right)\cdot 23^{166} + \left(4 a^{13} + 10 a^{12} + 6 a^{11} + 17 a^{10} + 13 a^{9} + 16 a^{8} + 18 a^{7} + 15 a^{6} + 21 a^{5} + 3 a^{4} + 21 a^{3} + 16 a^{2} + 11\right)\cdot 23^{167} + \left(6 a^{14} + 12 a^{13} + 3 a^{12} + 2 a^{11} + 12 a^{10} + 15 a^{9} + 9 a^{8} + 21 a^{7} + 22 a^{6} + 8 a^{5} + 12 a^{4} + 21 a^{3} + 15 a^{2} + 14 a + 14\right)\cdot 23^{168} + \left(9 a^{14} + 20 a^{13} + 15 a^{12} + 17 a^{11} + 10 a^{10} + 11 a^{9} + 21 a^{8} + 7 a^{7} + 20 a^{6} + 2 a^{5} + 4 a^{4} + 2 a^{3} + 21 a^{2} + 22 a + 22\right)\cdot 23^{169} + \left(22 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12 a + 7\right)\cdot 23^{175} + \left(14 a^{14} + 11 a^{13} + 3 a^{12} + 11 a^{11} + a^{9} + 15 a^{8} + 13 a^{7} + 7 a^{6} + 8 a^{5} + 19 a^{4} + 15 a^{3} + 15 a^{2} + 12 a + 4\right)\cdot 23^{176} + \left(7 a^{14} + 17 a^{13} + 13 a^{12} + 8 a^{11} + 8 a^{10} + 15 a^{9} + 3 a^{8} + 4 a^{7} + 8 a^{6} + 19 a^{5} + 14 a^{4} + 2 a^{3} + 20 a^{2} + a + 22\right)\cdot 23^{177} + \left(a^{14} + 5 a^{13} + 6 a^{12} + 17 a^{11} + 10 a^{10} + 11 a^{9} + 12 a^{8} + 2 a^{7} + 16 a^{6} + 12 a^{5} + 14 a^{4} + 17 a^{3} + 12 a^{2} + 11 a + 16\right)\cdot 23^{178} + \left(11 a^{14} + 12 a^{13} + 7 a^{12} + 4 a^{11} + 21 a^{10} + 9 a^{9} + 19 a^{8} + 16 a^{7} + 7 a^{6} + 21 a^{5} + 20 a^{4} + 3 a^{3} + 5 a^{2} + 21 a + 20\right)\cdot 23^{179} + \left(20 a^{14} + 5 a^{13} + 11 a^{12} + 15 a^{11} + 18 a^{10} + 15 a^{9} + 12 a^{8} + 15 a^{7} + 19 a^{6} + 8 a^{5} + 3 a^{4} + 10 a^{3} + 20 a^{2} + 21 a + 17\right)\cdot 23^{180} + \left(5 a^{14} + 21 a^{13} + a^{12} + 5 a^{11} + 4 a^{10} + 12 a^{9} + 20 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+ 10 a^{11} + 17 a^{10} + 7 a^{9} + a^{8} + 6 a^{7} + 16 a^{6} + 22 a^{5} + a^{4} + 3 a^{3} + 7 a^{2} + 9 a\right)\cdot 23^{187} + \left(12 a^{14} + 20 a^{13} + 2 a^{12} + 6 a^{11} + 3 a^{10} + 6 a^{9} + 15 a^{8} + 10 a^{7} + 6 a^{6} + 7 a^{4} + 13 a^{3} + 3 a^{2} + 12 a + 13\right)\cdot 23^{188} + \left(12 a^{14} + 15 a^{13} + 13 a^{12} + 12 a^{11} + 11 a^{10} + 20 a^{9} + 11 a^{8} + 5 a^{7} + 16 a^{6} + 5 a^{5} + 13 a^{3} + 22 a^{2} + a + 10\right)\cdot 23^{189} + \left(8 a^{14} + 15 a^{13} + 13 a^{11} + 11 a^{10} + 9 a^{9} + 14 a^{8} + 14 a^{6} + 2 a^{5} + 9 a^{4} + 17 a^{3} + 18 a^{2} + 6 a + 8\right)\cdot 23^{190} + \left(10 a^{14} + 8 a^{13} + 13 a^{12} + 2 a^{11} + 9 a^{10} + 10 a^{9} + a^{8} + 17 a^{7} + 10 a^{6} + 19 a^{5} + 12 a^{4} + 15 a^{3} + 18 a^{2} + 10 a + 7\right)\cdot 23^{191} + \left(8 a^{14} + 11 a^{13} + 19 a^{12} + 8 a^{11} + a^{10} + 15 a^{9} + 11 a^{8} + 13 a^{7} + 7 a^{6} + 6 a^{5} + 22 a^{4} + 8 a^{3} + 22 a^{2} + 10 a + 21\right)\cdot 23^{192} + \left(21 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23^{198} + \left(11 a^{14} + 11 a^{13} + 2 a^{12} + 18 a^{11} + 7 a^{9} + 10 a^{8} + 11 a^{7} + 14 a^{6} + 6 a^{5} + 21 a^{4} + 13 a^{3} + 9 a^{2} + 17 a + 4\right)\cdot 23^{199} + \left(14 a^{14} + 12 a^{13} + 17 a^{12} + 13 a^{11} + a^{10} + a^{9} + 18 a^{8} + 13 a^{7} + 5 a^{6} + 3 a^{5} + 9 a^{4} + 10 a^{3} + 21 a^{2} + 3 a + 12\right)\cdot 23^{200} + \left(13 a^{14} + 11 a^{13} + 4 a^{12} + 8 a^{11} + 5 a^{10} + a^{9} + 21 a^{8} + 15 a^{7} + 9 a^{6} + 19 a^{5} + 13 a^{4} + 3 a^{3} + 6 a^{2} + 8 a + 15\right)\cdot 23^{201} + \left(14 a^{14} + 15 a^{13} + 14 a^{12} + 5 a^{11} + 18 a^{10} + 21 a^{9} + 11 a^{8} + 3 a^{7} + 19 a^{6} + 15 a^{5} + 12 a^{4} + 22 a^{3} + 14 a^{2} + 12 a + 12\right)\cdot 23^{202} + \left(15 a^{14} + 8 a^{13} + 20 a^{12} + 9 a^{11} + 14 a^{10} + 12 a^{9} + a^{8} + 7 a^{7} + 4 a^{6} + 22 a^{5} + 7 a^{4} + 8 a^{3} + 6 a^{2} + 19 a + 15\right)\cdot 23^{203} + \left(3 a^{14} + 5 a^{13} + 18 a^{12} + 15 a^{11} + 20 a^{10} + 2 a^{9} + 11 a^{8} + 17 a^{7} + 19 a^{6} + 9 a^{5} + 6 a^{4} + 2 a^{3} + 6 a^{2} + 3 a + 12\right)\cdot 23^{204} + \left(21 a^{14} + 2 a^{13} + 9 a^{12} + 6 a^{11} + 18 a^{10} + 10 a^{9} + 5 a^{8} + 7 a^{7} + 4 a^{6} + 4 a^{5} + 21 a^{4} + 5 a^{2} + 10 a + 14\right)\cdot 23^{205} + \left(a^{14} + 17 a^{13} + 15 a^{12} + 7 a^{11} + 12 a^{10} + 12 a^{9} + 14 a^{8} + 13 a^{6} + 18 a^{5} + 5 a^{4} + 2 a^{3} + 8 a^{2} + 6 a + 1\right)\cdot 23^{206} + \left(19 a^{14} + 5 a^{13} + 22 a^{12} + 4 a^{11} + 14 a^{9} + 13 a^{8} + 4 a^{7} + 2 a^{6} + 22 a^{5} + 19 a^{4} + 22 a^{3} + 15 a^{2} + 3 a + 17\right)\cdot 23^{207} + \left(6 a^{14} + 14 a^{13} + 19 a^{12} + 11 a^{11} + 3 a^{10} + 2 a^{8} + 4 a^{7} + 20 a^{6} + 22 a^{5} + 20 a^{4} + 18 a^{3} + 16 a^{2} + 11 a + 17\right)\cdot 23^{208} + \left(8 a^{14} + 6 a^{13} + 11 a^{12} + 10 a^{11} + 13 a^{10} + 21 a^{9} + 10 a^{8} + 2 a^{7} + 22 a^{6} + 20 a^{5} + 17 a^{4} + 21 a^{3} + 13 a^{2} + 16 a + 4\right)\cdot 23^{209} + \left(4 a^{14} + 3 a^{13} + 19 a^{12} + 2 a^{11} + 12 a^{10} + 2 a^{9} + 22 a^{8} + 2 a^{7} + 13 a^{6} + 21 a^{5} + 9 a^{4} + 22 a^{3} + 20 a^{2} + 3 a + 6\right)\cdot 23^{210} + \left(12 a^{14} + 10 a^{13} + 14 a^{11} + 5 a^{10} + 16 a^{9} + 8 a^{8} + 12 a^{7} + 3 a^{6} + 14 a^{5} + 18 a^{4} + 3 a^{3} + 19 a^{2} + 7 a + 22\right)\cdot 23^{211} + \left(19 a^{14} + 20 a^{13} + a^{12} + 10 a^{11} + 2 a^{10} + 20 a^{9} + 2 a^{8} + 14 a^{7} + 13 a^{6} + 20 a^{5} + 20 a^{4} + 16 a^{3} + 5 a^{2} + 21 a + 3\right)\cdot 23^{212} + \left(12 a^{14} + 17 a^{13} + 4 a^{12} + 8 a^{11} + 22 a^{10} + a^{9} + 9 a^{8} + 19 a^{7} + 7 a^{5} + 5 a^{4} + 11 a^{3} + 9 a^{2} + a + 21\right)\cdot 23^{213} + \left(22 a^{14} + 15 a^{13} + 6 a^{12} + 7 a^{11} + 3 a^{10} + 19 a^{9} + 15 a^{8} + 3 a^{7} + 22 a^{6} + 20 a^{5} + 21 a^{4} + 12 a^{3} + 12 a^{2} + 9 a + 1\right)\cdot 23^{214} + \left(4 a^{14} + 17 a^{13} + 21 a^{12} + 12 a^{11} + 13 a^{10} + 22 a^{9} + 8 a^{8} + 4 a^{7} + 17 a^{6} + 11 a^{5} + 13 a^{4} + 21 a^{3} + 8 a^{2} + 21 a + 3\right)\cdot 23^{215} + \left(14 a^{14} + 12 a^{13} + 11 a^{12} + 14 a^{11} + a^{10} + 16 a^{9} + 2 a^{8} + 10 a^{7} + 8 a^{6} + 6 a^{5} + a^{4} + 6 a^{3} + 22 a^{2} + 18 a + 20\right)\cdot 23^{216} + \left(20 a^{14} + 10 a^{13} + 13 a^{12} + 6 a^{11} + 14 a^{10} + 2 a^{8} + 12 a^{7} + 16 a^{6} + 4 a^{5} + 12 a^{3} + 5 a^{2} + 11 a + 6\right)\cdot 23^{217} + \left(5 a^{14} + a^{13} + 12 a^{12} + 18 a^{11} + a^{10} + 4 a^{9} + 18 a^{8} + 9 a^{7} + 17 a^{6} + 21 a^{5} + 2 a^{4} + 5 a^{3} + 2 a^{2} + 3 a + 14\right)\cdot 23^{218} + \left(17 a^{14} + 16 a^{13} + 18 a^{12} + 5 a^{11} + 16 a^{10} + 18 a^{9} + 4 a^{8} + 8 a^{7} + 8 a^{6} + 3 a^{5} + 22 a^{4} + 8 a^{3} + 16 a^{2} + 19 a + 4\right)\cdot 23^{219} + \left(18 a^{14} + 2 a^{13} + a^{11} + 8 a^{9} + 16 a^{8} + 12 a^{7} + 14 a^{6} + 7 a^{5} + 9 a^{4} + 2 a^{3} + 21 a^{2} + 12 a + 19\right)\cdot 23^{220} + \left(15 a^{14} + 2 a^{13} + 4 a^{12} + 11 a^{11} + 14 a^{10} + 3 a^{9} + 3 a^{8} + a^{7} + a^{6} + 3 a^{5} + 18 a^{4} + 8 a^{3} + 3 a^{2} + 4 a\right)\cdot 23^{221} + \left(7 a^{14} + a^{13} + 18 a^{12} + 14 a^{11} + 7 a^{10} + 22 a^{9} + 19 a^{8} + 3 a^{7} + 16 a^{6} + 6 a^{5} + 22 a^{4} + 11 a^{3} + 7 a^{2} + 10 a + 20\right)\cdot 23^{222} + \left(15 a^{14} + 19 a^{13} + 8 a^{12} + 5 a^{11} + 6 a^{8} + 3 a^{7} + 9 a^{6} + 16 a^{5} + 22 a^{3} + 16 a^{2} + 12 a + 17\right)\cdot 23^{223} + \left(13 a^{14} + 9 a^{13} + 7 a^{12} + 2 a^{11} + 16 a^{10} + a^{9} + 8 a^{8} + 4 a^{7} + 6 a^{6} + 13 a^{5} + 5 a^{4} + 17 a^{3} + 5 a^{2} + 2 a + 9\right)\cdot 23^{224} + \left(16 a^{14} + 20 a^{13} + 14 a^{12} + 3 a^{11} + a^{10} + 10 a^{9} + 17 a^{8} + 10 a^{7} + 12 a^{6} + 3 a^{5} + 12 a^{4} + 22 a^{3} + 14 a^{2} + 11 a + 18\right)\cdot 23^{225} + \left(17 a^{14} + 2 a^{13} + 15 a^{12} + 20 a^{11} + 10 a^{10} + 15 a^{9} + 2 a^{8} + 4 a^{7} + a^{6} + 15 a^{5} + 13 a^{4} + 21 a^{3} + 16 a^{2} + 18 a + 18\right)\cdot 23^{226} + \left(18 a^{14} + 14 a^{13} + 17 a^{12} + 5 a^{10} + 15 a^{9} + 14 a^{8} + 18 a^{7} + 12 a^{6} + 11 a^{5} + 22 a^{4} + 12 a^{3} + 13 a^{2} + 20 a + 11\right)\cdot 23^{227} + \left(22 a^{14} + 22 a^{13} + 9 a^{12} + 3 a^{11} + 10 a^{10} + 19 a^{9} + 12 a^{8} + 16 a^{7} + 9 a^{6} + 21 a^{5} + 2 a^{4} + 3 a^{3} + 2 a + 8\right)\cdot 23^{228} + \left(20 a^{14} + 6 a^{13} + 7 a^{12} + 17 a^{11} + 11 a^{10} + 14 a^{9} + 7 a^{8} + 13 a^{7} + 5 a^{6} + 9 a^{5} + 19 a^{4} + 17 a^{3} + 5 a^{2} + 4 a + 19\right)\cdot 23^{229} + \left(14 a^{14} + 5 a^{13} + 14 a^{12} + 6 a^{11} + 6 a^{10} + 10 a^{9} + 9 a^{8} + 12 a^{7} + 17 a^{6} + 20 a^{5} + 10 a^{4} + 4 a^{3} + 10 a + 11\right)\cdot 23^{230} + \left(13 a^{13} + 16 a^{12} + 10 a^{11} + 9 a^{10} + 2 a^{9} + 9 a^{8} + 9 a^{7} + 15 a^{6} + 15 a^{5} + 17 a^{4} + a^{3} + 6 a^{2} + 8 a + 20\right)\cdot 23^{231} + \left(16 a^{14} + 19 a^{13} + 11 a^{12} + 13 a^{11} + 5 a^{10} + 20 a^{9} + 5 a^{8} + 19 a^{7} + 10 a^{6} + 6 a^{5} + 19 a^{4} + 22 a^{3} + 22 a^{2} + a + 12\right)\cdot 23^{232} + \left(2 a^{14} + 4 a^{13} + 4 a^{11} + 17 a^{10} + 21 a^{9} + 15 a^{8} + 16 a^{7} + 16 a^{6} + 9 a^{5} + 3 a^{4} + 20 a^{3} + 17 a^{2} + 22 a + 21\right)\cdot 23^{233} + \left(3 a^{14} + 8 a^{13} + a^{12} + 9 a^{11} + a^{10} + 4 a^{9} + 7 a^{8} + 2 a^{7} + a^{6} + 15 a^{5} + 18 a^{4} + 6 a^{3} + 4 a^{2} + a + 14\right)\cdot 23^{234} + \left(20 a^{14} + 5 a^{13} + 15 a^{12} + 15 a^{11} + 22 a^{10} + 3 a^{9} + 15 a^{8} + 5 a^{7} + 20 a^{6} + 18 a^{5} + 19 a^{4} + 13 a^{3} + 22 a^{2} + 20 a + 18\right)\cdot 23^{235} + \left(13 a^{14} + 16 a^{13} + 7 a^{12} + 19 a^{11} + 4 a^{10} + 11 a^{9} + 14 a^{8} + 16 a^{7} + a^{6} + 11 a^{5} + 5 a^{4} + 6 a^{3} + 13 a^{2} + 20 a + 15\right)\cdot 23^{236} + \left(a^{14} + 9 a^{13} + 8 a^{12} + 21 a^{11} + 6 a^{10} + 19 a^{9} + 21 a^{8} + 14 a^{7} + 5 a^{6} + 15 a^{4} + 15 a^{3} + 20 a^{2} + 8 a + 12\right)\cdot 23^{237} + \left(16 a^{14} + 7 a^{13} + 16 a^{12} + 18 a^{11} + 20 a^{10} + 4 a^{9} + 3 a^{8} + 11 a^{7} + 22 a^{6} + 11 a^{5} + 8 a^{4} + 13 a^{3} + 3 a^{2} + 10\right)\cdot 23^{238} + \left(15 a^{14} + 10 a^{13} + 17 a^{12} + 11 a^{11} + 2 a^{10} + a^{9} + 12 a^{8} + 16 a^{7} + 15 a^{6} + 9 a^{5} + 14 a^{4} + 10 a^{3} + 18 a^{2} + 3 a + 11\right)\cdot 23^{239} + \left(12 a^{14} + 12 a^{13} + 17 a^{12} + 3 a^{11} + 3 a^{10} + 8 a^{9} + 15 a^{8} + 14 a^{7} + 13 a^{6} + 20 a^{5} + 11 a^{4} + 4 a^{3} + 4 a^{2} + 19 a + 12\right)\cdot 23^{240} + \left(a^{14} + 4 a^{13} + 6 a^{12} + 12 a^{11} + 20 a^{10} + 8 a^{9} + 6 a^{8} + 3 a^{7} + 16 a^{6} + a^{5} + 9 a^{4} + 11 a^{3} + 10 a^{2} + 14 a + 8\right)\cdot 23^{241} + \left(13 a^{14} + 7 a^{12} + 11 a^{11} + 7 a^{10} + 13 a^{9} + 13 a^{8} + 9 a^{7} + 10 a^{6} + 11 a^{5} + 11 a^{4} + 5 a^{3} + 11 a^{2} + 5 a + 15\right)\cdot 23^{242} + \left(9 a^{14} + 12 a^{13} + 13 a^{12} + 8 a^{11} + 12 a^{10} + 3 a^{8} + 4 a^{7} + 18 a^{6} + 3 a^{5} + 11 a^{4} + 16 a^{3} + 18 a^{2} + 10\right)\cdot 23^{243} + \left(11 a^{13} + 12 a^{12} + 5 a^{11} + 15 a^{10} + 20 a^{9} + 11 a^{8} + 10 a^{7} + 17 a^{6} + 12 a^{5} + 10 a^{4} + a^{3} + 9 a^{2} + 18\right)\cdot 23^{244} + \left(2 a^{14} + 11 a^{12} + 8 a^{11} + 19 a^{10} + a^{9} + 12 a^{8} + 4 a^{7} + 5 a^{6} + 9 a^{5} + 17 a^{4} + 22 a^{3} + 8 a^{2} + 2 a + 3\right)\cdot 23^{245} + \left(21 a^{14} + 14 a^{13} + 13 a^{12} + 4 a^{11} + 10 a^{10} + 20 a^{9} + 21 a^{8} + 15 a^{7} + 14 a^{6} + 14 a^{4} + 14 a^{3} + 13 a^{2} + a\right)\cdot 23^{246} + \left(10 a^{14} + 22 a^{13} + 13 a^{12} + 12 a^{11} + 16 a^{10} + 11 a^{9} + 20 a^{8} + 13 a^{7} + 11 a^{6} + 14 a^{5} + 9 a^{4} + 12 a^{3} + 18 a^{2} + 22 a + 3\right)\cdot 23^{247} + \left(11 a^{14} + 3 a^{13} + 3 a^{12} + 21 a^{11} + 19 a^{10} + 9 a^{9} + 12 a^{8} + 17 a^{7} + 9 a^{6} + 11 a^{5} + 12 a^{4} + 3 a^{3} + 22 a + 21\right)\cdot 23^{248} + \left(3 a^{14} + 15 a^{13} + 5 a^{12} + 13 a^{11} + 13 a^{10} + 22 a^{9} + 12 a^{8} + 8 a^{7} + 9 a^{6} + 18 a^{5} + 17 a^{4} + 4 a^{3} + 18 a^{2} + 2 a + 16\right)\cdot 23^{249} + \left(19 a^{14} + 14 a^{13} + 20 a^{12} + 4 a^{11} + 19 a^{10} + 9 a^{9} + 8 a^{8} + 2 a^{7} + 3 a^{6} + 10 a^{5} + 10 a^{4} + 22 a^{2} + 3 a + 18\right)\cdot 23^{250} + \left(a^{14} + 12 a^{13} + 17 a^{12} + 14 a^{11} + 8 a^{10} + 21 a^{9} + 11 a^{8} + 20 a^{7} + 2 a^{6} + 20 a^{5} + 20 a^{4} + 4 a^{3} + 13 a^{2} + 4 a + 14\right)\cdot 23^{251} + \left(21 a^{14} + 10 a^{13} + 9 a^{12} + 20 a^{11} + 14 a^{10} + 9 a^{9} + 2 a^{8} + 15 a^{7} + 8 a^{6} + 18 a^{5} + 22 a^{4} + a^{3} + 5 a^{2} + 15 a + 2\right)\cdot 23^{252} + \left(9 a^{14} + 7 a^{13} + 3 a^{10} + 7 a^{9} + 8 a^{8} + 13 a^{7} + 7 a^{6} + 20 a^{5} + 13 a^{4} + 18 a^{3} + 10 a + 14\right)\cdot 23^{253} + \left(3 a^{14} + 10 a^{13} + 21 a^{12} + 8 a^{11} + 7 a^{10} + 5 a^{9} + 16 a^{8} + 10 a^{7} + 10 a^{6} + 3 a^{5} + 19 a^{4} + 2 a^{3} + 15 a^{2} + 12 a + 5\right)\cdot 23^{254} + \left(16 a^{14} + 21 a^{13} + 4 a^{12} + 21 a^{11} + 21 a^{10} + 8 a^{9} + 20 a^{8} + 2 a^{7} + 7 a^{6} + 5 a^{5} + 10 a^{3} + 10 a^{2} + 13 a + 20\right)\cdot 23^{255} + \left(19 a^{14} + 12 a^{13} + 3 a^{12} + 2 a^{11} + 13 a^{10} + 19 a^{9} + 8 a^{8} + 13 a^{7} + 14 a^{6} + a^{5} + 5 a^{4} + 12 a^{3} + 17 a^{2} + 22 a + 9\right)\cdot 23^{256} + \left(11 a^{14} + 6 a^{13} + 9 a^{12} + a^{11} + 14 a^{10} + 18 a^{9} + 2 a^{8} + 13 a^{7} + 5 a^{6} + 18 a^{5} + 12 a^{4} + 11 a^{3} + 3 a^{2} + 2 a + 16\right)\cdot 23^{257} + \left(3 a^{14} + 22 a^{13} + 15 a^{12} + 9 a^{11} + 7 a^{10} + 15 a^{9} + 2 a^{8} + 17 a^{7} + 2 a^{6} + 17 a^{5} + 13 a^{4} + a^{3} + 11 a^{2} + 12 a + 10\right)\cdot 23^{258} + \left(4 a^{14} + 6 a^{13} + 14 a^{12} + 15 a^{11} + 16 a^{10} + 16 a^{9} + 22 a^{8} + 21 a^{7} + 13 a^{6} + 21 a^{5} + 18 a^{4} + 16 a^{3} + 6 a^{2} + 3 a + 2\right)\cdot 23^{259} + \left(17 a^{13} + 5 a^{12} + 18 a^{11} + 10 a^{10} + 12 a^{9} + 13 a^{8} + 4 a^{7} + 2 a^{6} + 5 a^{5} + 14 a^{4} + a^{3} + 3 a^{2} + 10 a + 19\right)\cdot 23^{260} + \left(20 a^{14} + 10 a^{13} + 15 a^{12} + 21 a^{11} + 19 a^{10} + 2 a^{9} + 6 a^{8} + 12 a^{7} + 14 a^{6} + 2 a^{5} + 12 a^{4} + 11 a^{3} + 2 a^{2} + 17 a + 8\right)\cdot 23^{261} + \left(7 a^{14} + 7 a^{13} + 4 a^{12} + 4 a^{11} + 22 a^{10} + 7 a^{9} + 9 a^{8} + 16 a^{7} + a^{6} + 13 a^{5} + 13 a^{4} + 15 a^{3} + 13 a^{2} + 2 a + 2\right)\cdot 23^{262} + \left(14 a^{14} + 10 a^{13} + a^{12} + a^{11} + 19 a^{10} + 11 a^{9} + 3 a^{8} + 16 a^{7} + 22 a^{6} + 2 a^{5} + 5 a^{4} + 19 a^{3} + 4 a^{2} + 10 a + 21\right)\cdot 23^{263} + \left(6 a^{14} + 20 a^{13} + 9 a^{12} + 10 a^{11} + 12 a^{10} + 18 a^{9} + 13 a^{8} + 11 a^{7} + 14 a^{6} + 15 a^{5} + a^{4} + 10 a^{3} + 4 a^{2} + a + 20\right)\cdot 23^{264} + \left(18 a^{14} + 21 a^{13} + 4 a^{12} + 6 a^{11} + 2 a^{10} + 4 a^{8} + a^{7} + 12 a^{6} + 20 a^{5} + 20 a^{4} + 13 a^{3} + 14 a^{2} + 3 a + 6\right)\cdot 23^{265} + \left(8 a^{14} + 17 a^{13} + 3 a^{12} + 16 a^{11} + 21 a^{9} + 3 a^{8} + 11 a^{7} + 14 a^{6} + 9 a^{5} + 9 a^{4} + 14 a^{3} + 2 a^{2} + a + 7\right)\cdot 23^{266} + \left(19 a^{14} + a^{13} + 16 a^{12} + 17 a^{11} + 17 a^{9} + 20 a^{8} + a^{7} + 7 a^{6} + 7 a^{5} + 10 a^{4} + 9 a^{3} + 19 a^{2} + 3 a + 11\right)\cdot 23^{267} + \left(13 a^{14} + 14 a^{13} + 9 a^{12} + 3 a^{11} + 2 a^{10} + 15 a^{9} + 22 a^{8} + 18 a^{7} + 6 a^{6} + 11 a^{5} + 16 a^{4} + 14 a^{3} + 13 a^{2} + 5 a + 16\right)\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 13 }$ $=$ $3 a^{14} + 3 a^{13} + 9 a^{12} + 9 a^{11} + a^{10} + a^{9} + 5 a^{8} + 8 a^{7} + 21 a^{6} + 8 a^{5} + 22 a^{4} + a^{3} + 7 a^{2} + 8 a + 4 + \left(22 a^{14} + 3 a^{13} + 4 a^{12} + 15 a^{11} + 17 a^{10} + 18 a^{9} + 4 a^{8} + 8 a^{7} + 2 a^{6} + 5 a^{5} + 16 a^{4} + 22 a^{3} + 10 a^{2} + 19\right)\cdot 23 + \left(22 a^{14} + 8 a^{13} + a^{12} + 17 a^{11} + 11 a^{10} + 19 a^{9} + 12 a^{8} + 5 a^{7} + a^{6} + 2 a^{5} + 15 a^{4} + 16 a^{3} + 18 a^{2} + 2 a + 4\right)\cdot 23^{2} + \left(18 a^{14} + 10 a^{13} + 18 a^{12} + 16 a^{11} + 14 a^{10} + 11 a^{9} + 14 a^{7} + 12 a^{6} + 2 a^{5} + 13 a^{4} + 5 a^{3} + 12 a^{2} + a + 19\right)\cdot 23^{3} + \left(19 a^{14} + 13 a^{13} + 20 a^{12} + 17 a^{11} + 20 a^{9} + 12 a^{8} + 5 a^{7} + 4 a^{6} + 7 a^{5} + 12 a^{4} + 10 a^{3} + 5 a^{2} + 13 a + 11\right)\cdot 23^{4} + \left(22 a^{14} + 21 a^{13} + 7 a^{12} + 17 a^{11} + 18 a^{10} + 13 a^{9} + 11 a^{8} + 8 a^{7} + 8 a^{6} + 7 a^{5} + 10 a^{4} + 16 a^{3} + 13 a^{2} + 11 a + 18\right)\cdot 23^{5} + \left(6 a^{14} + 22 a^{13} + 9 a^{12} + 19 a^{11} + 17 a^{10} + 10 a^{9} + 16 a^{8} + 10 a^{7} + 5 a^{6} + 19 a^{5} + 4 a^{4} + 6 a^{3} + 14 a^{2} + 20 a + 16\right)\cdot 23^{6} + \left(22 a^{13} + 10 a^{12} + 12 a^{11} + 11 a^{10} + a^{9} + 8 a^{8} + a^{7} + 6 a^{6} + 7 a^{5} + a^{4} + 3 a^{3} + 5 a^{2} + 9 a + 8\right)\cdot 23^{7} + \left(5 a^{14} + 6 a^{13} + 19 a^{12} + 13 a^{11} + 3 a^{10} + 8 a^{9} + 5 a^{8} + 11 a^{7} + 10 a^{6} + 4 a^{5} + 10 a^{4} + 12 a^{3} + 4 a + 6\right)\cdot 23^{8} + \left(5 a^{14} + 7 a^{13} + 10 a^{12} + 19 a^{11} + 3 a^{10} + 18 a^{9} + 4 a^{8} + 4 a^{7} + 10 a^{6} + 11 a^{5} + 17 a^{4} + 7 a^{3} + 3 a^{2} + 5 a + 10\right)\cdot 23^{9} + \left(5 a^{14} + 8 a^{13} + 19 a^{12} + 8 a^{11} + 18 a^{10} + 16 a^{9} + 9 a^{8} + 3 a^{7} + 20 a^{6} + 14 a^{5} + 3 a^{4} + 3 a^{3} + 8 a^{2} + 14 a + 1\right)\cdot 23^{10} + \left(14 a^{14} + 22 a^{13} + 10 a^{12} + 11 a^{11} + 13 a^{10} + 20 a^{9} + 9 a^{8} + 17 a^{7} + 22 a^{6} + 14 a^{5} + 12 a^{4} + 4 a^{3} + 12 a^{2} + 12 a + 8\right)\cdot 23^{11} + \left(6 a^{14} + 22 a^{13} + 12 a^{12} + 6 a^{11} + 6 a^{10} + 10 a^{8} + 3 a^{7} + 3 a^{6} + 12 a^{5} + 4 a^{4} + 2 a^{3} + 4 a^{2} + 13 a + 1\right)\cdot 23^{12} + \left(12 a^{14} + 18 a^{13} + 13 a^{12} + a^{11} + 10 a^{10} + 7 a^{9} + 12 a^{8} + 18 a^{7} + 20 a^{6} + 15 a^{5} + 22 a^{4} + 5 a^{3} + 6 a^{2} + 12 a + 7\right)\cdot 23^{13} + \left(11 a^{14} + 3 a^{13} + 8 a^{12} + 18 a^{11} + 15 a^{10} + 5 a^{9} + 9 a^{7} + 20 a^{6} + 17 a^{5} + 9 a^{4} + 4 a^{2} + 13 a + 9\right)\cdot 23^{14} + \left(14 a^{14} + 10 a^{13} + a^{12} + 22 a^{11} + 7 a^{10} + 7 a^{9} + 12 a^{8} + 17 a^{7} + 7 a^{6} + 13 a^{5} + 18 a^{4} + 11 a^{3} + 9 a^{2} + 15 a + 10\right)\cdot 23^{15} + \left(17 a^{14} + 14 a^{13} + 20 a^{12} + 15 a^{11} + 4 a^{10} + 20 a^{9} + 8 a^{8} + 10 a^{7} + 9 a^{6} + 21 a^{5} + a^{4} + 12 a^{3} + 6 a^{2} + 4 a + 14\right)\cdot 23^{16} + \left(21 a^{14} + 16 a^{13} + 7 a^{12} + 4 a^{11} + 9 a^{10} + 13 a^{9} + 19 a^{8} + 2 a^{7} + 8 a^{6} + 22 a^{5} + 13 a^{4} + 6 a^{3} + 3 a^{2} + 19 a + 12\right)\cdot 23^{17} + \left(9 a^{14} + 11 a^{13} + 16 a^{12} + 10 a^{11} + 9 a^{10} + 5 a^{9} + 4 a^{8} + 10 a^{7} + 17 a^{6} + 3 a^{5} + a^{4} + 2 a^{3} + 17 a^{2} + 7 a + 12\right)\cdot 23^{18} + \left(16 a^{14} + 11 a^{13} + 5 a^{12} + 4 a^{11} + 19 a^{10} + 22 a^{9} + 20 a^{8} + 2 a^{7} + 20 a^{5} + 21 a^{4} + 3 a^{3} + 11 a^{2} + 13 a + 22\right)\cdot 23^{19} + \left(11 a^{14} + 7 a^{13} + 2 a^{12} + 15 a^{11} + 6 a^{10} + 17 a^{9} + 16 a^{8} + 2 a^{7} + 12 a^{6} + a^{5} + 19 a^{4} + 21 a^{3} + 10 a^{2} + 22\right)\cdot 23^{20} + \left(22 a^{14} + 20 a^{13} + 9 a^{12} + 14 a^{11} + 3 a^{10} + 9 a^{9} + 9 a^{8} + 13 a^{7} + 11 a^{6} + 22 a^{5} + 3 a^{4} + 16 a^{3} + 14 a^{2} + 17 a + 16\right)\cdot 23^{21} + \left(11 a^{14} + 17 a^{13} + 19 a^{12} + 10 a^{11} + 11 a^{10} + 5 a^{9} + 11 a^{8} + 5 a^{7} + 16 a^{6} + 4 a^{5} + 12 a^{4} + 3 a^{3} + 7 a^{2}\right)\cdot 23^{22} + \left(4 a^{14} + 4 a^{13} + 19 a^{12} + 13 a^{11} + 7 a^{10} + 12 a^{9} + 8 a^{8} + 15 a^{7} + 21 a^{6} + 8 a^{5} + 6 a^{4} + 17 a^{3} + 18 a^{2} + 20 a + 9\right)\cdot 23^{23} + \left(11 a^{14} + 10 a^{13} + 2 a^{12} + 14 a^{11} + 21 a^{10} + 12 a^{9} + 19 a^{8} + 22 a^{7} + 6 a^{6} + 22 a^{5} + 21 a^{4} + 22 a^{3} + 19 a^{2} + 20 a + 14\right)\cdot 23^{24} + \left(8 a^{14} + 7 a^{13} + 14 a^{12} + 22 a^{11} + 12 a^{10} + 13 a^{9} + 18 a^{8} + 7 a^{7} + a^{6} + 4 a^{5} + 4 a^{4} + 19 a^{3} + 22 a^{2} + 9 a + 18\right)\cdot 23^{25} + \left(10 a^{14} + 11 a^{13} + 3 a^{12} + 16 a^{11} + 12 a^{10} + 15 a^{8} + a^{7} + 9 a^{6} + 3 a^{5} + 19 a^{4} + a^{3} + 6 a^{2} + 4 a + 9\right)\cdot 23^{26} + \left(5 a^{14} + 14 a^{13} + 5 a^{12} + 8 a^{11} + 11 a^{9} + 16 a^{8} + 22 a^{7} + 18 a^{6} + 15 a^{5} + 5 a^{4} + 16 a^{3} + 21 a^{2} + 14 a + 14\right)\cdot 23^{27} + \left(3 a^{14} + 8 a^{13} + 22 a^{12} + 2 a^{11} + 15 a^{9} + 21 a^{8} + 3 a^{7} + 2 a^{6} + 13 a^{5} + 21 a^{4} + 19 a^{3} + 7 a^{2} + 16 a + 4\right)\cdot 23^{28} + \left(13 a^{14} + 14 a^{13} + 7 a^{12} + 2 a^{11} + 7 a^{10} + 5 a^{9} + 17 a^{8} + 16 a^{7} + 3 a^{6} + 8 a^{5} + 19 a^{4} + 18 a^{2} + 11 a + 8\right)\cdot 23^{29} + \left(13 a^{14} + 7 a^{13} + 22 a^{12} + 18 a^{11} + 16 a^{10} + 19 a^{9} + 19 a^{8} + 21 a^{7} + 8 a^{6} + 5 a^{5} + 5 a^{4} + a^{3} + 8 a^{2} + 6 a + 21\right)\cdot 23^{30} + \left(11 a^{14} + 7 a^{13} + 13 a^{12} + 12 a^{10} + 6 a^{9} + 4 a^{8} + 13 a^{7} + 17 a^{6} + 16 a^{5} + 12 a^{4} + 11 a^{3} + 8 a^{2} + 9 a\right)\cdot 23^{31} + \left(a^{14} + 7 a^{13} + 18 a^{12} + 9 a^{11} + 4 a^{10} + 16 a^{9} + 20 a^{8} + 13 a^{7} + 14 a^{6} + 11 a^{5} + 8 a^{4} + 8 a^{3} + 6 a^{2} + 19 a + 2\right)\cdot 23^{32} + \left(16 a^{14} + 12 a^{13} + 3 a^{12} + 17 a^{11} + 13 a^{10} + 2 a^{9} + 2 a^{8} + 4 a^{7} + 3 a^{6} + 11 a^{5} + 14 a^{4} + 16 a^{3} + 7 a^{2} + 13 a + 8\right)\cdot 23^{33} + \left(20 a^{14} + 10 a^{13} + 2 a^{12} + 14 a^{11} + 2 a^{10} + 13 a^{9} + 18 a^{8} + 6 a^{7} + 3 a^{6} + 21 a^{5} + 19 a^{4} + 8 a^{3} + 11 a^{2} + 3 a + 15\right)\cdot 23^{34} + \left(10 a^{14} + 14 a^{13} + 13 a^{12} + 6 a^{11} + 10 a^{10} + 3 a^{9} + 17 a^{8} + 11 a^{7} + 20 a^{6} + 22 a^{5} + 2 a^{4} + 3 a^{3} + 8 a^{2} + 5 a + 5\right)\cdot 23^{35} + \left(19 a^{14} + 6 a^{13} + 19 a^{12} + 14 a^{11} + a^{10} + 22 a^{9} + 22 a^{8} + 9 a^{7} + 15 a^{6} + 5 a^{5} + 19 a^{4} + 17 a^{3} + 21 a^{2} + 15 a + 15\right)\cdot 23^{36} + \left(2 a^{14} + 17 a^{13} + 8 a^{12} + 11 a^{11} + a^{10} + 2 a^{9} + 8 a^{8} + 22 a^{7} + 11 a^{6} + 10 a^{5} + 18 a^{4} + 13 a^{3} + 20 a^{2} + 4 a + 5\right)\cdot 23^{37} + \left(15 a^{14} + 15 a^{13} + 17 a^{12} + a^{11} + 12 a^{10} + 14 a^{9} + 20 a^{8} + 9 a^{7} + 22 a^{4} + 20 a^{3} + 18 a^{2} + 15 a + 7\right)\cdot 23^{38} + \left(6 a^{14} + 2 a^{13} + a^{12} + 4 a^{11} + 16 a^{10} + a^{9} + 11 a^{7} + 12 a^{6} + 21 a^{5} + 13 a^{4} + 10 a^{3} + 10 a^{2} + 5 a + 17\right)\cdot 23^{39} + \left(21 a^{14} + 3 a^{13} + 3 a^{12} + 11 a^{11} + 17 a^{10} + 6 a^{9} + 15 a^{8} + 2 a^{7} + 14 a^{6} + 11 a^{5} + 15 a^{4} + 14 a^{3} + 2 a^{2} + 6 a + 11\right)\cdot 23^{40} + \left(15 a^{14} + 10 a^{13} + 12 a^{12} + 2 a^{11} + 16 a^{10} + 21 a^{9} + 17 a^{8} + 16 a^{7} + 4 a^{6} + 19 a^{5} + 3 a^{4} + 5 a^{3} + 7 a^{2} + 14 a + 2\right)\cdot 23^{41} + \left(12 a^{14} + 6 a^{13} + a^{12} + 4 a^{11} + 14 a^{10} + 2 a^{9} + a^{8} + 12 a^{7} + 19 a^{6} + 9 a^{5} + 22 a^{4} + a^{3} + 17 a^{2} + 21 a + 22\right)\cdot 23^{42} + \left(3 a^{14} + 10 a^{13} + 18 a^{12} + 19 a^{11} + 18 a^{10} + 8 a^{9} + 2 a^{8} + 16 a^{7} + 5 a^{6} + 19 a^{5} + 3 a^{4} + 15 a^{3} + 19 a^{2} + 17 a + 20\right)\cdot 23^{43} + \left(20 a^{14} + 9 a^{13} + 20 a^{12} + a^{11} + 2 a^{10} + 20 a^{9} + 22 a^{8} + 16 a^{7} + 21 a^{6} + a^{5} + a^{4} + 8 a^{3} + 18 a^{2} + 18 a + 7\right)\cdot 23^{44} + \left(3 a^{14} + 3 a^{13} + 6 a^{12} + 21 a^{11} + 21 a^{10} + 14 a^{9} + 22 a^{8} + 21 a^{7} + 16 a^{5} + a^{4} + 9 a^{3} + 17 a^{2} + 10 a + 22\right)\cdot 23^{45} + \left(3 a^{14} + 10 a^{13} + 8 a^{12} + a^{11} + 17 a^{10} + 6 a^{9} + 16 a^{8} + 13 a^{6} + 14 a^{5} + 13 a^{4} + 7 a^{3} + 10 a^{2} + 4 a + 13\right)\cdot 23^{46} + \left(2 a^{14} + 3 a^{13} + 19 a^{12} + 8 a^{11} + a^{10} + 21 a^{9} + 5 a^{8} + a^{7} + 15 a^{6} + 6 a^{5} + 10 a^{4} + 4 a^{3} + 6 a^{2} + a + 2\right)\cdot 23^{47} + \left(6 a^{14} + 5 a^{13} + 19 a^{12} + 6 a^{11} + a^{10} + 6 a^{9} + 15 a^{8} + 4 a^{7} + a^{6} + 19 a^{5} + 18 a^{4} + 18 a^{3} + 6 a^{2} + 4 a + 8\right)\cdot 23^{48} + \left(19 a^{14} + 5 a^{13} + 19 a^{12} + 16 a^{11} + 20 a^{9} + 17 a^{8} + 18 a^{7} + 2 a^{6} + 3 a^{5} + 19 a^{4} + 2 a^{3} + 9 a^{2} + 18 a + 22\right)\cdot 23^{49} + \left(9 a^{14} + 2 a^{13} + 21 a^{11} + 18 a^{10} + 14 a^{9} + 17 a^{8} + a^{7} + 19 a^{6} + 3 a^{5} + 12 a^{4} + 19 a^{3} + 11 a^{2} + 22 a + 10\right)\cdot 23^{50} + \left(15 a^{13} + 4 a^{12} + 22 a^{11} + 20 a^{10} + 18 a^{9} + 11 a^{8} + 21 a^{7} + 14 a^{6} + 3 a^{5} + 12 a^{4} + 3 a^{3} + 2 a^{2} + 14 a + 8\right)\cdot 23^{51} + \left(8 a^{14} + 14 a^{13} + 3 a^{11} + 21 a^{10} + 21 a^{9} + 7 a^{8} + a^{7} + 15 a^{4} + 5 a^{3} + 20 a^{2} + 9 a + 14\right)\cdot 23^{52} + \left(17 a^{13} + 9 a^{12} + 22 a^{11} + 18 a^{10} + 8 a^{9} + 16 a^{8} + 4 a^{7} + 14 a^{6} + 6 a^{5} + 4 a^{4} + 5 a^{3} + 21 a^{2} + a + 10\right)\cdot 23^{53} + \left(11 a^{14} + 11 a^{13} + 22 a^{12} + 14 a^{11} + 15 a^{10} + 2 a^{9} + 15 a^{8} + 5 a^{7} + 11 a^{6} + a^{5} + 12 a^{4} + 4 a^{3} + 3 a^{2} + 10 a + 14\right)\cdot 23^{54} + \left(3 a^{14} + 21 a^{13} + 5 a^{12} + 18 a^{11} + 22 a^{10} + 7 a^{9} + 5 a^{8} + 15 a^{7} + 7 a^{6} + 18 a^{4} + 17 a^{3} + 15 a^{2} + 7 a + 13\right)\cdot 23^{55} + \left(18 a^{14} + 18 a^{13} + 3 a^{12} + 12 a^{11} + 18 a^{10} + 15 a^{9} + 13 a^{8} + 10 a^{7} + 20 a^{6} + 3 a^{5} + 10 a^{4} + 21 a^{3} + 19 a^{2} + 3 a + 14\right)\cdot 23^{56} + \left(15 a^{14} + 13 a^{13} + 20 a^{12} + 6 a^{11} + 2 a^{10} + 11 a^{9} + 11 a^{8} + 9 a^{7} + 20 a^{6} + 8 a^{5} + 8 a^{4} + 5 a^{3} + 4 a^{2} + 2 a + 13\right)\cdot 23^{57} + \left(3 a^{14} + 16 a^{13} + 17 a^{12} + 21 a^{11} + 20 a^{9} + 5 a^{8} + 11 a^{7} + 18 a^{6} + 12 a^{5} + 21 a^{4} + 16 a^{3} + 6 a^{2} + 8 a + 11\right)\cdot 23^{58} + \left(8 a^{14} + a^{13} + 18 a^{12} + 17 a^{11} + 10 a^{10} + 14 a^{9} + 15 a^{8} + 8 a^{7} + 5 a^{6} + 18 a^{5} + 2 a^{4} + 19 a^{3} + 21 a^{2} + 18 a + 19\right)\cdot 23^{59} + \left(20 a^{14} + 19 a^{13} + 13 a^{12} + 3 a^{11} + 19 a^{10} + 5 a^{9} + 3 a^{8} + 22 a^{7} + 17 a^{6} + 17 a^{5} + 17 a^{4} + 17 a^{3} + 16 a^{2} + 9 a + 18\right)\cdot 23^{60} + \left(8 a^{13} + 16 a^{12} + 13 a^{11} + 21 a^{10} + a^{9} + 12 a^{8} + 16 a^{7} + 3 a^{6} + 22 a^{5} + 15 a^{4} + 19 a + 6\right)\cdot 23^{61} + \left(a^{14} + 16 a^{13} + 3 a^{12} + 11 a^{11} + 2 a^{10} + 10 a^{9} + 15 a^{8} + a^{7} + 18 a^{6} + 19 a^{5} + 5 a^{4} + 20 a^{3} + 11 a^{2} + 3 a + 16\right)\cdot 23^{62} + \left(11 a^{14} + 17 a^{13} + 21 a^{12} + 19 a^{11} + 16 a^{10} + 18 a^{9} + 10 a^{8} + 18 a^{6} + 22 a^{5} + 20 a^{4} + 19 a^{3} + 4 a^{2} + 14 a + 19\right)\cdot 23^{63} + \left(4 a^{14} + 3 a^{13} + 21 a^{12} + 18 a^{11} + 5 a^{10} + 20 a^{9} + 15 a^{8} + 22 a^{7} + a^{5} + 19 a^{4} + 13 a^{3} + 14 a^{2} + 12 a + 17\right)\cdot 23^{64} + \left(15 a^{13} + 5 a^{12} + 15 a^{11} + 16 a^{10} + 19 a^{9} + 5 a^{8} + 13 a^{7} + 20 a^{6} + 10 a^{5} + a^{4} + 7 a^{3} + 7 a^{2} + 13 a + 16\right)\cdot 23^{65} + \left(10 a^{14} + 7 a^{13} + 15 a^{12} + 2 a^{11} + 7 a^{10} + 6 a^{9} + 16 a^{8} + 20 a^{7} + 19 a^{6} + 18 a^{5} + 20 a^{4} + 18 a^{2} + 5 a + 5\right)\cdot 23^{66} + \left(14 a^{14} + 19 a^{13} + 16 a^{12} + 19 a^{11} + 6 a^{10} + 4 a^{9} + 10 a^{8} + 13 a^{6} + 3 a^{5} + 3 a^{4} + 6 a^{3} + 15 a^{2} + 17\right)\cdot 23^{67} + \left(13 a^{14} + 16 a^{13} + 8 a^{12} + 14 a^{11} + 22 a^{10} + 13 a^{9} + 4 a^{8} + 2 a^{7} + 12 a^{6} + a^{5} + 3 a^{3} + 2 a^{2} + 9 a + 16\right)\cdot 23^{68} + \left(5 a^{14} + 14 a^{13} + 7 a^{12} + 3 a^{11} + 11 a^{10} + a^{9} + 6 a^{8} + 3 a^{7} + 20 a^{6} + 4 a^{5} + 7 a^{4} + 5 a^{3} + 9 a^{2} + 21 a + 7\right)\cdot 23^{69} + \left(6 a^{14} + a^{13} + 17 a^{12} + 6 a^{11} + 16 a^{10} + 17 a^{9} + 14 a^{8} + 19 a^{7} + 5 a^{5} + 20 a^{4} + 10 a^{3} + 7 a^{2} + 19 a + 4\right)\cdot 23^{70} + \left(10 a^{13} + 20 a^{12} + 17 a^{11} + 21 a^{10} + 6 a^{9} + 13 a^{7} + 4 a^{6} + 10 a^{5} + 9 a^{4} + 14 a^{3} + 20 a^{2} + 10 a + 20\right)\cdot 23^{71} + \left(7 a^{14} + 16 a^{13} + 20 a^{12} + 2 a^{11} + 12 a^{10} + 14 a^{9} + 18 a^{8} + 5 a^{7} + 16 a^{6} + 17 a^{5} + 20 a^{4} + 16 a^{3} + 14 a^{2} + 21 a + 1\right)\cdot 23^{72} + \left(17 a^{14} + 20 a^{13} + 12 a^{12} + 7 a^{11} + 5 a^{10} + 3 a^{9} + 16 a^{8} + 9 a^{7} + 3 a^{6} + 17 a^{4} + 8 a^{3} + 18 a^{2} + 16 a + 5\right)\cdot 23^{73} + \left(19 a^{14} + 13 a^{13} + 8 a^{12} + 15 a^{11} + 9 a^{10} + 6 a^{9} + 14 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19 a^{8} + 19 a^{7} + 10 a^{6} + 2 a^{5} + 2 a^{4} + 2 a^{3} + 12 a^{2} + 16 a + 18\right)\cdot 23^{80} + \left(13 a^{14} + 9 a^{13} + 5 a^{12} + 21 a^{11} + 19 a^{10} + 11 a^{9} + 18 a^{8} + 6 a^{7} + 17 a^{6} + 9 a^{5} + 10 a^{4} + 13 a^{3} + 13 a^{2} + 16 a + 3\right)\cdot 23^{81} + \left(8 a^{14} + 12 a^{13} + 14 a^{12} + 12 a^{11} + 14 a^{10} + 4 a^{9} + 2 a^{8} + 12 a^{7} + 15 a^{6} + 17 a^{5} + 7 a^{4} + 16 a^{3} + 12 a^{2} + 21 a + 1\right)\cdot 23^{82} + \left(3 a^{14} + 4 a^{13} + 12 a^{11} + 15 a^{10} + 10 a^{9} + 19 a^{8} + 10 a^{7} + 8 a^{6} + 13 a^{5} + a^{4} + 3 a^{3} + 2 a^{2} + 12 a + 3\right)\cdot 23^{83} + \left(19 a^{13} + 22 a^{12} + 18 a^{11} + 15 a^{10} + 15 a^{9} + 16 a^{8} + 5 a^{7} + 21 a^{6} + 16 a^{5} + 11 a^{4} + 2 a^{3} + 9 a^{2} + 17 a + 13\right)\cdot 23^{84} + \left(4 a^{13} + 20 a^{12} + a^{11} + 9 a^{10} + 19 a^{9} + 7 a^{8} + 3 a^{7} + 4 a^{6} + 11 a^{5} + 6 a^{4} + 9 a^{3} + 3 a^{2} + 3 a + 20\right)\cdot 23^{85} + \left(22 a^{14} + 15 a^{13} + 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a + 13\right)\cdot 23^{91} + \left(21 a^{14} + 19 a^{13} + 11 a^{12} + 14 a^{11} + 14 a^{10} + 9 a^{9} + 3 a^{8} + 21 a^{7} + 3 a^{6} + 11 a^{5} + 14 a^{4} + 10 a^{3} + 4 a^{2} + 21\right)\cdot 23^{92} + \left(12 a^{14} + 15 a^{13} + 21 a^{12} + 15 a^{11} + 7 a^{10} + 8 a^{9} + 4 a^{8} + 2 a^{7} + 17 a^{6} + 14 a^{5} + 15 a^{4} + 9 a^{3} + 8 a^{2} + 13 a + 12\right)\cdot 23^{93} + \left(6 a^{14} + a^{13} + 17 a^{12} + 7 a^{11} + 7 a^{9} + 6 a^{8} + a^{6} + 22 a^{4} + 17 a^{3} + 17 a^{2} + 11\right)\cdot 23^{94} + \left(14 a^{14} + 20 a^{13} + 8 a^{12} + 17 a^{11} + 7 a^{10} + 13 a^{9} + 21 a^{8} + 5 a^{7} + 21 a^{6} + 2 a^{5} + 2 a^{4} + 19 a^{3} + 6 a^{2} + 6\right)\cdot 23^{95} + \left(5 a^{14} + 16 a^{13} + 12 a^{12} + 5 a^{11} + 13 a^{10} + 13 a^{9} + 6 a^{8} + 20 a^{7} + 9 a^{6} + 2 a^{5} + 20 a^{4} + 7 a^{3} + 18 a^{2} + 22 a + 17\right)\cdot 23^{96} + \left(16 a^{14} + 15 a^{13} + 20 a^{12} + 16 a^{11} + 6 a^{10} + 15 a^{9} + 13 a^{8} + 18 a^{7} + 22 a^{6} + 18 a^{5} + 12 a^{4} + 19 a^{3} + 11 a^{2} + 14 a + 14\right)\cdot 23^{97} + \left(15 a^{14} + a^{13} + a^{12} + a^{11} + 14 a^{10} + 18 a^{9} + 4 a^{8} + 10 a^{7} + 2 a^{6} + 9 a^{5} + 16 a^{4} + 5 a^{3} + 5 a^{2} + 2 a + 10\right)\cdot 23^{98} + \left(20 a^{14} + 5 a^{13} + 22 a^{12} + a^{11} + 12 a^{10} + 2 a^{8} + 6 a^{7} + 17 a^{6} + 11 a^{5} + 18 a^{4} + 8 a^{3} + 16 a^{2} + 8 a\right)\cdot 23^{99} + \left(10 a^{14} + 20 a^{13} + 22 a^{12} + 15 a^{11} + 14 a^{10} + 21 a^{9} + 7 a^{8} + 22 a^{7} + 13 a^{6} + a^{5} + 14 a^{4} + 6 a^{3} + 21 a^{2} + 22 a + 19\right)\cdot 23^{100} + \left(4 a^{14} + 9 a^{13} + 2 a^{12} + 10 a^{11} + 19 a^{10} + 7 a^{9} + 11 a^{8} + 15 a^{7} + 21 a^{6} + a^{5} + 13 a^{3} + 22 a^{2} + a + 22\right)\cdot 23^{101} + \left(21 a^{14} + 18 a^{13} + 9 a^{12} + 21 a^{11} + 15 a^{10} + 19 a^{9} + 6 a^{8} + 6 a^{7} + 16 a^{6} + 4 a^{5} + 13 a^{4} + 3 a^{3} + 17 a^{2} + 4 a + 7\right)\cdot 23^{102} + \left(8 a^{14} + 10 a^{13} + 11 a^{12} + 9 a^{11} + 8 a^{10} + 15 a^{9} + 16 a^{8} 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7\right)\cdot 23^{120} + \left(6 a^{14} + 14 a^{13} + 15 a^{12} + 22 a^{11} + 18 a^{10} + 22 a^{9} + 22 a^{8} + 6 a^{7} + 7 a^{6} + 22 a^{5} + 18 a^{4} + 16 a^{3} + 13 a^{2} + 19 a + 18\right)\cdot 23^{121} + \left(21 a^{14} + a^{13} + 14 a^{12} + 15 a^{11} + 15 a^{9} + a^{8} + 13 a^{7} + 16 a^{6} + 7 a^{5} + 3 a^{4} + 14 a^{3} + 10 a^{2} + 6 a + 16\right)\cdot 23^{122} + \left(22 a^{14} + 19 a^{13} + 15 a^{12} + 15 a^{11} + 13 a^{10} + 21 a^{9} + 7 a^{8} + 14 a^{7} + 8 a^{6} + 20 a^{5} + 2 a^{4} + 13 a^{2} + 19 a + 8\right)\cdot 23^{123} + \left(14 a^{14} + 3 a^{13} + 16 a^{12} + 18 a^{11} + 7 a^{10} + 13 a^{9} + 2 a^{8} + 15 a^{7} + 5 a^{6} + 3 a^{5} + 21 a^{4} + 13 a^{3} + 5 a^{2} + 21 a + 1\right)\cdot 23^{124} + \left(18 a^{14} + 10 a^{13} + 16 a^{12} + a^{11} + 21 a^{9} + 15 a^{8} + a^{7} + 11 a^{6} + 6 a^{5} + 3 a^{4} + 6 a^{3} + a^{2} + 16 a + 22\right)\cdot 23^{125} + \left(6 a^{14} + 3 a^{13} + 17 a^{12} + 6 a^{11} + 2 a^{10} + 8 a^{9} + 15 a^{8} + 22 a^{7} + 21 a^{6} + 21 a^{5} + 16 a^{4} + 7 a^{3} + 16 a^{2} + 20 a + 9\right)\cdot 23^{126} + \left(2 a^{13} + 22 a^{12} + 11 a^{11} + 9 a^{10} + 10 a^{9} + 7 a^{8} + 19 a^{7} + 10 a^{6} + 20 a^{5} + 15 a^{4} + 20 a^{3} + 3 a^{2} + 21 a + 12\right)\cdot 23^{127} + \left(19 a^{14} + 2 a^{13} + 9 a^{12} + 7 a^{11} + 2 a^{10} + 11 a^{9} + 14 a^{8} + 5 a^{7} + 5 a^{6} + 16 a^{5} + 20 a^{4} + 19 a^{3} + 16 a^{2} + 19 a + 3\right)\cdot 23^{128} + \left(17 a^{14} + 7 a^{12} + a^{11} + 7 a^{10} + 17 a^{9} + 11 a^{8} + a^{6} + 7 a^{5} + 9 a^{4} + 18 a^{3} + 19 a^{2} + 8 a + 4\right)\cdot 23^{129} + \left(18 a^{13} + 14 a^{12} + 21 a^{11} + 2 a^{10} + 16 a^{9} + 16 a^{8} + 8 a^{7} + 21 a^{6} + 14 a^{5} + 15 a^{4} + 18 a^{3} + 18 a^{2} + 17 a + 3\right)\cdot 23^{130} + \left(11 a^{14} + 18 a^{13} + 17 a^{12} + 4 a^{11} + 4 a^{10} + 20 a^{9} + 19 a^{8} + 17 a^{7} + 5 a^{6} + 5 a^{5} + 13 a^{4} + 21 a^{3} + 2 a^{2} + 16 a + 21\right)\cdot 23^{131} + \left(19 a^{14} + 9 a^{13} + 17 a^{12} + 8 a^{11} + 20 a^{10} + 16 a^{9} + 10 a^{8} + 20 a^{7} + 7 a^{6} + 19 a^{5} + 7 a^{4} + 14 a^{3} + 11 a^{2} + 19 a + 13\right)\cdot 23^{132} + \left(19 a^{14} + 6 a^{13} + a^{12} + 6 a^{11} + 4 a^{10} + 7 a^{9} + 13 a^{8} + 15 a^{7} + 17 a^{6} + 5 a^{5} + 19 a^{4} + 6 a^{3} + 10 a^{2} + 7 a + 12\right)\cdot 23^{133} + \left(18 a^{14} + a^{13} + 3 a^{12} + 5 a^{11} + 14 a^{10} + 4 a^{9} + 5 a^{8} + 17 a^{7} + 16 a^{6} + 6 a^{5} + 2 a^{4} + 4 a^{3} + 6 a^{2} + 16 a + 8\right)\cdot 23^{134} + \left(2 a^{14} + 12 a^{13} + 15 a^{12} + 8 a^{11} + a^{10} + 22 a^{9} + 18 a^{8} + 6 a^{6} + 17 a^{5} + 8 a^{4} + 10 a^{3} + 7 a^{2} + 3 a + 7\right)\cdot 23^{135} + \left(4 a^{14} + 13 a^{13} + 15 a^{12} + 11 a^{11} + 19 a^{10} + 2 a^{9} + 12 a^{8} + 16 a^{6} + 14 a^{5} + 9 a^{4} + 19 a^{3} + 14 a^{2} + 22 a + 4\right)\cdot 23^{136} + \left(12 a^{14} + 2 a^{13} + 21 a^{12} + 10 a^{11} + 9 a^{10} + 12 a^{9} + 13 a^{8} + 2 a^{7} + 13 a^{6} + 5 a^{5} + 5 a^{4} + 2 a^{3} + 20 a^{2} + 18 a + 19\right)\cdot 23^{137} + \left(4 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\left(15 a^{13} + 10 a^{12} + a^{11} + 18 a^{10} + 16 a^{9} + 18 a^{8} + a^{7} + 7 a^{6} + 11 a^{5} + 18 a^{4} + 5 a^{3} + 4 a^{2} + 15\right)\cdot 23^{144} + \left(15 a^{14} + 19 a^{13} + 13 a^{12} + 17 a^{11} + 8 a^{10} + 9 a^{9} + 19 a^{8} + 4 a^{7} + 8 a^{6} + 9 a^{5} + 22 a^{4} + 18 a^{3} + 13 a^{2} + 6 a + 15\right)\cdot 23^{145} + \left(20 a^{14} + 5 a^{13} + a^{11} + 18 a^{10} + 9 a^{9} + 16 a^{8} + 18 a^{7} + 10 a^{6} + 19 a^{5} + 8 a^{4} + 4 a^{3} + 13 a^{2} + 4 a + 15\right)\cdot 23^{146} + \left(8 a^{14} + 9 a^{13} + 2 a^{12} + 4 a^{11} + 13 a^{10} + 8 a^{9} + 7 a^{8} + 2 a^{7} + 17 a^{6} + 3 a^{5} + 13 a^{4} + 5 a^{3} + 15 a^{2} + 14 a + 9\right)\cdot 23^{147} + \left(21 a^{14} + a^{13} + 9 a^{12} + 11 a^{11} + 5 a^{10} + 17 a^{9} + 6 a^{8} + 7 a^{7} + 12 a^{6} + a^{5} + 15 a^{4} + 12 a^{3} + 17 a^{2} + 18 a + 5\right)\cdot 23^{148} + \left(14 a^{14} + 7 a^{13} + 14 a^{12} + 3 a^{11} + 9 a^{10} + 18 a^{9} + a^{8} + 8 a^{7} + 10 a^{6} + 5 a^{5} + 9 a^{4} + 9 a^{3} + 20 a^{2} + 19 a + 18\right)\cdot 23^{149} + \left(7 a^{14} + 3 a^{13} + 7 a^{12} + 19 a^{11} + a^{10} + 15 a^{9} + 8 a^{8} + 10 a^{7} + 18 a^{6} + 10 a^{4} + 20 a^{3} + 11 a^{2} + 18 a + 11\right)\cdot 23^{150} + \left(5 a^{14} + 12 a^{13} + 10 a^{12} + 3 a^{11} + 11 a^{10} + 9 a^{9} + 3 a^{8} + 12 a^{7} + 13 a^{6} + 19 a^{5} + 18 a^{4} + 22 a^{3} + 16 a^{2} + 5 a + 22\right)\cdot 23^{151} + \left(17 a^{14} + 2 a^{13} + 18 a^{12} + 13 a^{11} + 16 a^{10} + 10 a^{9} + 8 a^{8} + 7 a^{7} + 5 a^{5} + 17 a^{4} + 14 a^{3} + 4 a^{2} + 18 a + 14\right)\cdot 23^{152} + \left(14 a^{13} + 21 a^{12} + 20 a^{11} + 14 a^{10} + 14 a^{9} + 3 a^{8} + 6 a^{7} + 11 a^{6} + 20 a^{5} + 4 a^{4} + a^{3} + 10 a^{2} + 9 a + 18\right)\cdot 23^{153} + \left(16 a^{14} + 12 a^{13} + 4 a^{12} + 22 a^{11} + 3 a^{10} + 13 a^{9} + 10 a^{8} + 5 a^{7} + 22 a^{6} + 21 a^{5} + 8 a^{4} + 18 a^{3} + 10 a^{2} + 19 a + 7\right)\cdot 23^{154} + \left(21 a^{14} + a^{13} + 14 a^{12} + 16 a^{11} + 2 a^{10} + a^{9} + 19 a^{8} + 20 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20\right)\cdot 23^{166} + \left(19 a^{14} + 18 a^{13} + 16 a^{12} + 16 a^{11} + 13 a^{10} + 22 a^{9} + 5 a^{8} + 11 a^{7} + 10 a^{6} + 10 a^{5} + 4 a^{4} + 5 a^{3} + 2 a^{2} + 4 a + 21\right)\cdot 23^{167} + \left(3 a^{14} + 16 a^{13} + 22 a^{12} + 4 a^{11} + 14 a^{10} + 21 a^{9} + 9 a^{8} + 13 a^{7} + 4 a^{6} + 11 a^{5} + 5 a^{4} + 7 a^{3} + 18 a^{2} + 20 a + 1\right)\cdot 23^{168} + \left(15 a^{14} + 3 a^{13} + 7 a^{11} + 3 a^{10} + 5 a^{9} + 22 a^{8} + 13 a^{7} + 17 a^{6} + 14 a^{5} + 7 a^{4} + 4 a^{3} + 17 a^{2} + 9 a + 6\right)\cdot 23^{169} + \left(15 a^{14} + 20 a^{13} + 16 a^{12} + 19 a^{11} + 18 a^{10} + a^{9} + 22 a^{8} + a^{7} + 3 a^{6} + 4 a^{5} + 9 a^{4} + 14 a^{3} + 11 a^{2} + 9 a + 12\right)\cdot 23^{170} + \left(5 a^{14} + 17 a^{13} + 19 a^{12} + 2 a^{10} + 19 a^{9} + 18 a^{8} + 10 a^{7} + 11 a^{5} + 21 a^{4} + 13 a^{3} + 5 a^{2} + a + 5\right)\cdot 23^{171} + \left(15 a^{13} + 7 a^{12} + 17 a^{11} + 15 a^{9} + 8 a^{8} + 2 a^{7} + 11 a^{6} + 7 a^{5} + 19 a^{4} + 2 a^{3} + 20 a^{2} + 7\right)\cdot 23^{172} + \left(11 a^{14} + 15 a^{13} + 12 a^{11} + 22 a^{10} + 7 a^{9} + 5 a^{8} + 22 a^{7} + 9 a^{6} + 12 a^{5} + 16 a^{4} + 21 a^{3} + 12 a^{2} + 18 a + 3\right)\cdot 23^{173} + \left(13 a^{14} + 21 a^{13} + 6 a^{12} + 8 a^{11} + 2 a^{10} + a^{9} + 21 a^{8} + 5 a^{7} + 5 a^{6} + 7 a^{5} + 16 a^{4} + 19 a^{3} + 13 a^{2} + 20 a\right)\cdot 23^{174} + \left(19 a^{14} + 4 a^{13} + 14 a^{12} + 21 a^{11} + 22 a^{10} + 6 a^{9} + 20 a^{8} + 16 a^{7} + 7 a^{6} + 16 a^{5} + 18 a^{4} + 20 a^{3} + 10 a^{2} + 12 a + 7\right)\cdot 23^{175} + \left(11 a^{14} + 15 a^{13} + 11 a^{12} + 20 a^{11} + 19 a^{10} + 17 a^{9} + 7 a^{8} + 12 a^{7} + 18 a^{6} + 5 a^{4} + 22 a^{3} + 12 a^{2} + 22 a\right)\cdot 23^{176} + \left(7 a^{14} + 3 a^{13} + 9 a^{12} + 7 a^{11} + 9 a^{10} + 18 a^{9} + 3 a^{8} + 17 a^{7} + 5 a^{6} + 4 a^{4} + 9 a^{3} + 6 a^{2} + 3 a + 2\right)\cdot 23^{177} + \left(2 a^{14} + 15 a^{13} + 13 a^{12} + 4 a^{11} + a^{10} + 15 a^{9} + 21 a^{8} + a^{7} + 8 a^{6} + 8 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+ 4 a^{10} + 13 a^{9} + 5 a^{8} + 3 a^{7} + 14 a^{6} + 18 a^{5} + 20 a^{4} + 18 a^{3} + 17 a^{2} + 21 a + 10\right)\cdot 23^{184} + \left(17 a^{14} + 6 a^{13} + 21 a^{12} + 9 a^{11} + a^{10} + 20 a^{9} + 18 a^{8} + 9 a^{7} + 16 a^{6} + 9 a^{5} + 16 a^{4} + 15 a^{3} + 4 a\right)\cdot 23^{185} + \left(11 a^{14} + 18 a^{13} + 16 a^{12} + 15 a^{11} + a^{10} + 20 a^{9} + 10 a^{8} + 18 a^{7} + 6 a^{6} + 22 a^{5} + a^{4} + 8 a^{3} + 18 a^{2} + 21 a + 9\right)\cdot 23^{186} + \left(13 a^{14} + 2 a^{13} + 10 a^{12} + a^{11} + 15 a^{10} + 18 a^{9} + 18 a^{8} + 18 a^{7} + 8 a^{6} + 19 a^{5} + 22 a^{4} + a^{2} + 19 a + 6\right)\cdot 23^{187} + \left(3 a^{14} + 11 a^{13} + 17 a^{12} + 8 a^{11} + 14 a^{10} + 9 a^{9} + 8 a^{8} + 2 a^{7} + 14 a^{6} + 14 a^{4} + 13 a^{3} + a^{2} + 21 a + 5\right)\cdot 23^{188} + \left(17 a^{14} + 7 a^{13} + 4 a^{12} + 2 a^{11} + 9 a^{10} + 7 a^{9} + 2 a^{8} + 3 a^{7} + 21 a^{6} + a^{5} + 9 a^{4} + 11 a^{3} + 20 a^{2} + 16 a + 16\right)\cdot 23^{189} + \left(15 a^{14} + 14 a^{13} + 10 a^{12} + 4 a^{11} + 2 a^{10} + 9 a^{9} + 3 a^{8} + 2 a^{7} + 12 a^{6} + 8 a^{4} + 4 a^{3} + 20 a^{2} + 3 a + 11\right)\cdot 23^{190} + \left(a^{14} + 6 a^{12} + 5 a^{11} + 11 a^{10} + 18 a^{9} + 4 a^{8} + 4 a^{7} + 3 a^{6} + 13 a^{5} + 9 a^{4} + 19 a^{3} + 12 a^{2} + a + 6\right)\cdot 23^{191} + \left(19 a^{14} + 19 a^{13} + 3 a^{12} + 5 a^{11} + 8 a^{10} + 4 a^{9} + 21 a^{8} + 17 a^{6} + 17 a^{5} + 2 a^{4} + 17 a^{3} + 6 a^{2} + 8 a + 16\right)\cdot 23^{192} + \left(a^{14} + 16 a^{13} + 10 a^{12} + 13 a^{11} + 14 a^{10} + 21 a^{9} + 18 a^{8} + 8 a^{7} + 20 a^{6} + a^{5} + 7 a^{4} + 21 a^{3} + 10 a + 1\right)\cdot 23^{193} + \left(3 a^{14} + 9 a^{13} + 14 a^{12} + 10 a^{11} + 10 a^{10} + 7 a^{9} + a^{8} + 9 a^{7} + 13 a^{6} + 11 a^{5} + 4 a^{4} + 7 a^{3} + 18 a^{2} + 14 a + 7\right)\cdot 23^{194} + \left(5 a^{14} + a^{13} + 21 a^{12} + 4 a^{11} + 8 a^{10} + 19 a^{9} + 4 a^{8} + a^{7} + 13 a^{6} + 7 a^{5} + 6 a^{4} + 15 a^{3} + 16 a^{2} + 10 a + 13\right)\cdot 23^{195} + \left(8 a^{14} + 22 a^{13} + 21 a^{12} + 7 a^{10} + 10 a^{9} + 19 a^{8} + 5 a^{7} + 11 a^{6} + 11 a^{5} + 2 a^{4} + 15 a^{3} + 8 a^{2} + 22 a + 1\right)\cdot 23^{196} + \left(2 a^{14} + 19 a^{13} + 5 a^{12} + 15 a^{11} + 13 a^{10} + 17 a^{9} + 15 a^{8} + 18 a^{7} + 5 a^{5} + 7 a^{4} + 3 a^{3} + 22 a^{2} + 22 a + 10\right)\cdot 23^{197} + \left(19 a^{14} + 9 a^{13} + 4 a^{12} + 4 a^{11} + 7 a^{10} + 6 a^{9} + 21 a^{8} + 16 a^{6} + 17 a^{5} + 6 a^{4} + 4 a^{3} + 12 a^{2} + 8 a + 18\right)\cdot 23^{198} + \left(14 a^{14} + 18 a^{13} + 2 a^{12} + 7 a^{11} + 5 a^{10} + 10 a^{9} + 8 a^{8} + a^{7} + 12 a^{6} + 4 a^{5} + 20 a^{4} + 2 a^{3} + 13 a^{2} + 22 a + 5\right)\cdot 23^{199} + \left(6 a^{13} + 4 a^{12} + 7 a^{11} + 9 a^{10} + 20 a^{9} + 8 a^{7} + 5 a^{6} + 18 a^{5} + 20 a^{4} + 16 a^{3} + 7 a^{2} + 19 a + 5\right)\cdot 23^{200} + \left(20 a^{14} + 13 a^{13} + 15 a^{12} + 11 a^{11} + 16 a^{10} + 10 a^{9} + 10 a^{8} + 9 a^{7} + 8 a^{5} + 6 a^{4} + 5 a^{3} + 16 a^{2} + 18 a + 21\right)\cdot 23^{201} + \left(16 a^{14} + 20 a^{13} + 21 a^{12} + 4 a^{11} + 11 a^{10} + 8 a^{9} + 3 a^{8} + 17 a^{7} + 19 a^{6} + 12 a^{5} + 19 a^{4} + 9 a^{3} + 21 a^{2} + 5 a + 6\right)\cdot 23^{202} + \left(14 a^{14} + 9 a^{13} + 13 a^{12} + 16 a^{11} + a^{10} + 11 a^{9} + 14 a^{8} + 5 a^{7} + a^{6} + 17 a^{5} + 22 a^{4} + 6 a^{2} + 10 a + 17\right)\cdot 23^{203} + \left(8 a^{14} + 5 a^{13} + 14 a^{12} + 22 a^{11} + 5 a^{10} + 20 a^{9} + 10 a^{8} + 2 a^{7} + 12 a^{6} + 9 a^{5} + a^{4} + 16 a^{3} + 2 a^{2} + 10 a + 16\right)\cdot 23^{204} + \left(9 a^{14} + 9 a^{13} + 5 a^{12} + 3 a^{11} + 13 a^{10} + 16 a^{9} + 2 a^{8} + 4 a^{7} + 15 a^{6} + 19 a^{5} + 16 a^{4} + 17 a^{3} + 18 a^{2} + 2 a + 5\right)\cdot 23^{205} + \left(16 a^{14} + 15 a^{13} + 9 a^{12} + 18 a^{11} + 22 a^{10} + 22 a^{8} + 3 a^{7} + 16 a^{6} + 6 a^{5} + 9 a^{4} + 22 a^{3} + 8 a^{2} + 5 a + 18\right)\cdot 23^{206} + \left(4 a^{14} + 22 a^{13} + 22 a^{12} + 9 a^{11} + 21 a^{10} + 9 a^{9} + 19 a^{8} + 8 a^{7} + 8 a^{6} + 15 a^{5} + 11 a^{4} + 20 a^{3} + 13 a^{2} + 22 a + 22\right)\cdot 23^{207} + \left(20 a^{14} + 6 a^{13} + 21 a^{12} + 14 a^{11} + 21 a^{10} + 9 a^{9} + 9 a^{8} + 10 a^{7} + 11 a^{6} + 10 a^{5} + 14 a^{4} + 16 a^{3} + 14 a\right)\cdot 23^{208} + \left(21 a^{14} + 21 a^{13} + 5 a^{12} + 4 a^{11} + 4 a^{10} + 5 a^{9} + 20 a^{8} + 4 a^{7} + 3 a^{6} + 19 a^{4} + 18 a^{3} + 4 a^{2} + 11 a + 10\right)\cdot 23^{209} + \left(8 a^{14} + a^{12} + 10 a^{11} + 7 a^{10} + 7 a^{9} + 15 a^{8} + 17 a^{7} + 17 a^{6} + 7 a^{5} + 20 a^{4} + 5 a^{3} + 11 a^{2} + 3 a + 2\right)\cdot 23^{210} + \left(17 a^{13} + 22 a^{12} + 6 a^{11} + 21 a^{10} + 22 a^{9} + 15 a^{8} + 3 a^{7} + 2 a^{6} + 15 a^{4} + 17 a^{3} + 18 a^{2} + 22 a + 18\right)\cdot 23^{211} + \left(17 a^{14} + a^{13} + 4 a^{12} + 12 a^{11} + 12 a^{10} + 13 a^{9} + 21 a^{8} + 6 a^{7} + 11 a^{6} + 14 a^{5} + 5 a^{4} + 20 a^{3} + 4 a^{2} + 10 a + 12\right)\cdot 23^{212} + \left(19 a^{14} + 18 a^{12} + 5 a^{11} + 15 a^{10} + 15 a^{9} + 15 a^{8} + 20 a^{7} + 3 a^{6} + 6 a^{5} + 18 a^{4} + 19 a^{3} + 10 a^{2} + 8 a\right)\cdot 23^{213} + \left(4 a^{14} + a^{13} + 12 a^{12} + 19 a^{11} + 16 a^{10} + 14 a^{9} + 17 a^{8} + 10 a^{7} + 20 a^{5} + 11 a^{4} + 22 a^{3} + 13 a^{2} + 22 a + 9\right)\cdot 23^{214} + \left(22 a^{14} + 20 a^{13} + 14 a^{12} + 18 a^{11} + 6 a^{10} + 19 a^{9} + 4 a^{8} + 18 a^{7} + 11 a^{6} + 19 a^{5} + 20 a^{4} + 2 a^{3} + 3 a^{2} + 3 a + 11\right)\cdot 23^{215} + \left(16 a^{14} + 7 a^{13} + 4 a^{12} + 10 a^{11} + 18 a^{10} + 13 a^{9} + 21 a^{8} + 17 a^{7} + 4 a^{6} + 20 a^{5} + 13 a^{4} + 18 a^{3} + 10 a^{2} + 7 a + 22\right)\cdot 23^{216} + \left(12 a^{13} + 17 a^{12} + 22 a^{11} + 17 a^{10} + 9 a^{9} + 22 a^{8} + 8 a^{7} + 15 a^{6} + 12 a^{5} + 19 a^{4} + 3 a^{3} + 16 a^{2} + 22 a + 6\right)\cdot 23^{217} + \left(2 a^{14} + 2 a^{13} + 16 a^{11} + 21 a^{10} + 3 a^{9} + 19 a^{8} + 5 a^{7} + 20 a^{6} + 7 a^{5} + 11 a^{4} + 4 a^{3} + 14 a^{2} + 9 a + 21\right)\cdot 23^{218} + \left(8 a^{14} + 19 a^{13} + 10 a^{12} + 7 a^{11} + 21 a^{10} + 14 a^{9} + 20 a^{8} + 11 a^{7} + 16 a^{5} + 21 a^{4} + 16 a^{3} + 3 a^{2} + 13 a + 5\right)\cdot 23^{219} + \left(21 a^{14} + 11 a^{13} + 15 a^{12} + 14 a^{11} + 22 a^{10} + 9 a^{9} + 11 a^{8} + 11 a^{7} + a^{6} + 16 a^{5} + 8 a^{4} + 12 a^{3} + a^{2} + 6 a + 4\right)\cdot 23^{220} + \left(10 a^{14} + 10 a^{13} + a^{12} + 4 a^{11} + 9 a^{10} + 7 a^{9} + 12 a^{8} + 6 a^{7} + 12 a^{6} + 17 a^{5} + 15 a^{4} + 7 a^{3} + 3 a^{2} + 11 a\right)\cdot 23^{221} + \left(3 a^{14} + 4 a^{13} + 17 a^{12} + 15 a^{11} + 12 a^{10} + 14 a^{9} + 11 a^{8} + 11 a^{7} + 18 a^{6} + a^{5} + 22 a^{4} + 5 a^{3} + 12 a^{2} + 2 a + 9\right)\cdot 23^{222} + \left(19 a^{14} + 20 a^{13} + 6 a^{12} + 14 a^{11} + 11 a^{10} + 18 a^{9} + 11 a^{8} + 15 a^{7} + 4 a^{6} + 5 a^{5} + 7 a^{4} + 16 a^{3} + 21 a^{2} + 18 a + 18\right)\cdot 23^{223} + \left(7 a^{14} + 12 a^{13} + 6 a^{12} + 5 a^{11} + 5 a^{10} + 16 a^{9} + 10 a^{8} + a^{7} + 11 a^{6} + 11 a^{5} + 15 a^{4} + 15 a^{3} + 3 a^{2} + 7 a\right)\cdot 23^{224} + \left(10 a^{14} + 10 a^{13} + 17 a^{12} + 3 a^{11} + 9 a^{10} + 15 a^{9} + 22 a^{8} + 11 a^{7} + 4 a^{6} + 2 a^{5} + 10 a^{4} + 20 a^{3} + 17 a^{2} + 14 a + 20\right)\cdot 23^{225} + \left(8 a^{14} + a^{13} + 4 a^{12} + 15 a^{11} + 14 a^{10} + 22 a^{9} + 22 a^{8} + 21 a^{6} + a^{5} + 5 a^{4} + 22 a^{3} + 4 a^{2} + 14 a + 2\right)\cdot 23^{226} + \left(17 a^{14} + 17 a^{13} + a^{12} + 16 a^{11} + a^{10} + a^{9} + 14 a^{8} + 11 a^{7} + 9 a^{6} + 21 a^{5} + 6 a^{4} + 20 a^{3} + 14 a^{2} + 18 a + 15\right)\cdot 23^{227} + \left(3 a^{14} + 14 a^{12} + 3 a^{11} + 6 a^{10} + 15 a^{9} + 11 a^{8} + 18 a^{7} + 5 a^{6} + 17 a^{5} + 8 a^{4} + 14 a^{3} + 5 a^{2} + 6 a + 21\right)\cdot 23^{228} + \left(a^{14} + 21 a^{13} + 7 a^{12} + 15 a^{11} + 22 a^{10} + 10 a^{9} + a^{8} + 13 a^{7} + 7 a^{6} + 22 a^{5} + 16 a^{4} + 21 a^{3} + 6 a^{2} + 15 a + 16\right)\cdot 23^{229} + \left(19 a^{14} + 2 a^{13} + 13 a^{12} + 21 a^{11} + 8 a^{10} + 5 a^{9} + 12 a^{8} + 6 a^{7} + 14 a^{6} + 18 a^{5} + 8 a^{4} + a^{3} + 18 a^{2} + 11 a\right)\cdot 23^{230} + \left(15 a^{14} + 6 a^{13} + a^{12} + 5 a^{11} + 19 a^{10} + 15 a^{9} + 4 a^{8} + 9 a^{7} + 21 a^{6} + 14 a^{5} + 21 a^{4} + 5 a^{3} + 21 a^{2} + 13 a + 9\right)\cdot 23^{231} + \left(12 a^{14} + 15 a^{13} + 4 a^{12} + 19 a^{11} + 7 a^{10} + a^{9} + 7 a^{8} + 15 a^{7} + 16 a^{6} + 18 a^{5} + 9 a^{4} + 14 a^{3} + 19 a^{2} + 4 a + 2\right)\cdot 23^{232} + \left(22 a^{14} + 16 a^{13} + 6 a^{12} + 8 a^{11} + 10 a^{9} + 22 a^{8} + 17 a^{7} + 16 a^{6} + 13 a^{5} + 20 a^{4} + 7 a^{3} + 6 a^{2} + 13 a + 19\right)\cdot 23^{233} + \left(14 a^{14} + 4 a^{13} + 9 a^{12} + 16 a^{11} + 22 a^{10} + 14 a^{9} + 4 a^{8} + a^{7} + 11 a^{6} + 16 a^{5} + a^{4} + 8 a^{3} + 15 a^{2} + 10 a + 7\right)\cdot 23^{234} + \left(5 a^{14} + 11 a^{13} + 10 a^{12} + 4 a^{11} + a^{9} + 18 a^{8} + 2 a^{7} + 15 a^{6} + 22 a^{5} + 19 a^{4} + 14 a^{3} + 14 a^{2} + 18 a + 8\right)\cdot 23^{235} + \left(20 a^{14} + 2 a^{13} + 21 a^{12} + a^{11} + 12 a^{10} + 12 a^{8} + 17 a^{7} + 8 a^{6} + 7 a^{4} + 22 a^{3} + 20 a^{2} + 19 a + 15\right)\cdot 23^{236} + \left(22 a^{14} + 5 a^{13} + 3 a^{12} + 22 a^{11} + 11 a^{10} + 18 a^{9} + 11 a^{8} + 5 a^{7} + 17 a^{6} + 3 a^{5} + 15 a^{4} + 6 a^{3} + a^{2} + 7 a + 12\right)\cdot 23^{237} + \left(17 a^{14} + 5 a^{13} + 13 a^{12} + 5 a^{11} + 10 a^{10} + 17 a^{9} + 10 a^{8} + 21 a^{7} + 2 a^{6} + 18 a^{5} + 15 a^{3} + 17 a^{2} + 8 a + 9\right)\cdot 23^{238} + \left(2 a^{14} + 2 a^{13} + 14 a^{11} + 16 a^{10} + 15 a^{9} + 16 a^{8} + 15 a^{7} + 21 a^{6} + 16 a^{5} + 18 a^{4} + 16 a^{3} + 21 a^{2} + 17 a + 5\right)\cdot 23^{239} + \left(3 a^{14} + 10 a^{13} + 16 a^{12} + 6 a^{11} + 11 a^{10} + 20 a^{9} + 8 a^{8} + 4 a^{7} + 8 a^{6} + a^{5} + 11 a^{4} + 7 a^{3} + 19 a^{2} + 11 a + 15\right)\cdot 23^{240} + \left(12 a^{14} + 22 a^{13} + 8 a^{12} + 5 a^{11} + 8 a^{9} + 5 a^{8} + 18 a^{7} + 17 a^{6} + 16 a^{5} + 8 a^{4} + 13 a^{3} + 14 a^{2} + 3 a + 9\right)\cdot 23^{241} + \left(19 a^{14} + 15 a^{13} + a^{12} + 14 a^{11} + 17 a^{10} + 15 a^{9} + 4 a^{8} + 17 a^{7} + 16 a^{6} + 19 a^{5} + 5 a^{4} + 17 a^{3} + 9 a^{2} + 6 a + 3\right)\cdot 23^{242} + \left(9 a^{14} + 13 a^{13} + 14 a^{12} + 21 a^{11} + 19 a^{10} + 4 a^{9} + 3 a^{8} + a^{7} + 2 a^{6} + 11 a^{5} + 6 a^{4} + 22 a^{3} + 10 a^{2} + 2 a + 14\right)\cdot 23^{243} + \left(15 a^{14} + 9 a^{13} + 6 a^{12} + 10 a^{11} + 2 a^{10} + 19 a^{9} + 8 a^{8} + 5 a^{7} + 12 a^{5} + 9 a^{4} + 12 a^{3} + 8 a^{2} + 6 a + 21\right)\cdot 23^{244} + \left(15 a^{14} + 3 a^{13} + 8 a^{12} + 17 a^{11} + 18 a^{10} + 19 a^{9} + 7 a^{8} + 15 a^{7} + 7 a^{6} + 2 a^{5} + 18 a^{4} + 18 a^{3} + 12 a^{2} + 7 a + 1\right)\cdot 23^{245} + \left(19 a^{14} + 17 a^{13} + 7 a^{12} + 19 a^{11} + 9 a^{10} + 17 a^{9} + 9 a^{8} + 17 a^{7} + 10 a^{6} + 4 a^{5} + 15 a^{4} + a^{3} + 11 a^{2} + 14 a + 8\right)\cdot 23^{246} + \left(4 a^{14} + 15 a^{13} + 15 a^{12} + 4 a^{11} + 2 a^{10} + 13 a^{9} + 9 a^{8} + 21 a^{7} + a^{6} + 18 a^{5} + 9 a^{4} + 10 a^{3} + 21 a^{2} + 5 a + 22\right)\cdot 23^{247} + \left(9 a^{14} + 6 a^{13} + 7 a^{12} + 13 a^{11} + 21 a^{10} + 18 a^{9} + 12 a^{8} + 13 a^{7} + 3 a^{6} + 12 a^{5} + 11 a^{4} + 9 a^{3} + 21 a^{2} + 5 a + 16\right)\cdot 23^{248} + \left(5 a^{13} + 5 a^{12} + 22 a^{11} + 18 a^{10} + 9 a^{9} + 17 a^{8} + 3 a^{7} + a^{6} + 4 a^{5} + 16 a^{4} + 22 a^{3} + 4 a + 20\right)\cdot 23^{249} + \left(4 a^{14} + 7 a^{13} + 16 a^{11} + 10 a^{10} + 9 a^{9} + 16 a^{8} + 5 a^{7} + 2 a^{6} + 16 a^{5} + 5 a^{4} + 12 a^{3} + 9 a^{2} + 12 a + 22\right)\cdot 23^{250} + \left(4 a^{14} + 3 a^{13} + 17 a^{12} + 13 a^{11} + 5 a^{10} + 8 a^{9} + 4 a^{8} + 22 a^{7} + 10 a^{6} + 7 a^{5} + 4 a^{4} + 17 a^{3} + 4 a^{2} + 10 a + 16\right)\cdot 23^{251} + \left(a^{14} + 20 a^{13} + 14 a^{12} + a^{11} + 12 a^{10} + 15 a^{9} + 21 a^{8} + 13 a^{7} + 8 a^{6} + 21 a^{5} + 9 a^{4} + a^{2} + 11 a + 12\right)\cdot 23^{252} + \left(14 a^{14} + 13 a^{13} + 11 a^{12} + 9 a^{11} + 8 a^{10} + 21 a^{9} + 15 a^{8} + 12 a^{7} + 20 a^{6} + 22 a^{5} + 2 a^{4} + 22 a^{3} + 5 a^{2} + 13 a + 20\right)\cdot 23^{253} + \left(3 a^{14} + 13 a^{13} + 19 a^{12} + 22 a^{11} + 21 a^{10} + 11 a^{9} + 3 a^{8} + 9 a^{6} + 2 a^{4} + 12 a^{3} + 5 a^{2} + 20 a + 10\right)\cdot 23^{254} + \left(12 a^{14} + 15 a^{13} + 2 a^{12} + 2 a^{11} + 15 a^{10} + 3 a^{9} + 12 a^{8} + 13 a^{7} + 20 a^{6} + 21 a^{5} + 12 a^{4} + 14 a^{3} + 17 a^{2} + a + 20\right)\cdot 23^{255} + \left(18 a^{14} + 7 a^{13} + 15 a^{12} + 22 a^{11} + 12 a^{10} + 13 a^{9} + 11 a^{8} + 22 a^{7} + 21 a^{6} + 2 a^{5} + 17 a^{4} + 9 a^{3} + 14 a^{2} + 7 a + 18\right)\cdot 23^{256} + \left(19 a^{14} + 6 a^{13} + 6 a^{11} + 11 a^{10} + 4 a^{9} + 9 a^{8} + a^{7} + 19 a^{6} + 18 a^{5} + 15 a^{4} + 18 a^{3} + 3 a + 4\right)\cdot 23^{257} + \left(3 a^{14} + 6 a^{13} + 2 a^{12} + 3 a^{11} + 3 a^{10} + 20 a^{9} + 12 a^{8} + 13 a^{6} + 17 a^{5} + 18 a^{4} + 15 a^{3} + 3 a^{2} + 6 a + 22\right)\cdot 23^{258} + \left(3 a^{14} + 5 a^{13} + 18 a^{12} + 13 a^{11} + 16 a^{10} + 9 a^{9} + 18 a^{8} + 11 a^{7} + 18 a^{6} + a^{5} + 21 a^{4} + 9 a^{3} + 13 a^{2} + 17 a + 16\right)\cdot 23^{259} + \left(19 a^{14} + 4 a^{13} + 13 a^{12} + 16 a^{11} + 7 a^{10} + 10 a^{9} + 9 a^{8} + 22 a^{7} + 21 a^{6} + 19 a^{5} + 15 a^{4} + 4 a^{3} + 8 a^{2} + 15 a + 6\right)\cdot 23^{260} + \left(6 a^{14} + 22 a^{13} + a^{12} + 11 a^{11} + 20 a^{10} + 2 a^{9} + 19 a^{8} + 6 a^{7} + 21 a^{6} + 19 a^{5} + 13 a^{4} + 18 a^{3} + 15 a^{2} + 18 a + 6\right)\cdot 23^{261} + \left(a^{14} + 15 a^{13} + 7 a^{12} + 8 a^{11} + 9 a^{10} + a^{9} + 10 a^{8} + 21 a^{7} + 12 a^{6} + 20 a^{5} + 7 a^{4} + 9 a^{3} + a^{2} + 4 a + 20\right)\cdot 23^{262} + \left(10 a^{14} + 7 a^{13} + 20 a^{12} + 20 a^{10} + 14 a^{9} + 21 a^{8} + 21 a^{7} + 22 a^{6} + 16 a^{5} + 7 a^{4} + 9 a^{3} + 17 a^{2} + 11 a + 14\right)\cdot 23^{263} + \left(14 a^{14} + 8 a^{13} + 3 a^{12} + 4 a^{11} + 3 a^{10} + 17 a^{9} + a^{8} + 6 a^{7} + 18 a^{6} + 2 a^{5} + 19 a^{4} + 15 a^{3} + 15 a^{2} + 5 a + 19\right)\cdot 23^{264} + \left(4 a^{14} + 15 a^{13} + 4 a^{12} + 15 a^{11} + 16 a^{10} + 2 a^{9} + 5 a^{8} + a^{7} + a^{6} + 21 a^{5} + 7 a^{4} + a^{3} + 15 a^{2} + 2 a + 3\right)\cdot 23^{265} + \left(22 a^{14} + 12 a^{13} + a^{12} + 13 a^{11} + 3 a^{10} + 3 a^{9} + 9 a^{8} + 19 a^{7} + 5 a^{6} + a^{5} + 18 a^{4} + 10 a^{3} + 10 a^{2} + 15 a + 9\right)\cdot 23^{266} + \left(13 a^{14} + 20 a^{13} + 17 a^{12} + 13 a^{11} + 8 a^{10} + 7 a^{9} + 14 a^{8} + 17 a^{7} + 6 a^{6} + a^{5} + 22 a^{4} + 12 a^{3} + 12 a^{2} + 16 a + 18\right)\cdot 23^{267} + \left(21 a^{14} + 18 a^{13} + 17 a^{12} + 6 a^{11} + 19 a^{10} + 9 a^{9} + 16 a^{8} + 17 a^{7} + 17 a^{6} + 8 a^{5} + 13 a^{4} + 5 a^{3} + 13 a^{2} + 11 a + 1\right)\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 14 }$ $=$ $16 a^{14} + a^{13} + 15 a^{12} + 19 a^{11} + 13 a^{10} + 15 a^{9} + 5 a^{8} + 8 a^{7} + 11 a^{6} + 7 a^{5} + 22 a^{4} + 22 a^{3} + 21 a^{2} + 12 a + 18 + \left(21 a^{14} + 19 a^{13} + a^{12} + 20 a^{11} + 2 a^{10} + 21 a^{9} + 22 a^{8} + 11 a^{7} + 19 a^{6} + a^{5} + 13 a^{4} + 20 a^{3} + 19 a^{2} + 11 a + 15\right)\cdot 23 + \left(20 a^{14} + 12 a^{13} + 9 a^{12} + 19 a^{11} + 14 a^{10} + 10 a^{9} + 14 a^{8} + 20 a^{7} + 4 a^{6} + a^{5} + 11 a^{4} + 11 a^{3} + 3 a^{2} + 17 a + 10\right)\cdot 23^{2} + \left(9 a^{14} + 6 a^{13} + 2 a^{11} + 2 a^{10} + 17 a^{9} + 21 a^{8} + 7 a^{7} + 16 a^{6} + 22 a^{5} + 2 a^{4} + 9 a^{3} + 3 a^{2} + 11 a\right)\cdot 23^{3} + \left(13 a^{14} + 5 a^{13} + 3 a^{12} + 6 a^{11} + 13 a^{10} + 5 a^{9} + a^{8} + 5 a^{7} + 21 a^{6} + 18 a^{5} + a^{4} + a^{3} + 3 a^{2} + 10\right)\cdot 23^{4} + \left(4 a^{14} + 16 a^{13} + 19 a^{12} + 15 a^{11} + 21 a^{10} + a^{9} + 9 a^{8} + 12 a^{7} + 20 a^{6} + 19 a^{5} + 6 a^{4} + 20 a^{3} + a^{2} + 20 a + 10\right)\cdot 23^{5} + \left(18 a^{14} + 19 a^{13} + 7 a^{12} + 20 a^{11} + 15 a^{10} + 9 a^{9} + 16 a^{8} + 7 a^{7} + 20 a^{6} + 10 a^{5} + 7 a^{4} + 20 a^{3} + 4 a^{2} + 6 a + 2\right)\cdot 23^{6} + \left(19 a^{14} + 15 a^{13} + 13 a^{12} + 10 a^{11} + 8 a^{10} + 17 a^{8} + 3 a^{7} + 18 a^{6} + 14 a^{5} + 3 a^{4} + 12 a^{3} + 21 a^{2} + 20 a + 20\right)\cdot 23^{7} + \left(3 a^{13} + 16 a^{12} + 2 a^{11} + 22 a^{10} + 8 a^{9} + 2 a^{8} + 3 a^{7} + 15 a^{6} + 11 a^{5} + 4 a^{4} + 17 a^{3} + 5 a^{2} + 14 a + 7\right)\cdot 23^{8} + \left(22 a^{14} + 3 a^{13} + 15 a^{12} + a^{11} + 15 a^{10} + 16 a^{9} + 10 a^{7} + 21 a^{6} + 14 a^{5} + 6 a^{4} + 7 a^{3} + a^{2} + 16 a + 21\right)\cdot 23^{9} + \left(20 a^{13} + 17 a^{12} + 3 a^{11} + 11 a^{9} + 8 a^{8} + 8 a^{7} + 7 a^{6} + 15 a^{5} + 3 a^{4} + 9 a^{3} + 8 a^{2} + 13 a + 1\right)\cdot 23^{10} + \left(13 a^{14} + 5 a^{13} + 5 a^{12} + 10 a^{11} + 17 a^{10} + 18 a^{9} + 3 a^{8} + 4 a^{7} + 21 a^{6} + 15 a^{5} + 12 a^{4} + 7 a^{3} + 11 a^{2} + 2 a + 21\right)\cdot 23^{11} + \left(13 a^{14} + 14 a^{13} + 5 a^{12} + 5 a^{11} + 17 a^{9} + 4 a^{8} + 12 a^{7} + 5 a^{6} + 22 a^{5} + 13 a^{4} + 13 a^{3} + 18 a + 2\right)\cdot 23^{12} + \left(19 a^{14} + 4 a^{13} + 16 a^{12} + 4 a^{11} + 17 a^{10} + 2 a^{9} + 13 a^{8} + 19 a^{7} + 22 a^{6} + a^{5} + 12 a^{4} + 2 a^{3} + 4 a^{2} + 21 a + 7\right)\cdot 23^{13} + \left(17 a^{14} + 4 a^{12} + 19 a^{11} + 9 a^{10} + 14 a^{9} + 11 a^{8} + 16 a^{7} + 13 a^{6} + 14 a^{5} + 6 a^{4} + 17 a^{3} + 5 a^{2} + 14 a + 1\right)\cdot 23^{14} + \left(5 a^{14} + 10 a^{13} + 10 a^{12} + 3 a^{11} + 11 a^{10} + 16 a^{9} + 5 a^{8} + 12 a^{7} + 20 a^{6} + 11 a^{5} + 17 a^{3} + 17 a^{2} + 15 a + 16\right)\cdot 23^{15} + \left(2 a^{14} + 8 a^{12} + 11 a^{11} + 12 a^{9} + 14 a^{8} + 4 a^{7} + 14 a^{6} + 13 a^{5} + 11 a^{4} + 22 a^{3} + 13 a^{2} + 3 a + 4\right)\cdot 23^{16} + \left(14 a^{14} + a^{13} + 8 a^{12} + 5 a^{11} + 21 a^{10} + 9 a^{9} + 21 a^{8} + 19 a^{7} + 16 a^{5} + 2 a^{4} + 22 a^{3} + 4 a^{2} + 16 a + 17\right)\cdot 23^{17} + \left(5 a^{14} + 14 a^{12} + 19 a^{11} + 19 a^{10} + 14 a^{9} + 16 a^{8} + 6 a^{7} + 19 a^{6} + 6 a^{5} + 13 a^{4} + 4 a^{3} + 16 a^{2} + 15 a\right)\cdot 23^{18} + \left(22 a^{14} + 2 a^{13} + 2 a^{12} + 20 a^{11} + 14 a^{10} + 12 a^{9} + 18 a^{8} + 13 a^{7} + 7 a^{6} + 12 a^{5} + 9 a^{4} + 15 a^{3} + 9 a^{2} + 6 a + 10\right)\cdot 23^{19} + \left(a^{14} + 20 a^{13} + 19 a^{12} + 16 a^{10} + 18 a^{9} + a^{8} + 14 a^{7} + 21 a^{6} + 2 a^{5} + 17 a^{4} + 15 a^{3} + 9 a^{2} + 20 a + 3\right)\cdot 23^{20} + \left(16 a^{14} + 15 a^{13} + 7 a^{12} + 10 a^{11} + 13 a^{10} + 20 a^{9} + 16 a^{8} + 22 a^{7} + 21 a^{6} + 18 a^{5} + 8 a^{4} + 6 a^{3} + 12 a^{2} + 11 a + 15\right)\cdot 23^{21} + \left(3 a^{14} + 4 a^{13} + 10 a^{12} + 15 a^{11} + 6 a^{10} + 22 a^{9} + 19 a^{7} + 11 a^{6} + 5 a^{5} + 4 a^{4} + 15 a^{3} + 16 a^{2} + 21 a + 6\right)\cdot 23^{22} + \left(a^{14} + 4 a^{13} + 18 a^{12} + 3 a^{11} + 13 a^{10} + 7 a^{9} + a^{8} + 20 a^{7} + 20 a^{6} + 9 a^{5} + 15 a^{4} + 22 a^{3} + 14 a^{2} + 8 a + 21\right)\cdot 23^{23} + \left(17 a^{14} + 14 a^{13} + 7 a^{12} + 11 a^{11} + 5 a^{10} + 7 a^{9} + 22 a^{8} + 15 a^{7} + a^{6} + 21 a^{5} + 22 a^{4} + 16 a^{3} + 2 a^{2} + 14 a + 16\right)\cdot 23^{24} + \left(18 a^{14} + 17 a^{13} + 21 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7\right)\cdot 23^{36} + \left(10 a^{14} + 7 a^{13} + 16 a^{12} + 12 a^{11} + 2 a^{10} + 6 a^{9} + 11 a^{8} + 17 a^{6} + 14 a^{5} + 22 a^{4} + a^{3} + 7 a^{2} + 12 a + 1\right)\cdot 23^{37} + \left(17 a^{13} + 20 a^{12} + a^{11} + 13 a^{10} + 16 a^{9} + 19 a^{8} + 11 a^{7} + 18 a^{6} + 22 a^{5} + 10 a^{4} + 4 a^{3} + 10 a^{2} + 7 a + 4\right)\cdot 23^{38} + \left(a^{14} + 18 a^{13} + 18 a^{12} + 12 a^{11} + 3 a^{10} + a^{9} + 4 a^{7} + 11 a^{6} + 19 a^{5} + 8 a^{4} + 10 a^{3} + 17 a^{2} + 12 a + 6\right)\cdot 23^{39} + \left(18 a^{14} + 14 a^{13} + 17 a^{12} + 3 a^{11} + 15 a^{10} + 7 a^{9} + 7 a^{8} + 19 a^{7} + 15 a^{6} + 8 a^{5} + 7 a^{4} + 7 a^{3} + 20 a^{2} + 19 a + 16\right)\cdot 23^{40} + \left(4 a^{14} + 12 a^{13} + 7 a^{12} + 3 a^{11} + 17 a^{10} + 15 a^{9} + 12 a^{8} + 8 a^{7} + 17 a^{6} + 22 a^{5} + 7 a^{4} + 19 a^{3} + 5 a^{2} + 16 a + 15\right)\cdot 23^{41} + \left(7 a^{14} + 2 a^{12} + 17 a^{10} + 20 a^{9} + 10 a^{8} + 19 a^{7} + 16 a^{6} + 9 a^{5} + 22 a^{4} + 12 a^{3} + 8 a^{2} + 3 a + 22\right)\cdot 23^{42} + \left(a^{14} + 19 a^{13} + 12 a^{12} + 18 a^{11} + 14 a^{10} + 8 a^{9} + 8 a^{8} + 5 a^{7} + 9 a^{6} + 7 a^{5} + 16 a^{4} + 21 a^{3} + 12 a^{2} + 2 a + 12\right)\cdot 23^{43} + \left(3 a^{14} + 9 a^{13} + 4 a^{12} + a^{11} + 10 a^{10} + 19 a^{9} + 20 a^{8} + 3 a^{7} + 8 a^{6} + 3 a^{5} + 6 a^{4} + 4 a^{3} + a^{2} + 21 a + 18\right)\cdot 23^{44} + \left(13 a^{14} + 13 a^{13} + a^{12} + 9 a^{11} + 6 a^{10} + 19 a^{9} + 3 a^{8} + 10 a^{7} + 19 a^{6} + 13 a^{5} + 11 a^{4} + 10 a^{3} + 5 a^{2} + 13 a + 17\right)\cdot 23^{45} + \left(12 a^{14} + 3 a^{13} + 14 a^{12} + 7 a^{11} + 20 a^{10} + 16 a^{9} + 5 a^{8} + 9 a^{7} + 5 a^{6} + 6 a^{5} + 15 a^{4} + 4 a^{3} + 13 a^{2} + 9 a + 11\right)\cdot 23^{46} + \left(4 a^{14} + 17 a^{13} + 18 a^{12} + 4 a^{11} + 16 a^{10} + 3 a^{9} + 11 a^{8} + 12 a^{7} + 6 a^{6} + 3 a^{5} + 5 a^{4} + 9 a^{3} + 9 a^{2} + 7 a + 2\right)\cdot 23^{47} + \left(a^{14} + 15 a^{13} + 15 a^{12} + 8 a^{11} + 15 a^{10} + 11 a^{9} + 18 a^{8} + 2 a^{7} + 19 a^{5} + a^{4} + a^{3} + 22 a^{2} + 16 a + 20\right)\cdot 23^{48} + \left(13 a^{14} + 22 a^{13} + 18 a^{12} + 8 a^{11} + a^{10} + 3 a^{9} + 22 a^{8} + 13 a^{7} + 12 a^{6} + 22 a^{5} + 12 a^{4} + 6 a^{3} + 11 a^{2} + 3 a + 8\right)\cdot 23^{49} + \left(10 a^{14} + 6 a^{13} + 7 a^{12} + 4 a^{11} + 16 a^{10} + 3 a^{9} + 18 a^{7} + 16 a^{5} + 16 a^{4} + 20 a^{3} + 17 a^{2} + 13 a + 20\right)\cdot 23^{50} + \left(5 a^{14} + 2 a^{13} + 19 a^{12} + 11 a^{11} + 8 a^{10} + 6 a^{9} + 13 a^{8} + 22 a^{7} + 22 a^{6} + 9 a^{5} + 7 a^{4} + 14 a^{3} + 11 a + 11\right)\cdot 23^{51} + \left(21 a^{14} + 7 a^{13} + 3 a^{12} + 4 a^{11} + 17 a^{10} + 9 a^{9} + 6 a^{8} + 20 a^{7} + 22 a^{6} + 12 a^{5} + 18 a^{3} + 16 a^{2} + 19 a + 14\right)\cdot 23^{52} + \left(9 a^{14} + 9 a^{13} + a^{12} + 5 a^{11} + 15 a^{10} + 19 a^{9} + 9 a^{8} + 19 a^{7} + 6 a^{6} + 7 a^{5} + 17 a^{4} + 8 a^{3} + 22 a + 6\right)\cdot 23^{53} + \left(8 a^{14} + 20 a^{13} + 19 a^{12} + 14 a^{11} + 13 a^{10} + 5 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+ 12\right)\cdot 23^{71} + \left(14 a^{14} + 5 a^{13} + 21 a^{12} + 2 a^{11} + 22 a^{10} + 6 a^{9} + 14 a^{8} + 14 a^{7} + 4 a^{6} + 5 a^{5} + 2 a^{4} + 16 a^{3} + 22 a^{2} + 6 a + 10\right)\cdot 23^{72} + \left(9 a^{14} + 8 a^{13} + 6 a^{12} + 20 a^{11} + a^{10} + 6 a^{9} + 19 a^{8} + 10 a^{7} + 5 a^{6} + 6 a^{5} + 18 a^{4} + 2 a^{3} + 12 a^{2} + 6 a + 2\right)\cdot 23^{73} + \left(4 a^{14} + 12 a^{13} + 4 a^{12} + 8 a^{11} + 16 a^{10} + 19 a^{9} + 3 a^{8} + 2 a^{7} + 7 a^{6} + 3 a^{5} + a^{4} + 18 a^{2} + 6 a + 22\right)\cdot 23^{74} + \left(10 a^{13} + 13 a^{12} + 17 a^{11} + 5 a^{10} + 22 a^{9} + 18 a^{8} + 6 a^{7} + 22 a^{6} + 11 a^{5} + 4 a^{4} + 22 a^{3} + 9 a^{2} + 13 a + 14\right)\cdot 23^{75} + \left(21 a^{14} + a^{13} + a^{12} + 8 a^{11} + 6 a^{10} + 5 a^{9} + 19 a^{8} + 13 a^{7} + a^{6} + 22 a^{5} + 9 a^{4} + 7 a^{3} + 5 a^{2} + 16 a + 22\right)\cdot 23^{76} + \left(14 a^{14} + 4 a^{13} + 16 a^{12} + 6 a^{11} + 13 a^{10} + 6 a^{9} + 12 a^{8} + 17 a^{7} + 13 a^{6} + a^{5} + 6 a^{4} + 6 a^{3} + a^{2} + 19 a + 22\right)\cdot 23^{77} + \left(11 a^{14} + 20 a^{13} + a^{12} + 22 a^{11} + 12 a^{10} + 21 a^{9} + a^{8} + 10 a^{7} + 14 a^{6} + 5 a^{5} + 6 a^{4} + 10 a^{3} + 4 a^{2} + 21 a + 5\right)\cdot 23^{78} + \left(6 a^{14} + 6 a^{13} + 2 a^{12} + 6 a^{11} + 17 a^{10} + a^{9} + 7 a^{8} + 16 a^{7} + 21 a^{6} + 9 a^{5} + 2 a^{4} + 19 a^{3} + 6 a^{2} + 20 a + 12\right)\cdot 23^{79} + \left(12 a^{14} + 17 a^{13} + 12 a^{12} + 11 a^{11} + 17 a^{10} + 8 a^{9} + 14 a^{8} + 15 a^{7} + 13 a^{5} + 4 a^{4} + 12 a^{3} + 12 a^{2} + 18 a + 1\right)\cdot 23^{80} + \left(8 a^{14} + 22 a^{13} + 6 a^{12} + a^{10} + 5 a^{9} + a^{8} + 22 a^{7} + 19 a^{6} + 5 a^{5} + 13 a^{4} + 6 a^{3} + 11 a^{2} + 10 a + 18\right)\cdot 23^{81} + \left(3 a^{14} + 15 a^{13} + 19 a^{12} + 4 a^{11} + 3 a^{10} + 11 a^{9} + 10 a^{8} + 14 a^{7} + 8 a^{6} + 14 a^{5} + 14 a^{4} + 13 a^{3} + 15 a^{2} + 13 a + 21\right)\cdot 23^{82} + \left(13 a^{14} + 4 a^{13} + 15 a^{12} + 16 a^{11} + 5 a^{10} + 12 a^{9} + 14 a^{8} + 10 a^{7} + 9 a^{6} + 21 a^{5} + 11 a^{4} + 11 a^{3} + a^{2} + 12 a + 21\right)\cdot 23^{83} + \left(13 a^{14} + 4 a^{13} + 21 a^{12} + 15 a^{11} + 22 a^{10} + 6 a^{9} + 17 a^{8} + 16 a^{7} + 12 a^{6} + 13 a^{5} + 15 a^{4} + 14 a^{3} + 18 a^{2} + 10 a + 10\right)\cdot 23^{84} + \left(16 a^{14} + 19 a^{13} + 16 a^{12} + 18 a^{11} + 17 a^{10} + 22 a^{9} + a^{8} + 6 a^{7} + 18 a^{6} + 3 a^{5} + 12 a^{4} + 8 a^{3} + 16 a^{2} + 9 a + 1\right)\cdot 23^{85} + \left(8 a^{14} + 20 a^{12} + 2 a^{10} + 3 a^{9} + 16 a^{8} + 20 a^{7} + 21 a^{6} + 4 a^{5} + 8 a^{4} + 19 a^{3} + 11 a^{2} + 17 a + 3\right)\cdot 23^{86} + \left(5 a^{14} + 8 a^{13} + a^{12} + 17 a^{11} + 16 a^{10} + 7 a^{9} + 3 a^{8} + 10 a^{6} + 8 a^{5} + 19 a^{4} + 10 a^{3} + 7 a^{2} + 3 a + 5\right)\cdot 23^{87} + \left(20 a^{14} + 13 a^{13} + 22 a^{12} + 11 a^{11} + 18 a^{10} + 13 a^{9} + 16 a^{8} + 10 a^{7} + 4 a^{6} + 12 a^{5} + 8 a^{4} + 3 a^{3} + 22 a^{2} + 7 a + 11\right)\cdot 23^{88} + \left(7 a^{14} + 12 a^{13} + 5 a^{12} + 3 a^{11} + 4 a^{10} + 11 a^{9} + 7 a^{8} + 21 a^{7} + 6 a^{6} + 4 a^{5} + 16 a^{4} + 7 a^{3} + 3 a^{2} + 17 a\right)\cdot 23^{89} + \left(8 a^{14} + 12 a^{13} + 14 a^{12} + 7 a^{11} + 11 a^{10} + 10 a^{9} + 2 a^{6} + 22 a^{5} + 9 a^{4} + 3 a^{3} + 9 a^{2} + 3 a + 14\right)\cdot 23^{90} + \left(20 a^{14} + 21 a^{13} + 7 a^{12} + 20 a^{10} + 9 a^{9} + 12 a^{8} + 13 a^{7} + 10 a^{6} + 10 a^{5} + 3 a^{4} + 20 a^{2} + 13 a + 19\right)\cdot 23^{91} + \left(3 a^{14} + 18 a^{13} + 8 a^{11} + a^{10} + 10 a^{9} + 5 a^{8} + 13 a^{7} + 6 a^{6} + 13 a^{5} + 22 a^{4} + 13 a^{3} + 2 a^{2} + 14 a\right)\cdot 23^{92} + \left(2 a^{14} + 19 a^{13} + 3 a^{12} + a^{11} + 12 a^{10} + 17 a^{9} + 18 a^{8} + 18 a^{5} + 7 a^{4} + 7 a^{3} + 6 a^{2} + 2 a + 4\right)\cdot 23^{93} + \left(3 a^{14} + 19 a^{13} + 8 a^{12} + 9 a^{11} + 12 a^{10} + 14 a^{9} + 5 a^{8} + 12 a^{7} + 2 a^{6} + 17 a^{5} + 20 a^{4} + 15 a^{3} + 2 a^{2} + 20 a + 19\right)\cdot 23^{94} + \left(19 a^{14} + 10 a^{13} + 20 a^{12} + 6 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\left(22 a^{13} + 12 a^{12} + 13 a^{11} + 11 a^{10} + 14 a^{9} + 14 a^{8} + 16 a^{7} + 14 a^{6} + 19 a^{5} + 8 a^{4} + 6 a^{3} + 17 a^{2} + 3 a + 3\right)\cdot 23^{101} + \left(16 a^{13} + 18 a^{12} + 19 a^{11} + 13 a^{10} + 17 a^{9} + 8 a^{8} + 11 a^{7} + 22 a^{6} + 20 a^{5} + 4 a^{4} + 18 a^{3} + 2 a^{2} + 4 a + 15\right)\cdot 23^{102} + \left(2 a^{14} + 9 a^{13} + 22 a^{12} + 9 a^{11} + 10 a^{10} + 21 a^{9} + 5 a^{8} + 11 a^{7} + a^{6} + 8 a^{5} + 16 a^{4} + 8 a^{3} + 14 a^{2} + 12 a + 5\right)\cdot 23^{103} + \left(5 a^{14} + 11 a^{13} + 7 a^{12} + 8 a^{11} + 13 a^{10} + 6 a^{9} + 18 a^{8} + 21 a^{7} + 15 a^{6} + 18 a^{5} + 9 a^{4} + 21 a^{3} + 2 a^{2} + 6 a + 2\right)\cdot 23^{104} + \left(10 a^{14} + 22 a^{13} + 2 a^{11} + 18 a^{10} + 21 a^{9} + 3 a^{8} + 8 a^{7} + 10 a^{6} + 4 a^{4} + 18 a^{3} + 21 a + 3\right)\cdot 23^{105} + \left(2 a^{14} + 3 a^{13} + 9 a^{12} + 16 a^{11} + 15 a^{10} + 7 a^{9} + 2 a^{8} + a^{7} + 10 a^{6} + 12 a^{5} + 16 a^{4} + 18 a^{3} + 14 a^{2} + 7 a + 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9 a^{2} + 21 a + 9\right)\cdot 23^{112} + \left(16 a^{14} + 5 a^{13} + 4 a^{12} + 3 a^{11} + 21 a^{10} + 15 a^{9} + 13 a^{8} + 15 a^{6} + 17 a^{5} + 21 a^{4} + 8 a^{3} + 21 a^{2} + 9 a + 6\right)\cdot 23^{113} + \left(21 a^{14} + a^{13} + 12 a^{12} + a^{11} + 5 a^{10} + 17 a^{8} + 6 a^{7} + 21 a^{6} + 6 a^{5} + 14 a^{4} + 10 a^{3} + 16 a^{2} + 8 a + 2\right)\cdot 23^{114} + \left(7 a^{14} + 3 a^{13} + 13 a^{12} + 11 a^{11} + 22 a^{10} + 13 a^{9} + 14 a^{8} + 17 a^{7} + 17 a^{6} + 3 a^{5} + 19 a^{4} + 4 a^{3} + 14 a^{2} + 9 a + 22\right)\cdot 23^{115} + \left(9 a^{14} + 16 a^{13} + 20 a^{12} + 19 a^{11} + 18 a^{10} + 9 a^{8} + 16 a^{7} + 4 a^{6} + a^{5} + 22 a^{4} + 18 a^{3} + 20 a^{2} + 16 a + 12\right)\cdot 23^{116} + \left(6 a^{14} + 9 a^{13} + 2 a^{12} + 17 a^{11} + 21 a^{10} + 12 a^{9} + 16 a^{8} + 22 a^{7} + 21 a^{6} + 16 a^{5} + 15 a^{4} + 6 a^{3} + 21 a^{2} + 22 a + 15\right)\cdot 23^{117} + \left(5 a^{14} + 20 a^{13} + 18 a^{12} + 20 a^{11} + 6 a^{10} + 20 a^{9} + 2 a^{8} + 9 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a^{7} + 12 a^{5} + 18 a^{4} + 10 a^{3} + 4 a^{2} + 19 a + 2\right)\cdot 23^{130} + \left(5 a^{13} + 22 a^{11} + 8 a^{10} + 8 a^{9} + 13 a^{8} + a^{7} + 3 a^{6} + 21 a^{5} + 11 a^{4} + 2 a^{3} + 18 a^{2} + 7 a + 10\right)\cdot 23^{131} + \left(8 a^{14} + 11 a^{13} + 13 a^{12} + 13 a^{11} + 4 a^{10} + 17 a^{9} + 15 a^{8} + 11 a^{7} + 12 a^{6} + 16 a^{5} + 18 a^{4} + 18 a^{3} + 19 a^{2} + 19 a + 2\right)\cdot 23^{132} + \left(4 a^{14} + 11 a^{13} + 9 a^{12} + 21 a^{10} + 3 a^{9} + 19 a^{8} + 19 a^{7} + 15 a^{6} + 11 a^{5} + 15 a^{4} + 19 a^{3} + 7 a^{2} + 7 a + 22\right)\cdot 23^{133} + \left(4 a^{14} + 8 a^{13} + 4 a^{12} + 5 a^{11} + 10 a^{10} + 4 a^{9} + 13 a^{8} + 5 a^{7} + 19 a^{6} + 11 a^{5} + 7 a^{4} + 6 a^{2} + 19 a + 1\right)\cdot 23^{134} + \left(11 a^{14} + 13 a^{13} + 5 a^{12} + 13 a^{11} + 17 a^{10} + a^{9} + 16 a^{8} + 4 a^{7} + 8 a^{6} + 13 a^{5} + 18 a^{4} + 22 a^{3} + 12 a^{2} + 22 a + 19\right)\cdot 23^{135} + \left(5 a^{14} + 19 a^{13} + 14 a^{12} + 22 a^{11} + 4 a^{10} + 13 a^{9} + 2 a^{8} + 3 a^{7} + a^{6} + 10 a^{5} + 14 a^{4} + 5 a^{3} + 6 a^{2} + 17 a + 20\right)\cdot 23^{136} + \left(5 a^{14} + 17 a^{13} + 12 a^{12} + 19 a^{11} + 15 a^{10} + 7 a^{9} + 22 a^{8} + 15 a^{7} + 4 a^{6} + 4 a^{5} + 20 a^{4} + 22 a^{3} + 3 a^{2} + a + 6\right)\cdot 23^{137} + \left(9 a^{14} + 8 a^{13} + 16 a^{12} + 15 a^{10} + 6 a^{9} + 17 a^{8} + 17 a^{7} + 16 a^{6} + 3 a^{5} + 10 a^{3} + a^{2} + 6 a + 15\right)\cdot 23^{138} + \left(18 a^{14} + a^{12} + 4 a^{11} + 15 a^{10} + 2 a^{9} + 9 a^{8} + 7 a^{7} + 6 a^{6} + 11 a^{4} + 20 a^{3} + 7 a^{2} + 14 a + 5\right)\cdot 23^{139} + \left(a^{14} + 22 a^{13} + 7 a^{12} + 4 a^{11} + 16 a^{10} + 15 a^{9} + 20 a^{8} + 12 a^{7} + 13 a^{6} + 12 a^{5} + 19 a^{4} + a^{3} + 16 a^{2} + 5\right)\cdot 23^{140} + \left(a^{14} + 10 a^{13} + 9 a^{12} + 5 a^{11} + 4 a^{10} + 13 a^{9} + 14 a^{8} + 5 a^{7} + 3 a^{6} + 14 a^{5} + 13 a^{4} + 15 a^{3} + 21 a^{2} + 6 a + 15\right)\cdot 23^{141} + \left(22 a^{14} + 22 a^{13} + 22 a^{12} + 12 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\left(8 a^{14} + 21 a^{13} + 5 a^{12} + 4 a^{11} + 13 a^{10} + 17 a^{9} + 2 a^{8} + 18 a^{7} + 2 a^{6} + 9 a^{5} + 17 a^{4} + 10 a^{3} + 15 a^{2} + 4 a + 17\right)\cdot 23^{160} + \left(15 a^{14} + 22 a^{13} + 15 a^{12} + 8 a^{11} + 2 a^{10} + 7 a^{9} + 12 a^{8} + 4 a^{7} + a^{6} + 8 a^{5} + 10 a^{4} + 16 a^{3} + 22 a^{2} + 16 a + 10\right)\cdot 23^{161} + \left(15 a^{14} + 3 a^{13} + 16 a^{12} + a^{11} + 19 a^{10} + 10 a^{9} + 13 a^{8} + 7 a^{7} + 20 a^{6} + 6 a^{5} + 5 a^{4} + 8 a^{3} + 8 a^{2} + 4 a + 4\right)\cdot 23^{162} + \left(8 a^{14} + 21 a^{13} + 19 a^{12} + 9 a^{11} + 7 a^{10} + 16 a^{9} + 9 a^{8} + 21 a^{7} + 16 a^{6} + 6 a^{5} + 3 a^{4} + 16 a^{3} + 22 a^{2} + 8 a + 21\right)\cdot 23^{163} + \left(17 a^{14} + 10 a^{13} + 9 a^{12} + 19 a^{11} + 5 a^{10} + 3 a^{9} + 14 a^{8} + 2 a^{7} + 15 a^{6} + 20 a^{5} + 12 a^{4} + 2 a^{3} + 7 a^{2} + 19 a + 22\right)\cdot 23^{164} + \left(21 a^{14} + 22 a^{13} + 15 a^{12} + 16 a^{11} + 17 a^{10} + 20 a^{9} + 14 a^{8} + 13 a^{7} + 6 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23^{176} + \left(4 a^{14} + 16 a^{13} + 18 a^{12} + 14 a^{11} + 14 a^{10} + 8 a^{9} + 5 a^{8} + 2 a^{7} + 12 a^{6} + 13 a^{5} + 2 a^{4} + 7 a^{3} + 4 a^{2} + 10 a + 18\right)\cdot 23^{177} + \left(9 a^{14} + 17 a^{13} + 2 a^{12} + 6 a^{11} + 18 a^{10} + 17 a^{9} + 3 a^{8} + 3 a^{7} + 11 a^{6} + 8 a^{5} + 13 a^{4} + 14 a^{3} + 15 a^{2} + 20 a + 13\right)\cdot 23^{178} + \left(13 a^{14} + 6 a^{13} + 18 a^{12} + 12 a^{11} + 19 a^{10} + 15 a^{9} + 22 a^{8} + 14 a^{7} + 3 a^{6} + 18 a^{5} + 20 a^{4} + 14 a^{3} + 6 a^{2} + 6 a + 8\right)\cdot 23^{179} + \left(16 a^{14} + 6 a^{13} + 5 a^{12} + 10 a^{11} + 7 a^{10} + 18 a^{9} + 11 a^{8} + 11 a^{7} + 12 a^{6} + 3 a^{5} + 13 a^{4} + 3 a^{3} + 8 a + 2\right)\cdot 23^{180} + \left(14 a^{14} + 12 a^{13} + 19 a^{12} + 2 a^{11} + 13 a^{10} + 18 a^{8} + 3 a^{7} + 14 a^{6} + 20 a^{5} + 18 a^{4} + 13 a^{3} + 4 a^{2} + 22 a + 17\right)\cdot 23^{181} + \left(8 a^{14} + 4 a^{13} + 6 a^{12} + 10 a^{11} + 2 a^{10} + 3 a^{9} + 9 a^{8} + 7 a^{7} + a^{5} + 6 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10 a^{11} + 22 a^{10} + 13 a^{9} + 12 a^{8} + 22 a^{7} + 5 a^{6} + a^{5} + 19 a^{4} + 12 a^{3} + 5 a^{2} + 16 a\right)\cdot 23^{194} + \left(19 a^{14} + 18 a^{13} + 16 a^{12} + 12 a^{11} + 6 a^{10} + 20 a^{9} + 15 a^{8} + 22 a^{6} + 9 a^{5} + 6 a^{4} + 5 a^{3} + a^{2} + 3 a + 20\right)\cdot 23^{195} + \left(18 a^{14} + 18 a^{13} + 4 a^{12} + 13 a^{11} + 5 a^{10} + 3 a^{9} + 7 a^{8} + 4 a^{7} + 3 a^{6} + 14 a^{5} + 14 a^{4} + 20 a^{3} + 7 a^{2} + 22 a + 11\right)\cdot 23^{196} + \left(17 a^{14} + 6 a^{13} + 2 a^{12} + 19 a^{11} + 10 a^{10} + 10 a^{9} + 8 a^{8} + 16 a^{7} + 7 a^{5} + 16 a^{3} + 13 a^{2} + 3 a + 16\right)\cdot 23^{197} + \left(10 a^{14} + 2 a^{13} + 12 a^{12} + 6 a^{11} + 5 a^{10} + 9 a^{9} + 12 a^{8} + 8 a^{7} + 11 a^{6} + 10 a^{5} + 8 a^{4} + 5 a^{3} + 21 a^{2} + 11 a + 4\right)\cdot 23^{198} + \left(20 a^{14} + 18 a^{13} + 5 a^{12} + 10 a^{11} + 2 a^{10} + 6 a^{9} + a^{8} + 19 a^{7} + 12 a^{6} + 2 a^{5} + 20 a^{4} + 18 a^{3} + 9 a^{2} + a + 8\right)\cdot 23^{199} + \left(19 a^{14} + 14 a^{13} + 20 a^{12} + 21 a^{11} + a^{9} + 7 a^{8} + 10 a^{7} + 15 a^{6} + 22 a^{5} + 15 a^{4} + 17 a^{3} + 7 a^{2} + 14 a + 17\right)\cdot 23^{200} + \left(12 a^{14} + 2 a^{13} + 14 a^{12} + 4 a^{11} + 15 a^{10} + 15 a^{9} + 17 a^{8} + 5 a^{6} + 10 a^{5} + 8 a^{4} + 15 a^{3} + 3 a^{2} + 6\right)\cdot 23^{201} + \left(16 a^{14} + 12 a^{13} + 22 a^{12} + 14 a^{10} + 15 a^{9} + 14 a^{8} + a^{7} + 14 a^{6} + 10 a^{5} + 19 a^{4} + 2 a^{3} + 4 a^{2} + 11 a + 8\right)\cdot 23^{202} + \left(13 a^{14} + 20 a^{13} + 20 a^{12} + 13 a^{11} + 19 a^{10} + 10 a^{9} + 17 a^{8} + 8 a^{7} + 19 a^{6} + 14 a^{5} + 2 a^{4} + 5 a^{3} + a^{2} + a + 18\right)\cdot 23^{203} + \left(18 a^{14} + 22 a^{13} + 9 a^{12} + 12 a^{10} + 15 a^{9} + 20 a^{8} + 10 a^{7} + 7 a^{6} + 11 a^{5} + 7 a^{4} + 7 a^{3} + a^{2} + 20 a + 9\right)\cdot 23^{204} + \left(2 a^{14} + 17 a^{13} + a^{12} + 18 a^{11} + 13 a^{10} + 15 a^{9} + 4 a^{8} + 10 a^{7} + 12 a^{6} + 18 a^{5} + 7 a^{4} + 9 a^{3} + 16 a^{2} + 12 a + 22\right)\cdot 23^{205} + \left(15 a^{14} + 16 a^{13} + 12 a^{12} + 15 a^{11} + 12 a^{10} + 18 a^{9} + 13 a^{8} + 18 a^{6} + 2 a^{5} + 20 a^{4} + 12 a^{2} + 13 a + 20\right)\cdot 23^{206} + \left(11 a^{14} + 6 a^{13} + 11 a^{12} + 8 a^{11} + 9 a^{10} + 21 a^{9} + 16 a^{8} + 7 a^{7} + 11 a^{6} + 22 a^{5} + 21 a^{4} + 5 a^{3} + 6 a^{2} + 13 a + 20\right)\cdot 23^{207} + \left(a^{14} + 20 a^{13} + 11 a^{12} + 12 a^{11} + 4 a^{10} + 3 a^{9} + 4 a^{8} + 21 a^{7} + 3 a^{6} + 13 a^{5} + 14 a^{4} + 4 a^{3} + 9 a^{2} + 6 a + 21\right)\cdot 23^{208} + \left(15 a^{14} + 13 a^{12} + 13 a^{11} + 22 a^{10} + 13 a^{9} + 3 a^{8} + 20 a^{7} + 4 a^{6} + 5 a^{5} + 16 a^{4} + 9 a^{2} + 9 a + 9\right)\cdot 23^{209} + \left(19 a^{14} + 6 a^{13} + 8 a^{12} + 11 a^{11} + 12 a^{10} + 14 a^{9} + 18 a^{8} + 22 a^{7} + 16 a^{6} + 12 a^{5} + 3 a^{4} + 12 a^{3} + 5 a^{2} + 8 a + 6\right)\cdot 23^{210} + \left(8 a^{14} + a^{13} + 14 a^{12} + 2 a^{11} + 16 a^{10} + 6 a^{9} + 5 a^{8} + 10 a^{7} + 21 a^{6} + 3 a^{5} + 2 a^{4} + 10 a^{3} + 20 a^{2} + 4 a + 12\right)\cdot 23^{211} + \left(21 a^{14} + 19 a^{13} + 6 a^{12} + 20 a^{11} + 4 a^{10} + 10 a^{9} + 7 a^{8} + 13 a^{7} + 17 a^{6} + a^{5} + 18 a^{4} + 20 a^{3} + 4 a^{2} + 5 a + 6\right)\cdot 23^{212} + \left(20 a^{14} + 4 a^{13} + 2 a^{12} + 19 a^{11} + 6 a^{10} + 16 a^{9} + 15 a^{8} + 6 a^{7} + 2 a^{6} + 12 a^{5} + 9 a^{4} + 22 a^{3} + 12 a^{2} + 8\right)\cdot 23^{213} + \left(11 a^{14} + 18 a^{13} + 15 a^{12} + 15 a^{11} + 19 a^{10} + 2 a^{9} + 19 a^{8} + 5 a^{7} + 19 a^{5} + 3 a^{4} + 17 a^{3} + 15 a^{2} + 11 a + 5\right)\cdot 23^{214} + \left(7 a^{14} + 16 a^{13} + 9 a^{12} + 15 a^{11} + 3 a^{10} + 15 a^{9} + 15 a^{8} + 6 a^{7} + 12 a^{6} + 19 a^{5} + 13 a^{4} + 19 a^{3} + 16 a^{2} + 13 a + 5\right)\cdot 23^{215} + \left(17 a^{14} + 15 a^{13} + 18 a^{12} + 12 a^{11} + 21 a^{10} + 11 a^{9} + 18 a^{8} + 4 a^{7} + 3 a^{6} + 6 a^{5} + 20 a^{4} + 16 a^{3} + 10 a^{2} + 19 a + 3\right)\cdot 23^{216} + \left(3 a^{14} + 21 a^{13} + 22 a^{12} + 3 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\left(14 a^{14} + 22 a^{13} + 12 a^{12} + 21 a^{11} + 9 a^{10} + 4 a^{9} + 2 a^{8} + 22 a^{7} + 21 a^{6} + 2 a^{5} + 14 a^{4} + 21 a^{3} + 2 a^{2} + 3 a + 21\right)\cdot 23^{223} + \left(19 a^{13} + 8 a^{12} + 9 a^{11} + 4 a^{10} + 4 a^{9} + 3 a^{8} + 2 a^{7} + 8 a^{6} + 3 a^{5} + 20 a^{4} + 3 a^{3} + 15 a^{2} + 14 a + 4\right)\cdot 23^{224} + \left(2 a^{14} + 8 a^{13} + 3 a^{12} + 19 a^{11} + 4 a^{10} + 4 a^{8} + 10 a^{7} + 5 a^{6} + 21 a^{5} + 8 a^{4} + 22 a^{3} + 7 a^{2} + 11 a + 5\right)\cdot 23^{225} + \left(13 a^{14} + 20 a^{13} + 14 a^{12} + 8 a^{11} + 4 a^{10} + 2 a^{9} + 13 a^{8} + 9 a^{7} + 5 a^{6} + 12 a^{5} + 21 a^{4} + 16 a^{3} + 2 a^{2} + 11 a + 1\right)\cdot 23^{226} + \left(a^{14} + 5 a^{13} + 6 a^{11} + 13 a^{10} + 22 a^{9} + 17 a^{8} + 12 a^{7} + 3 a^{6} + 21 a^{5} + 10 a^{4} + 11 a^{3} + 8 a^{2} + 2 a + 3\right)\cdot 23^{227} + \left(8 a^{14} + 9 a^{13} + 15 a^{12} + 10 a^{11} + a^{10} + 20 a^{9} + 10 a^{8} + 14 a^{7} + 15 a^{6} + 6 a^{5} + 2 a^{4} + 20 a^{3} + 20 a^{2} + 20 a + 10\right)\cdot 23^{228} + \left(4 a^{14} + 6 a^{13} + 17 a^{12} + 22 a^{11} + 5 a^{10} + 13 a^{9} + 4 a^{8} + 19 a^{7} + 20 a^{6} + 6 a^{5} + 6 a^{4} + 20 a^{3} + 7 a^{2} + 21 a + 3\right)\cdot 23^{229} + \left(6 a^{14} + 16 a^{13} + 15 a^{12} + 21 a^{11} + a^{10} + 11 a^{9} + 17 a^{8} + 13 a^{7} + 19 a^{6} + 17 a^{5} + 5 a^{4} + 6 a^{3} + 18 a^{2} + 19 a + 6\right)\cdot 23^{230} + \left(4 a^{13} + 11 a^{12} + 2 a^{11} + 4 a^{10} + 10 a^{9} + a^{8} + 9 a^{7} + 16 a^{6} + 13 a^{5} + 8 a^{4} + 7 a^{3} + 21 a^{2} + 2 a + 4\right)\cdot 23^{231} + \left(21 a^{14} + 6 a^{13} + a^{12} + 16 a^{11} + 3 a^{10} + 4 a^{9} + 11 a^{8} + 20 a^{7} + 13 a^{6} + 9 a^{5} + 7 a^{4} + 22 a^{3} + 12 a^{2} + 4 a + 5\right)\cdot 23^{232} + \left(6 a^{14} + 6 a^{13} + 20 a^{12} + 11 a^{11} + 14 a^{10} + 21 a^{9} + 19 a^{8} + 9 a^{6} + 20 a^{5} + 9 a^{4} + 2 a^{3} + 22 a + 2\right)\cdot 23^{233} + \left(2 a^{14} + 13 a^{13} + 16 a^{12} + 9 a^{11} + 15 a^{9} + 9 a^{8} + 3 a^{7} + 6 a^{6} + 16 a^{5} + 20 a^{4} + 19 a^{3} + 17 a^{2} + 19 a + 20\right)\cdot 23^{234} + \left(12 a^{14} + 21 a^{13} + 6 a^{12} + 7 a^{11} + 15 a^{10} + 19 a^{9} + 19 a^{8} + 16 a^{7} + 10 a^{6} + 22 a^{5} + 4 a^{4} + 9 a^{3} + 5 a^{2} + 14 a + 15\right)\cdot 23^{235} + \left(8 a^{14} + 6 a^{13} + 5 a^{12} + 16 a^{11} + 20 a^{10} + 15 a^{9} + 5 a^{8} + 14 a^{7} + 5 a^{6} + 7 a^{5} + 12 a^{3} + a^{2} + 7 a + 1\right)\cdot 23^{236} + \left(8 a^{14} + a^{13} + 12 a^{12} + 3 a^{10} + 18 a^{9} + 3 a^{8} + 16 a^{7} + 5 a^{6} + 22 a^{5} + 15 a^{3} + 6 a^{2} + 3 a + 20\right)\cdot 23^{237} + \left(22 a^{14} + a^{13} + 5 a^{12} + 11 a^{11} + 22 a^{10} + 7 a^{9} + a^{8} + 14 a^{7} + 22 a^{6} + 13 a^{5} + 7 a^{4} + 19 a^{3} + 2 a^{2} + 20 a + 22\right)\cdot 23^{238} + \left(18 a^{14} + 16 a^{13} + 16 a^{12} + 11 a^{11} + 18 a^{10} + 9 a^{9} + 22 a^{8} + 5 a^{7} + 9 a^{6} + 9 a^{5} + a^{4} + 11 a^{3} + 16 a^{2} + 9 a + 5\right)\cdot 23^{239} + \left(21 a^{14} + 11 a^{13} + 16 a^{12} + 11 a^{11} + a^{10} + 5 a^{9} + a^{8} + 4 a^{7} + 20 a^{6} + 17 a^{5} + 12 a^{4} + 4 a^{3} + 19 a^{2} + 22 a + 9\right)\cdot 23^{240} + \left(5 a^{14} + 14 a^{12} + 14 a^{11} + 4 a^{10} + 2 a^{9} + 3 a^{8} + 20 a^{7} + 6 a^{6} + 12 a^{5} + 12 a^{4} + 15 a^{3} + 13 a^{2} + 3 a + 17\right)\cdot 23^{241} + \left(5 a^{14} + 4 a^{13} + 5 a^{12} + 13 a^{11} + 17 a^{10} + 5 a^{9} + 11 a^{8} + 22 a^{7} + 15 a^{5} + 7 a^{4} + 14 a^{3} + 18 a^{2} + 13 a + 2\right)\cdot 23^{242} + \left(8 a^{14} + 19 a^{13} + 16 a^{12} + 6 a^{11} + 17 a^{10} + 10 a^{9} + 14 a^{8} + 8 a^{7} + 18 a^{6} + 2 a^{5} + 7 a^{4} + 7 a^{3} + 18 a^{2} + 10 a + 3\right)\cdot 23^{243} + \left(9 a^{14} + 8 a^{13} + 9 a^{12} + 5 a^{11} + 12 a^{10} + 16 a^{9} + 9 a^{8} + 19 a^{7} + 7 a^{6} + 19 a^{5} + 11 a^{4} + 21 a^{3} + 7 a^{2} + 15 a + 15\right)\cdot 23^{244} + \left(19 a^{14} + 12 a^{13} + 16 a^{12} + 14 a^{11} + 20 a^{10} + 16 a^{9} + 19 a^{8} + 5 a^{7} + 13 a^{6} + 15 a^{5} + 15 a^{4} + a^{3} + 21 a^{2} + 18 a + 11\right)\cdot 23^{245} + \left(10 a^{14} + 5 a^{13} + 9 a^{12} + 2 a^{11} + 17 a^{10} + 5 a^{9} + 21 a^{8} + a^{6} + 13 a^{5} + 12 a^{4} + 6 a^{3} + 9 a^{2} + 3\right)\cdot 23^{246} + \left(13 a^{14} + 20 a^{13} + 13 a^{12} + 7 a^{11} + 5 a^{10} + 16 a^{9} + 2 a^{8} + 15 a^{7} + 21 a^{6} + 9 a^{5} + 10 a^{4} + 15 a^{3} + 10 a^{2} + 13 a + 10\right)\cdot 23^{247} + \left(13 a^{14} + a^{13} + 10 a^{11} + 22 a^{10} + 13 a^{9} + 12 a^{8} + 19 a^{7} + 6 a^{6} + 15 a^{5} + 10 a^{4} + 6 a^{3} + 7 a^{2} + 14 a + 15\right)\cdot 23^{248} + \left(7 a^{14} + 10 a^{13} + 10 a^{12} + 21 a^{11} + 12 a^{10} + 10 a^{9} + 13 a^{8} + 3 a^{7} + 8 a^{6} + 11 a^{5} + 4 a^{4} + 18 a^{3} + 8 a^{2} + 17 a + 21\right)\cdot 23^{249} + \left(16 a^{14} + 16 a^{13} + 22 a^{12} + 12 a^{11} + 14 a^{10} + 21 a^{9} + 10 a^{8} + 9 a^{7} + a^{6} + 9 a^{5} + 6 a^{4} + 13 a^{3} + 3 a^{2} + 11 a + 2\right)\cdot 23^{250} + \left(11 a^{12} + 12 a^{11} + 2 a^{10} + 10 a^{9} + 4 a^{8} + 12 a^{7} + 7 a^{6} + 9 a^{5} + 8 a^{4} + 7 a^{3} + 17 a^{2} + 4 a + 15\right)\cdot 23^{251} + \left(9 a^{14} + 18 a^{13} + 12 a^{12} + a^{11} + 19 a^{10} + 22 a^{9} + 14 a^{8} + 14 a^{7} + 11 a^{6} + 2 a^{5} + 5 a^{4} + a^{3} + 15 a^{2} + 13 a + 8\right)\cdot 23^{252} + \left(19 a^{14} + 18 a^{13} + 12 a^{12} + 20 a^{11} + 12 a^{10} + 7 a^{9} + 3 a^{8} + 7 a^{7} + 7 a^{6} + 19 a^{5} + a^{4} + 20 a^{3} + 16 a^{2} + 17 a + 13\right)\cdot 23^{253} + \left(14 a^{14} + 22 a^{13} + a^{12} + 4 a^{11} + 14 a^{10} + a^{9} + 11 a^{8} + 11 a^{7} + 14 a^{6} + 11 a^{5} + 13 a^{4} + 2 a^{3} + 15 a^{2} + 20 a + 8\right)\cdot 23^{254} + \left(9 a^{14} + 4 a^{13} + 18 a^{12} + 11 a^{11} + 6 a^{10} + 18 a^{9} + 22 a^{8} + 21 a^{7} + 7 a^{6} + 2 a^{5} + 22 a^{3} + 14 a^{2} + 5 a + 5\right)\cdot 23^{255} + \left(14 a^{14} + 9 a^{13} + 3 a^{12} + 6 a^{11} + 12 a^{10} + 11 a^{9} + a^{8} + 3 a^{7} + 3 a^{6} + 3 a^{5} + 10 a^{4} + a^{3} + 10 a^{2} + 5 a + 2\right)\cdot 23^{256} + \left(18 a^{14} + 13 a^{13} + 3 a^{12} + 13 a^{11} + 17 a^{10} + 15 a^{9} + 2 a^{8} + 21 a^{7} + 13 a^{6} + 14 a^{5} + 5 a^{4} + 10 a^{3} + 7 a^{2} + 20 a + 4\right)\cdot 23^{257} + \left(2 a^{14} + 4 a^{13} + 12 a^{12} + 12 a^{11} + 13 a^{10} + 21 a^{9} + 12 a^{8} + 20 a^{7} + 17 a^{6} + 13 a^{5} + 8 a^{4} + 15 a^{3} + 17 a^{2} + 17 a\right)\cdot 23^{258} + \left(11 a^{14} + 12 a^{13} + 12 a^{12} + 4 a^{11} + 21 a^{10} + 16 a^{9} + 17 a^{8} + 8 a^{7} + 5 a^{6} + 19 a^{5} + 17 a^{4} + 6 a^{3} + 19 a^{2} + 14 a + 2\right)\cdot 23^{259} + \left(11 a^{14} + 2 a^{13} + 13 a^{12} + 11 a^{11} + 14 a^{10} + 18 a^{9} + 12 a^{8} + 20 a^{7} + a^{6} + a^{5} + 6 a^{4} + 14 a^{3} + 12 a^{2} + 22 a + 18\right)\cdot 23^{260} + \left(a^{14} + 4 a^{13} + 11 a^{12} + a^{11} + 12 a^{10} + 2 a^{9} + 21 a^{8} + 18 a^{7} + 6 a^{6} + 4 a^{5} + 16 a^{4} + 8 a^{3} + 17 a^{2} + 19 a + 21\right)\cdot 23^{261} + \left(7 a^{14} + 6 a^{13} + 11 a^{12} + 15 a^{11} + 19 a^{10} + 19 a^{9} + a^{8} + 18 a^{7} + 18 a^{6} + 6 a^{5} + 8 a^{4} + 10 a^{3} + 8 a^{2} + 11 a + 1\right)\cdot 23^{262} + \left(20 a^{14} + 2 a^{13} + 9 a^{12} + a^{11} + 13 a^{10} + 12 a^{9} + 11 a^{8} + 20 a^{7} + 19 a^{6} + a^{5} + 9 a^{4} + 20 a^{3} + 17 a^{2} + 21 a + 14\right)\cdot 23^{263} + \left(2 a^{14} + a^{13} + 9 a^{12} + 12 a^{11} + 5 a^{10} + 14 a^{9} + 19 a^{8} + a^{7} + 4 a^{5} + 9 a^{4} + 18 a^{3} + 12 a^{2} + 19 a + 10\right)\cdot 23^{264} + \left(3 a^{13} + 9 a^{12} + 12 a^{11} + 10 a^{10} + 11 a^{9} + 4 a^{8} + 8 a^{7} + 6 a^{6} + 11 a^{5} + 15 a^{4} + 7 a^{3} + 16 a^{2} + 2 a + 17\right)\cdot 23^{265} + \left(13 a^{14} + 21 a^{13} + 17 a^{12} + 8 a^{11} + 19 a^{10} + 22 a^{9} + 4 a^{8} + 7 a^{7} + 17 a^{6} + 3 a^{5} + 16 a^{4} + 22 a^{3} + 2 a^{2} + a + 1\right)\cdot 23^{266} + \left(22 a^{14} + 11 a^{13} + 16 a^{12} + 4 a^{10} + 18 a^{9} + 22 a^{8} + 5 a^{7} + 12 a^{6} + 2 a^{5} + 3 a^{4} + 21 a^{3} + a^{2} + 15 a + 10\right)\cdot 23^{267} + \left(22 a^{14} + 14 a^{13} + 5 a^{12} + 20 a^{11} + 13 a^{10} + 3 a^{9} + 7 a^{8} + 22 a^{7} + 5 a^{6} + 10 a^{5} + 6 a^{4} + 7 a^{3} + 14 a^{2} + 12 a + 9\right)\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 15 }$ $=$ $6 a^{14} + a^{13} + 8 a^{12} + 12 a^{11} + 10 a^{10} + 9 a^{9} + 7 a^{8} + 2 a^{7} + 16 a^{6} + 12 a^{5} + 14 a^{4} + 17 a^{3} + 22 a + 3 + \left(11 a^{14} + 18 a^{13} + 12 a^{12} + 6 a^{11} + 19 a^{10} + 8 a^{9} + 6 a^{8} + 13 a^{7} + 15 a^{6} + 8 a^{5} + 10 a^{4} + 16 a^{3} + 22 a^{2} + 8 a + 12\right)\cdot 23 + \left(16 a^{14} + 16 a^{13} + 20 a^{12} + 11 a^{10} + 7 a^{9} + 16 a^{8} + 19 a^{7} + 10 a^{6} + 3 a^{5} + 22 a^{4} + 22 a^{3} + 13 a^{2} + 2 a + 9\right)\cdot 23^{2} + \left(18 a^{14} + 8 a^{13} + 6 a^{12} + 17 a^{11} + 20 a^{10} + 13 a^{9} + 16 a^{8} + 12 a^{7} + 15 a^{6} + 18 a^{5} + 17 a^{4} + 19 a^{3} + 5 a^{2} + 15 a + 14\right)\cdot 23^{3} + \left(8 a^{14} + 18 a^{13} + 2 a^{12} + 21 a^{11} + 4 a^{10} + 7 a^{9} + a^{8} + 16 a^{7} + 11 a^{6} + 20 a^{5} + 21 a^{4} + 14 a^{3} + 19 a^{2} + 9 a + 12\right)\cdot 23^{4} + \left(17 a^{14} + 11 a^{13} + 19 a^{12} + 18 a^{11} + 10 a^{10} + 3 a^{9} + 16 a^{8} + 15 a^{7} + 22 a^{6} + 7 a^{5} + a^{4} + 10 a^{3} + 6 a^{2} + 16 a + 10\right)\cdot 23^{5} + \left(a^{14} + 4 a^{13} + 5 a^{12} + a^{11} + 21 a^{10} + 15 a^{9} + 9 a^{8} + 19 a^{7} + 20 a^{6} + 22 a^{5} + 6 a^{4} + 17 a^{3} + 15 a^{2} + 19 a + 19\right)\cdot 23^{6} + \left(13 a^{14} + 12 a^{13} + 4 a^{12} + 22 a^{11} + 17 a^{10} + 16 a^{9} + 22 a^{8} + 10 a^{7} + 18 a^{6} + 13 a^{5} + 9 a^{4} + 14 a^{3} + 17 a^{2} + 13 a + 6\right)\cdot 23^{7} + \left(4 a^{14} + 13 a^{13} + 6 a^{12} + 11 a^{11} + 14 a^{10} + 18 a^{9} + 10 a^{8} + 18 a^{7} + 19 a^{6} + 2 a^{5} + 5 a^{4} + 12 a^{3} + 10 a^{2} + 7 a + 21\right)\cdot 23^{8} + \left(9 a^{14} + 2 a^{13} + 20 a^{12} + 18 a^{11} + 20 a^{10} + 9 a^{9} + 19 a^{6} + 11 a^{5} + 22 a^{4} + 4 a^{3} + 6 a^{2} + 21 a + 15\right)\cdot 23^{9} + \left(a^{14} + 20 a^{13} + 7 a^{12} + 11 a^{11} + 16 a^{10} + 5 a^{9} + a^{8} + 10 a^{7} + 15 a^{6} + 17 a^{5} + 22 a^{4} + 17 a^{3} + 15 a^{2} + 7 a + 15\right)\cdot 23^{10} + \left(20 a^{14} + 22 a^{13} + 17 a^{12} + 7 a^{11} + 10 a^{10} + 18 a^{9} + 4 a^{8} + 15 a^{7} + 3 a^{6} + 11 a^{5} + 15 a^{3} + 5 a^{2} + 4 a + 6\right)\cdot 23^{11} + \left(16 a^{14} + a^{13} + 15 a^{12} + 6 a^{11} + a^{10} + 8 a^{9} + 15 a^{8} + 16 a^{7} + 9 a^{6} + 10 a^{5} + 16 a^{4} + 9 a^{3} + 10 a^{2} + 7 a + 7\right)\cdot 23^{12} + \left(9 a^{14} + 9 a^{13} + 2 a^{12} + 12 a^{11} + 14 a^{10} + 16 a^{9} + a^{8} + a^{7} + 5 a^{6} + 9 a^{5} + 13 a^{4} + 6 a^{3} + 7 a^{2} + a + 10\right)\cdot 23^{13} + \left(8 a^{14} + 13 a^{13} + 14 a^{12} + 11 a^{11} + 12 a^{10} + 11 a^{9} + 15 a^{8} + 9 a^{7} + 11 a^{6} + 20 a^{5} + 20 a^{4} + 11 a^{3} + 5 a^{2} + 18 a + 9\right)\cdot 23^{14} + \left(3 a^{13} + 18 a^{12} + 5 a^{11} + 16 a^{10} + 15 a^{9} + 8 a^{8} + 10 a^{7} + 22 a^{6} + 10 a^{5} + 12 a^{4} + 16 a^{3} + 16 a + 6\right)\cdot 23^{15} + \left(22 a^{14} + 6 a^{13} + a^{12} + 3 a^{11} + 17 a^{10} + 6 a^{9} + 17 a^{8} + 18 a^{7} + 9 a^{6} + 14 a^{4} + 17 a^{3} + 21 a + 8\right)\cdot 23^{16} + \left(2 a^{14} + 21 a^{13} + 11 a^{12} + 20 a^{11} + 14 a^{10} + a^{9} + 12 a^{8} + 17 a^{7} + 19 a^{6} + 15 a^{5} + 22 a^{4} + 13 a^{3} + 9 a^{2} + 12 a + 21\right)\cdot 23^{17} + \left(22 a^{14} + 11 a^{13} + 15 a^{12} + 13 a^{11} + 14 a^{10} + 4 a^{9} + 18 a^{8} + a^{7} + 21 a^{6} + a^{5} + 6 a^{4} + 8 a^{3} + 5 a^{2} + 18 a + 11\right)\cdot 23^{18} + \left(19 a^{13} + 13 a^{12} + 15 a^{11} + 20 a^{10} + 4 a^{9} + 16 a^{8} + 8 a^{7} + 7 a^{6} + 11 a^{5} + 12 a^{4} + 20 a^{3} + 18 a^{2} + 15 a + 14\right)\cdot 23^{19} + \left(11 a^{14} + 14 a^{13} + 16 a^{12} + a^{11} + 8 a^{10} + 13 a^{9} + 4 a^{8} + 15 a^{7} + 4 a^{6} + 17 a^{5} + 2 a^{4} + 20 a^{3} + 6 a^{2} + 21 a + 17\right)\cdot 23^{20} + \left(19 a^{14} + 16 a^{13} + 22 a^{12} + 13 a^{11} + 5 a^{10} + 14 a^{9} + 16 a^{8} + 10 a^{7} + 18 a^{6} + 3 a^{5} + 16 a^{4} + 22 a^{3} + 22 a^{2} + 13 a + 16\right)\cdot 23^{21} + \left(3 a^{14} + 2 a^{13} + 7 a^{12} + 18 a^{11} + 16 a^{10} + 12 a^{9} + 7 a^{8} + 17 a^{7} + 20 a^{6} + 19 a^{5} + 4 a^{4} + 6 a^{3} + 10 a^{2} + 4 a + 4\right)\cdot 23^{22} + \left(16 a^{14} + 3 a^{13} + 14 a^{12} + 19 a^{11} + 13 a^{10} + 2 a^{9} + a^{8} + 11 a^{7} + 13 a^{6} + 8 a^{5} + 19 a^{4} + 2 a^{3} + 14 a^{2} + 20 a + 7\right)\cdot 23^{23} + \left(5 a^{14} + 21 a^{13} + 15 a^{12} + 3 a^{11} + 15 a^{10} + 15 a^{9} + 4 a^{8} + 6 a^{7} + 12 a^{6} + a^{5} + 22 a^{4} + 4 a^{3} + 10 a^{2} + 19 a + 2\right)\cdot 23^{24} + \left(5 a^{14} + 16 a^{13} + 12 a^{11} + 4 a^{10} + 4 a^{9} + 3 a^{8} + 14 a^{7} + 3 a^{6} + 18 a^{5} + 6 a^{4} + 5 a^{3} + 15 a^{2} + 19 a + 8\right)\cdot 23^{25} + \left(15 a^{14} + 22 a^{13} + a^{12} + 21 a^{11} + 15 a^{10} + 22 a^{9} + 19 a^{7} + 15 a^{6} + 8 a^{5} + 12 a^{4} + 5 a^{3} + 18 a^{2} + 14 a + 14\right)\cdot 23^{26} + \left(14 a^{14} + 3 a^{12} + 7 a^{11} + 22 a^{10} + 12 a^{9} + 10 a^{7} + 17 a^{6} + 15 a^{5} + 7 a^{4} + 19 a^{2} + 8 a + 5\right)\cdot 23^{27} + \left(19 a^{14} + 3 a^{13} + 19 a^{12} + 17 a^{11} + 4 a^{10} + 3 a^{9} + 11 a^{8} + 21 a^{7} + 7 a^{6} + 19 a^{5} + 13 a^{4} + 19 a^{3} + 4 a^{2} + 12 a + 8\right)\cdot 23^{28} + \left(19 a^{14} + 9 a^{13} + 9 a^{12} + 13 a^{11} + 9 a^{9} + 10 a^{8} + 15 a^{7} + 2 a^{6} + 17 a^{5} + 4 a^{4} + 20 a^{3} + 14 a^{2} + 15 a + 19\right)\cdot 23^{29} + \left(11 a^{14} + 3 a^{13} + 7 a^{12} + 14 a^{11} + 18 a^{10} + 2 a^{9} + a^{8} + 10 a^{7} + 9 a^{6} + 7 a^{5} + 14 a^{4} + 21 a^{3} + 16 a^{2} + 8 a + 7\right)\cdot 23^{30} + \left(19 a^{14} + 22 a^{13} + 16 a^{12} + 16 a^{11} + 14 a^{9} + 17 a^{8} + 7 a^{7} + 22 a^{6} + 2 a^{5} + 21 a^{4} + 8 a^{3} + 15 a^{2} + 6 a + 15\right)\cdot 23^{31} + \left(7 a^{14} + 14 a^{12} + 9 a^{11} + 14 a^{10} + 5 a^{9} + 21 a^{8} + 3 a^{7} + 20 a^{6} + 16 a^{5} + a^{4} + 15 a^{3} + 7 a^{2} + 22 a + 10\right)\cdot 23^{32} + \left(17 a^{14} + 22 a^{13} + 21 a^{12} + 12 a^{11} + 6 a^{10} + 13 a^{9} + 8 a^{8} + 10 a^{7} + 2 a^{6} + 22 a^{5} + 7 a^{4} + 5 a^{3} + 14 a^{2} + 12\right)\cdot 23^{33} + \left(9 a^{14} + 15 a^{12} + 18 a^{11} + 6 a^{10} + 17 a^{9} + 3 a^{8} + 10 a^{7} + 9 a^{6} + 10 a^{5} + 9 a^{4} + 14 a^{3} + 21 a^{2} + 4 a + 17\right)\cdot 23^{34} + \left(4 a^{14} + 9 a^{13} + 8 a^{12} + 13 a^{11} + 5 a^{10} + 17 a^{9} + 16 a^{8} + 6 a^{7} + 9 a^{5} + 16 a^{4} + a^{3} + 3 a^{2} + 5 a + 12\right)\cdot 23^{35} + \left(6 a^{14} + 20 a^{13} + 18 a^{12} + 15 a^{11} + 11 a^{10} + 21 a^{9} + 16 a^{8} + 12 a^{7} + 3 a^{6} + 4 a^{5} + 7 a^{4} + 6 a^{3} + 11 a^{2} + 15 a + 17\right)\cdot 23^{36} + \left(19 a^{14} + 8 a^{13} + 12 a^{12} + 16 a^{11} + 12 a^{10} + 5 a^{9} + 4 a^{8} + 14 a^{7} + 10 a^{6} + 12 a^{5} + 13 a^{4} + 17 a^{3} + 20 a^{2} + 17 a + 15\right)\cdot 23^{37} + \left(2 a^{14} + 9 a^{13} + 14 a^{12} + 15 a^{11} + 4 a^{10} + 18 a^{9} + 16 a^{8} + 11 a^{7} + 20 a^{6} + 2 a^{5} + 8 a^{4} + 19 a^{3} + 18 a^{2} + 10 a + 22\right)\cdot 23^{38} + \left(7 a^{14} + 5 a^{13} + 5 a^{12} + 14 a^{11} + 21 a^{10} + 5 a^{9} + 3 a^{8} + 16 a^{7} + 11 a^{6} + 9 a^{5} + 19 a^{4} + 7 a^{3} + 7 a^{2} + 16 a + 16\right)\cdot 23^{39} + \left(18 a^{14} + 15 a^{13} + 18 a^{12} + 11 a^{11} + 19 a^{10} + 2 a^{9} + 12 a^{8} + 19 a^{7} + 20 a^{6} + 15 a^{5} + 4 a^{4} + 10 a^{3} + 14 a^{2} + 8 a + 21\right)\cdot 23^{40} + \left(4 a^{14} + 16 a^{13} + 18 a^{12} + 6 a^{11} + 22 a^{10} + 18 a^{9} + 13 a^{8} + 16 a^{7} + 20 a^{6} + 14 a^{5} + 7 a^{4} + 9 a^{3} + 19 a^{2} + 12 a + 18\right)\cdot 23^{41} + \left(15 a^{14} + 18 a^{13} + a^{12} + a^{11} + 16 a^{10} + 17 a^{9} + 15 a^{8} + a^{7} + 4 a^{6} + 19 a^{4} + 15 a^{3} + 19 a^{2} + 8 a + 1\right)\cdot 23^{42} + \left(4 a^{14} + 14 a^{13} + 15 a^{12} + 20 a^{11} + 13 a^{10} + 2 a^{9} + a^{8} + 7 a^{7} + 15 a^{6} + 9 a^{5} + 10 a^{4} + 12 a^{3} + 12 a^{2} + 12 a + 18\right)\cdot 23^{43} + \left(17 a^{14} + 13 a^{13} + 11 a^{12} + 18 a^{11} + 18 a^{10} + 16 a^{9} + 9 a^{8} + 15 a^{7} + 11 a^{6} + 10 a^{5} + 3 a^{4} + 19 a^{3} + 22 a^{2} + 21 a + 9\right)\cdot 23^{44} + \left(20 a^{14} + 6 a^{13} + 2 a^{12} + 6 a^{11} + 15 a^{10} + 6 a^{9} + 19 a^{8} + 6 a^{7} + 13 a^{6} + 11 a^{5} + 16 a^{4} + a^{3} + 6 a^{2} + 10 a + 6\right)\cdot 23^{45} + \left(3 a^{14} + 16 a^{13} + 12 a^{12} + 9 a^{11} + 9 a^{10} + 13 a^{9} + 20 a^{8} + 16 a^{6} + 5 a^{5} + 21 a^{4} + 21 a^{3} + 10 a^{2} + 5 a + 1\right)\cdot 23^{46} + \left(8 a^{14} + 7 a^{13} + 15 a^{12} + 2 a^{11} + 16 a^{10} + 12 a^{9} + 4 a^{8} + 21 a^{7} + 22 a^{6} + 16 a^{5} + 12 a^{4} + 19 a^{3} + 10 a^{2} + 9 a\right)\cdot 23^{47} + \left(15 a^{14} + 3 a^{13} + 9 a^{12} + 13 a^{11} + 9 a^{10} + 6 a^{9} + 9 a^{8} + 12 a^{7} + 20 a^{5} + 8 a^{3} + 2 a^{2} + 17 a + 20\right)\cdot 23^{48} + \left(4 a^{14} + 9 a^{13} + 13 a^{12} + a^{11} + 8 a^{10} + 16 a^{9} + 4 a^{8} + a^{7} + 6 a^{6} + 21 a^{5} + 10 a^{4} + 3 a^{3} + 6 a^{2} + 9 a + 12\right)\cdot 23^{49} + \left(18 a^{14} + 2 a^{13} + 8 a^{12} + 3 a^{11} + 13 a^{10} + 2 a^{9} + 16 a^{8} + 20 a^{7} + 21 a^{6} + 20 a^{5} + 17 a^{4} + 3 a^{3} + 17 a^{2} + 7 a\right)\cdot 23^{50} + \left(9 a^{14} + 13 a^{13} + 15 a^{12} + 8 a^{11} + 7 a^{10} + 13 a^{9} + 6 a^{8} + 5 a^{7} + 6 a^{6} + 10 a^{5} + 5 a^{4} + 12 a^{3} + 20 a^{2} + 17 a + 3\right)\cdot 23^{51} + \left(13 a^{14} + 22 a^{13} + a^{12} + 11 a^{11} + a^{10} + 18 a^{9} + 11 a^{8} + 8 a^{7} + 4 a^{6} + 11 a^{5} + 12 a^{4} + 10 a^{3} + 22 a^{2} + 3 a + 10\right)\cdot 23^{52} + \left(20 a^{14} + 22 a^{13} + 3 a^{12} + 22 a^{11} + 15 a^{10} + 12 a^{9} + 17 a^{8} + 13 a^{7} + 8 a^{6} + 20 a^{5} + 14 a^{3} + 3 a^{2} + 15 a + 19\right)\cdot 23^{53} + \left(17 a^{14} + a^{13} + 13 a^{12} + 18 a^{11} + 12 a^{10} + 18 a^{9} + 19 a^{8} + 3 a^{7} + 16 a^{6} + 14 a^{5} + 2 a^{4} + 5 a^{3} + 13 a^{2} + 17 a + 7\right)\cdot 23^{54} + \left(9 a^{14} + 15 a^{13} + 3 a^{12} + 20 a^{11} + 20 a^{10} + 11 a^{9} + 8 a^{8} + 12 a^{7} + 12 a^{6} + 3 a^{5} + 2 a^{4} + 22 a^{3} + 22 a^{2} + 18 a + 12\right)\cdot 23^{55} + \left(9 a^{14} + 4 a^{13} + 12 a^{12} + 17 a^{11} + 15 a^{10} + 8 a^{9} + 6 a^{8} + 13 a^{7} + 8 a^{6} + 7 a^{5} + 18 a^{4} + 6 a^{3} + 6 a^{2} + 18 a + 2\right)\cdot 23^{56} + \left(19 a^{14} + 21 a^{13} + 8 a^{12} + 4 a^{11} 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a^{4} + 7 a^{3} + 7 a^{2} + 2 a + 13\right)\cdot 23^{74} + \left(18 a^{14} + a^{13} + 18 a^{12} + 8 a^{11} + 11 a^{10} + 11 a^{9} + 18 a^{8} + 11 a^{7} + 8 a^{6} + 9 a^{5} + 14 a^{4} + 6 a^{3} + 13 a^{2} + 15 a + 21\right)\cdot 23^{75} + \left(13 a^{14} + 4 a^{12} + 7 a^{11} + 3 a^{10} + 12 a^{8} + 12 a^{7} + 8 a^{6} + 10 a^{5} + 15 a^{4} + 22 a^{3} + 9 a^{2} + 5 a + 11\right)\cdot 23^{76} + \left(3 a^{13} + 6 a^{12} + 18 a^{11} + 4 a^{10} + 14 a^{9} + 11 a^{8} + 8 a^{7} + 10 a^{6} + 2 a^{5} + a^{4} + 21 a^{3} + 13 a^{2} + 22 a + 22\right)\cdot 23^{77} + \left(8 a^{14} + 8 a^{13} + 8 a^{12} + 10 a^{11} + 19 a^{10} + 18 a^{9} + 11 a^{8} + 8 a^{7} + 11 a^{6} + 11 a^{5} + 5 a^{4} + a^{3} + 7 a^{2} + 10 a + 2\right)\cdot 23^{78} + \left(18 a^{14} + 2 a^{13} + 4 a^{12} + 3 a^{11} + 12 a^{10} + 4 a^{9} + 9 a^{8} + 3 a^{7} + 5 a^{6} + 9 a^{5} + 3 a^{4} + 17 a^{3} + 15 a^{2} + 4 a + 5\right)\cdot 23^{79} + \left(6 a^{14} + 11 a^{13} + 12 a^{12} + 8 a^{11} + 19 a^{10} + 8 a^{9} + 9 a^{8} + 20 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21 a^{10} + 19 a^{9} + 2 a^{8} + 4 a^{7} + 12 a^{6} + 10 a^{5} + 6 a^{4} + a^{3} + 3 a^{2} + 14 a + 6\right)\cdot 23^{86} + \left(20 a^{14} + 5 a^{13} + 11 a^{12} + a^{11} + 16 a^{10} + 12 a^{9} + a^{8} + 12 a^{7} + 5 a^{6} + 16 a^{5} + 16 a^{4} + 22 a^{3} + 14 a^{2} + 13 a + 1\right)\cdot 23^{87} + \left(a^{13} + 20 a^{12} + 17 a^{11} + 10 a^{9} + 17 a^{8} + 17 a^{7} + 7 a^{6} + 15 a^{5} + 8 a^{4} + 14 a^{3} + 9 a^{2} + 8 a + 2\right)\cdot 23^{88} + \left(21 a^{14} + 10 a^{13} + 6 a^{12} + 19 a^{11} + 12 a^{10} + 8 a^{9} + 20 a^{8} + 15 a^{7} + 22 a^{6} + 3 a^{5} + 7 a^{4} + 6 a^{3} + 12 a^{2} + 17 a + 6\right)\cdot 23^{89} + \left(14 a^{14} + 21 a^{13} + 7 a^{12} + 19 a^{11} + 18 a^{10} + 7 a^{9} + 13 a^{8} + 2 a^{7} + 5 a^{5} + 22 a^{4} + 16 a^{3} + 9 a^{2} + 10 a + 4\right)\cdot 23^{90} + \left(2 a^{14} + 13 a^{13} + 10 a^{12} + 6 a^{11} + 18 a^{10} + 13 a^{9} + 22 a^{8} + 6 a^{7} + 9 a^{6} + 2 a^{5} + 19 a^{4} + 20 a^{3} + 11 a^{2} + 2\right)\cdot 23^{91} + \left(10 a^{14} + 15 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+ 19 a^{3} + 21 a^{2} + 5 a + 12\right)\cdot 23^{97} + \left(19 a^{14} + 17 a^{13} + 2 a^{12} + 16 a^{11} + 13 a^{10} + 8 a^{9} + 3 a^{8} + 9 a^{7} + 6 a^{6} + 19 a^{4} + 4 a^{3} + 5 a^{2} + 21 a + 7\right)\cdot 23^{98} + \left(11 a^{14} + 10 a^{13} + 9 a^{12} + 9 a^{11} + 14 a^{10} + a^{9} + 2 a^{8} + 18 a^{7} + 20 a^{6} + 9 a^{5} + 22 a^{4} + a^{3} + 4 a^{2} + 9 a + 13\right)\cdot 23^{99} + \left(3 a^{14} + 4 a^{13} + 6 a^{12} + 18 a^{11} + 10 a^{10} + 2 a^{9} + 20 a^{8} + 5 a^{7} + 15 a^{6} + 5 a^{5} + 2 a^{4} + 9 a^{3} + 20 a^{2} + 18 a + 15\right)\cdot 23^{100} + \left(5 a^{14} + 6 a^{13} + 22 a^{12} + 9 a^{11} + 10 a^{10} + 7 a^{9} + 5 a^{8} + a^{7} + 15 a^{6} + 10 a^{5} + 20 a^{4} + 5 a^{3} + 16 a^{2} + 3 a + 11\right)\cdot 23^{101} + \left(9 a^{14} + 20 a^{13} + 12 a^{12} + 19 a^{11} + 8 a^{10} + 21 a^{9} + 13 a^{8} + 11 a^{7} + 7 a^{6} + 4 a^{5} + 14 a^{4} + 15 a^{3} + 6 a^{2} + 8 a + 8\right)\cdot 23^{102} + \left(22 a^{14} + a^{13} + 3 a^{12} + 17 a^{11} + 17 a^{10} + 9 a^{9} + a^{8} + 22 a^{7} + a^{5} + 10 a^{4} + 4 a^{3} + 18 a^{2} + 6 a + 11\right)\cdot 23^{103} + \left(14 a^{14} + 8 a^{13} + 4 a^{12} + 18 a^{11} + 19 a^{10} + 21 a^{9} + 2 a^{8} + 13 a^{7} + 19 a^{6} + 2 a^{5} + 14 a^{4} + 11 a^{3} + 9 a^{2} + 2 a + 21\right)\cdot 23^{104} + \left(22 a^{14} + 22 a^{13} + 13 a^{12} + 20 a^{11} + 18 a^{10} + 19 a^{9} + 11 a^{8} + 19 a^{7} + 20 a^{6} + 14 a^{5} + 9 a^{4} + 12 a^{3} + 8 a^{2} + 18 a + 10\right)\cdot 23^{105} + \left(17 a^{14} + 7 a^{13} + 3 a^{12} + a^{11} + 15 a^{10} + 5 a^{9} + 6 a^{8} + 8 a^{7} + 14 a^{6} + 14 a^{5} + 16 a^{4} + 14 a^{3} + 12 a^{2} + 17 a + 13\right)\cdot 23^{106} + \left(22 a^{14} + 5 a^{13} + 22 a^{12} + 3 a^{11} + 5 a^{10} + 16 a^{9} + 15 a^{8} + 3 a^{7} + 10 a^{6} + 17 a^{5} + 17 a^{4} + 2 a^{3} + 2 a^{2} + 6 a + 2\right)\cdot 23^{107} + \left(21 a^{13} + 8 a^{12} + 5 a^{11} + 20 a^{10} + 6 a^{9} + 18 a^{8} + 5 a^{7} + 22 a^{6} + 12 a^{5} + 10 a^{4} + 6 a^{2} + 19 a + 15\right)\cdot 23^{108} + \left(3 a^{14} + 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5\right)\cdot 23^{114} + \left(19 a^{14} + 22 a^{13} + 10 a^{12} + 22 a^{11} + 5 a^{10} + 9 a^{9} + 14 a^{8} + 2 a^{7} + 3 a^{6} + 19 a^{5} + 11 a^{4} + 16 a^{3} + 9 a^{2} + 4 a + 18\right)\cdot 23^{115} + \left(17 a^{14} + 12 a^{13} + 19 a^{12} + 20 a^{11} + 13 a^{10} + 8 a^{9} + 16 a^{8} + 22 a^{7} + 7 a^{6} + 16 a^{5} + 12 a^{4} + 2 a^{3} + 17 a^{2} + 17 a + 3\right)\cdot 23^{116} + \left(5 a^{14} + a^{12} + 5 a^{11} + 20 a^{10} + 12 a^{9} + 11 a^{8} + a^{6} + 13 a^{5} + 17 a^{4} + 2 a^{3} + a^{2} + 20 a + 22\right)\cdot 23^{117} + \left(22 a^{14} + 18 a^{13} + 7 a^{12} + 19 a^{11} + a^{10} + 8 a^{9} + 7 a^{8} + 12 a^{7} + 8 a^{6} + 3 a^{5} + 10 a^{4} + 9 a^{3} + 19 a^{2} + 14 a + 11\right)\cdot 23^{118} + \left(19 a^{14} + 12 a^{13} + 17 a^{12} + 18 a^{11} + 21 a^{10} + 18 a^{9} + 2 a^{8} + a^{7} + 17 a^{6} + 7 a^{5} + 14 a^{4} + 6 a^{3} + 21 a^{2} + 6 a\right)\cdot 23^{119} + \left(3 a^{14} + 15 a^{13} + 7 a^{12} + 22 a^{11} + 17 a^{10} + 21 a^{9} + 12 a^{8} + a^{7} + 12 a^{6} + 3 a^{4} + 19 a^{3} + 4 a^{2} + 21 a + 19\right)\cdot 23^{120} + \left(a^{14} + 21 a^{13} + 4 a^{12} + 22 a^{11} + 3 a^{10} + 17 a^{9} + 19 a^{8} + a^{7} + 15 a^{6} + 9 a^{5} + 20 a^{4} + 2 a^{3} + 14 a^{2} + 15 a + 12\right)\cdot 23^{121} + \left(16 a^{14} + 8 a^{13} + 17 a^{12} + 8 a^{11} + 21 a^{10} + 16 a^{9} + 13 a^{8} + 3 a^{7} + 8 a^{6} + a^{5} + 3 a^{4} + 16 a^{3} + 11 a^{2} + 13 a + 19\right)\cdot 23^{122} + \left(13 a^{14} + a^{13} + 18 a^{12} + 12 a^{11} + 11 a^{10} + 19 a^{9} + 22 a^{8} + 18 a^{7} + 6 a^{6} + 7 a^{5} + 6 a^{4} + 2 a^{3} + 10 a^{2} + 5 a + 7\right)\cdot 23^{123} + \left(20 a^{14} + 19 a^{12} + a^{11} + 10 a^{10} + 22 a^{9} + 12 a^{8} + 15 a^{6} + 10 a^{5} + 17 a^{4} + 3 a^{3} + 8 a^{2} + 13 a + 17\right)\cdot 23^{124} + \left(8 a^{14} + 10 a^{13} + 13 a^{12} + 17 a^{11} + 11 a^{10} + 6 a^{9} + 19 a^{8} + 2 a^{6} + 16 a^{5} + 9 a^{4} + 11 a^{3} + 6 a^{2} + 5 a + 4\right)\cdot 23^{125} + \left(13 a^{14} + 2 a^{13} + 10 a^{12} + 6 a^{11} + 17 a^{10} + a^{9} + 20 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8 a^{12} + 4 a^{10} + 2 a^{9} + 18 a^{8} + 21 a^{7} + 13 a^{6} + 7 a^{5} + 11 a^{4} + 15 a^{3} + 5 a^{2} + 13 a + 5\right)\cdot 23^{132} + \left(20 a^{14} + 2 a^{13} + 9 a^{12} + 18 a^{11} + 21 a^{10} + 21 a^{9} + 7 a^{8} + 20 a^{7} + 5 a^{6} + 8 a^{5} + 10 a^{4} + 16 a^{3} + 18 a^{2} + 10 a + 17\right)\cdot 23^{133} + \left(5 a^{14} + 11 a^{13} + 22 a^{12} + 11 a^{11} + 20 a^{10} + 22 a^{9} + 18 a^{8} + 8 a^{7} + a^{6} + 14 a^{5} + 18 a^{4} + 8 a^{3} + 2 a + 16\right)\cdot 23^{134} + \left(3 a^{14} + 20 a^{12} + 19 a^{11} + 17 a^{10} + 11 a^{9} + 9 a^{8} + 21 a^{6} + 15 a^{5} + 4 a^{4} + 7 a^{3} + 22 a^{2} + 12 a + 19\right)\cdot 23^{135} + \left(13 a^{14} + 8 a^{13} + 5 a^{12} + 21 a^{11} + 19 a^{10} + 16 a^{9} + 7 a^{8} + 12 a^{7} + 15 a^{6} + 12 a^{5} + 22 a^{4} + 15 a^{3} + 9 a^{2} + 8 a + 9\right)\cdot 23^{136} + \left(3 a^{14} + a^{13} + 15 a^{12} + 3 a^{11} + 2 a^{10} + 15 a^{9} + a^{8} + 4 a^{7} + 6 a^{6} + 12 a^{5} + 5 a^{4} + 16 a^{3} + 8 a^{2} + 21 a + 13\right)\cdot 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+ 8 a^{4} + 15 a^{3} + 9 a^{2} + 19\right)\cdot 23^{143} + \left(20 a^{14} + 8 a^{13} + 6 a^{12} + 13 a^{11} + 10 a^{10} + 5 a^{9} + 4 a^{8} + 15 a^{7} + 6 a^{6} + 3 a^{5} + 11 a^{4} + 5 a^{3} + 4 a^{2} + 5\right)\cdot 23^{144} + \left(15 a^{14} + 5 a^{13} + 13 a^{12} + 14 a^{11} + 2 a^{10} + 8 a^{8} + 7 a^{6} + 2 a^{5} + 4 a^{4} + 21 a^{3} + 9 a^{2} + 18 a + 15\right)\cdot 23^{145} + \left(12 a^{14} + 8 a^{13} + 5 a^{12} + 17 a^{11} + 22 a^{10} + 21 a^{9} + 10 a^{8} + 17 a^{7} + 6 a^{6} + 6 a^{5} + 2 a^{4} + 11 a^{3} + 11 a^{2} + 12 a + 16\right)\cdot 23^{146} + \left(18 a^{14} + 9 a^{13} + 13 a^{12} + 10 a^{11} + 13 a^{10} + 14 a^{9} + 4 a^{8} + 7 a^{7} + 6 a^{6} + 4 a^{5} + 12 a^{4} + 16 a^{3} + 11 a^{2} + 7 a + 22\right)\cdot 23^{147} + \left(17 a^{14} + 21 a^{13} + 18 a^{12} + 11 a^{11} + 17 a^{10} + 21 a^{9} + 6 a^{8} + 14 a^{7} + 16 a^{6} + 15 a^{5} + 2 a^{4} + 16 a^{3} + 4 a + 7\right)\cdot 23^{148} + \left(20 a^{14} + 21 a^{13} + 21 a^{12} + 16 a^{11} + 21 a^{10} + 13 a^{9} + 11 a^{8} + 10 a^{7} + 11 a^{6} + 9 a^{5} + 20 a^{4} + 2 a^{3} + 17 a^{2} + 22 a + 13\right)\cdot 23^{149} + \left(10 a^{14} + 9 a^{13} + 6 a^{12} + 15 a^{11} + 11 a^{9} + 22 a^{8} + 7 a^{7} + 8 a^{6} + 22 a^{5} + 18 a^{4} + 22 a^{3} + 11 a^{2} + 6 a + 7\right)\cdot 23^{150} + \left(21 a^{14} + a^{13} + 13 a^{12} + 15 a^{11} + 9 a^{10} + a^{9} + 20 a^{8} + 4 a^{7} + 8 a^{6} + 13 a^{5} + 10 a^{4} + 19 a^{3} + 19 a^{2} + 12 a + 13\right)\cdot 23^{151} + \left(6 a^{14} + 21 a^{13} + 18 a^{12} + 5 a^{11} + 11 a^{10} + a^{9} + 3 a^{8} + 10 a^{7} + 15 a^{6} + 11 a^{5} + 18 a^{4} + 21 a^{3} + 16 a^{2} + 19 a + 15\right)\cdot 23^{152} + \left(5 a^{14} + 12 a^{13} + 20 a^{12} + 8 a^{11} + 13 a^{10} + 18 a^{9} + 20 a^{8} + 22 a^{7} + 11 a^{6} + 5 a^{5} + 3 a^{4} + 10 a^{2} + 14 a + 22\right)\cdot 23^{153} + \left(a^{14} + 2 a^{13} + 22 a^{12} + 3 a^{11} + 21 a^{10} + 4 a^{9} + 15 a^{8} + 18 a^{6} + a^{5} + 6 a^{4} + 4 a^{3} + 14 a^{2} + 19 a + 11\right)\cdot 23^{154} + \left(8 a^{14} + 4 a^{13} + 9 a^{12} + 12 a^{11} + 4 a^{10} + a^{9} + 9 a^{8} + 17 a^{7} + 13 a^{6} + 3 a^{4} + 14 a^{3} + 13 a^{2} + 22 a + 8\right)\cdot 23^{155} + \left(16 a^{14} + 17 a^{13} + 22 a^{12} + 9 a^{10} + 12 a^{9} + 20 a^{8} + 11 a^{7} + 14 a^{6} + 17 a^{5} + 15 a^{4} + 3 a^{3} + 20 a^{2} + 18 a + 10\right)\cdot 23^{156} + \left(22 a^{14} + 3 a^{13} + 21 a^{12} + 8 a^{11} + a^{10} + 18 a^{9} + 14 a^{8} + 6 a^{7} + 11 a^{6} + 20 a^{5} + 8 a^{4} + 11 a^{3} + 11 a^{2} + 19\right)\cdot 23^{157} + \left(15 a^{14} + 13 a^{13} + 8 a^{12} + 3 a^{11} + 16 a^{10} + 19 a^{9} + 7 a^{8} + 7 a^{7} + 17 a^{6} + 16 a^{5} + 13 a^{4} + a^{3} + 6 a^{2} + 3 a + 3\right)\cdot 23^{158} + \left(17 a^{13} + 5 a^{12} + 22 a^{11} + 7 a^{10} + 3 a^{9} + 17 a^{8} + 17 a^{7} + 6 a^{6} + 19 a^{5} + a^{4} + 4 a^{3} + 8 a^{2} + a + 10\right)\cdot 23^{159} + \left(a^{14} + 11 a^{13} + 3 a^{12} + 17 a^{11} + 9 a^{10} + 5 a^{9} + a^{8} + a^{7} + 7 a^{6} + 13 a^{5} + 6 a^{4} + 18 a^{3} + 12 a^{2} + 6 a + 15\right)\cdot 23^{160} + \left(20 a^{14} + 18 a^{13} + a^{12} + 21 a^{11} + 7 a^{10} + 17 a^{9} + 22 a^{8} + 5 a^{7} + 10 a^{6} + 20 a^{5} + 14 a^{4} + a^{3} + 8 a^{2} + 4 a + 21\right)\cdot 23^{161} + \left(13 a^{14} + 7 a^{13} + 16 a^{12} + 3 a^{11} + 7 a^{10} + 20 a^{9} + a^{8} + 5 a^{7} + 13 a^{6} + 4 a^{5} + 5 a^{4} + 18 a^{3} + 11 a^{2} + 9 a + 10\right)\cdot 23^{162} + \left(15 a^{14} + 14 a^{13} + 14 a^{12} + 22 a^{11} + 2 a^{10} + 13 a^{9} + 22 a^{8} + 7 a^{7} + 9 a^{6} + 20 a^{5} + 9 a^{4} + 3 a^{3} + 12 a + 4\right)\cdot 23^{163} + \left(22 a^{14} + 19 a^{13} + 5 a^{12} + 8 a^{11} + 11 a^{10} + 21 a^{9} + 4 a^{8} + 3 a^{6} + 3 a^{5} + 19 a^{4} + 3 a^{3} + 19 a^{2} + 4 a + 18\right)\cdot 23^{164} + \left(13 a^{14} + 3 a^{13} + 3 a^{12} + 18 a^{10} + 9 a^{9} + 18 a^{8} + 8 a^{7} + 13 a^{6} + 14 a^{5} + 7 a^{4} + 21 a^{3} + 20 a^{2} + 9 a + 7\right)\cdot 23^{165} + \left(22 a^{14} + 15 a^{13} + 15 a^{12} + 22 a^{11} + 22 a^{10} + 5 a^{9} + 17 a^{8} + 19 a^{7} + 8 a^{6} + 3 a^{5} + a^{4} + 16 a^{3} + 18 a^{2} + 18 a + 13\right)\cdot 23^{166} + \left(4 a^{14} + 4 a^{13} + a^{12} + 6 a^{11} + 22 a^{10} + 6 a^{9} + 16 a^{8} + 10 a^{7} + 21 a^{6} + 17 a^{5} + 15 a^{4} + 4 a^{3} + 21 a + 15\right)\cdot 23^{167} + \left(6 a^{14} + 6 a^{13} + 9 a^{12} + 22 a^{11} + 10 a^{10} + 20 a^{9} + 12 a^{8} + 16 a^{7} + 6 a^{6} + 8 a^{5} + 19 a^{4} + 2 a^{3} + 12 a + 5\right)\cdot 23^{168} + \left(a^{14} + 16 a^{13} + 19 a^{12} + 6 a^{10} + 7 a^{9} + 10 a^{8} + 7 a^{7} + 4 a^{6} + 13 a^{5} + 14 a^{4} + 10 a^{3} + 13 a^{2} + 9 a + 5\right)\cdot 23^{169} + \left(9 a^{14} + 21 a^{13} + 7 a^{12} + 11 a^{11} + 11 a^{10} + 8 a^{8} + 16 a^{7} + 7 a^{6} + 16 a^{5} + 2 a^{4} + 16 a^{3} + 19 a^{2} + 19 a + 14\right)\cdot 23^{170} + \left(13 a^{14} + a^{13} + 2 a^{12} + 4 a^{11} + 7 a^{10} + 4 a^{9} + 3 a^{8} + 16 a^{7} + 11 a^{6} + 7 a^{5} + 4 a^{4} + 20 a^{3} + 10 a^{2} + 18 a + 9\right)\cdot 23^{171} + \left(17 a^{14} + 3 a^{13} + 8 a^{12} + 6 a^{11} + 18 a^{10} + 4 a^{9} + 11 a^{8} + 16 a^{7} + 14 a^{6} + 3 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17 a^{8} + 16 a^{7} + 5 a^{6} + 17 a^{4} + a^{3} + 4 a^{2} + 21 a + 18\right)\cdot 23^{178} + \left(12 a^{14} + a^{13} + 20 a^{12} + 20 a^{11} + 6 a^{10} + 5 a^{9} + 6 a^{8} + 4 a^{7} + 19 a^{6} + 22 a^{5} + 9 a^{4} + 18 a^{3} + 11 a^{2} + 5 a + 11\right)\cdot 23^{179} + \left(11 a^{14} + 9 a^{13} + a^{12} + 11 a^{11} + 10 a^{10} + 3 a^{9} + 5 a^{8} + 7 a^{7} + 17 a^{6} + 6 a^{5} + 13 a^{4} + 2 a^{3} + 5 a^{2} + 4 a + 1\right)\cdot 23^{180} + \left(16 a^{14} + 19 a^{13} + 7 a^{12} + 20 a^{11} + 19 a^{10} + 4 a^{9} + 18 a^{8} + 4 a^{6} + 18 a^{5} + 10 a^{4} + 4 a^{3} + 16 a^{2} + 16 a + 20\right)\cdot 23^{181} + \left(14 a^{14} + 7 a^{12} + 10 a^{11} + 8 a^{9} + 8 a^{8} + 16 a^{7} + 15 a^{6} + 12 a^{5} + 13 a^{4} + 4 a^{3} + 3 a^{2} + 21 a + 13\right)\cdot 23^{182} + \left(12 a^{14} + 2 a^{13} + 2 a^{12} + 18 a^{11} + 6 a^{10} + 9 a^{9} + 4 a^{8} + 16 a^{7} + 8 a^{6} + 12 a^{5} + 6 a^{4} + 4 a^{3} + 5 a^{2} + 12 a + 12\right)\cdot 23^{183} + \left(6 a^{14} + 20 a^{13} + 16 a^{12} + 18 a^{11} + 15 a^{10} + 14 a^{9} + 5 a^{8} + 18 a^{7} + 13 a^{6} + 21 a^{5} + 4 a^{4} + 13 a^{3} + 13 a + 11\right)\cdot 23^{184} + \left(16 a^{14} + 16 a^{13} + 10 a^{12} + 12 a^{11} + 20 a^{10} + 6 a^{9} + 15 a^{8} + 13 a^{7} + 20 a^{6} + 4 a^{5} + 17 a^{4} + 16 a^{3} + 16 a^{2} + 10 a + 3\right)\cdot 23^{185} + \left(17 a^{14} + 18 a^{13} + 5 a^{12} + 22 a^{11} + 16 a^{10} + 20 a^{9} + 20 a^{7} + 20 a^{6} + 14 a^{5} + 8 a^{4} + 14 a^{3} + 21 a^{2} + 4 a + 1\right)\cdot 23^{186} + \left(16 a^{14} + 18 a^{12} + 10 a^{11} + 3 a^{10} + 13 a^{9} + 5 a^{8} + 19 a^{7} + 9 a^{6} + 21 a^{5} + 6 a^{3} + 6 a^{2} + 3 a + 18\right)\cdot 23^{187} + \left(21 a^{14} + 6 a^{13} + 21 a^{12} + 14 a^{11} + 22 a^{10} + 15 a^{9} + 22 a^{8} + 13 a^{7} + 12 a^{6} + 3 a^{5} + 12 a^{4} + 16 a^{3} + 5 a + 13\right)\cdot 23^{188} + \left(5 a^{14} + a^{13} + 19 a^{12} + 8 a^{11} + 20 a^{10} + 16 a^{9} + 10 a^{8} + 10 a^{7} + 4 a^{6} + 9 a^{5} + a^{4} + 6 a^{3} + 16 a^{2} + 5 a + 18\right)\cdot 23^{189} + \left(7 a^{14} + 20 a^{13} + 19 a^{12} + a^{11} + 11 a^{10} + 2 a^{9} + 4 a^{8} + 18 a^{7} + 5 a^{6} + 22 a^{5} + 20 a^{4} + 13 a^{3} + 22 a^{2} + 9 a + 16\right)\cdot 23^{190} + \left(5 a^{14} + 15 a^{13} + 6 a^{12} + 17 a^{11} + 8 a^{10} + 18 a^{9} + 17 a^{8} + 4 a^{7} + 10 a^{6} + 5 a^{4} + 13 a^{3} + 20 a^{2} + 22 a + 9\right)\cdot 23^{191} + \left(19 a^{14} + 19 a^{13} + 16 a^{12} + 17 a^{11} + 9 a^{10} + a^{9} + 7 a^{8} + 10 a^{7} + 19 a^{6} + 13 a^{5} + 7 a^{4} + 11 a^{3} + 6 a^{2} + 8 a + 22\right)\cdot 23^{192} + \left(20 a^{12} + 10 a^{11} + 19 a^{10} + 10 a^{9} + a^{8} + 6 a^{7} + 12 a^{6} + 22 a^{5} + 21 a^{4} + 11 a^{3} + 9 a^{2} + 18 a + 8\right)\cdot 23^{193} + \left(8 a^{14} + 19 a^{13} + 5 a^{12} + 5 a^{11} + 4 a^{10} + 8 a^{9} + 9 a^{8} + 7 a^{7} + 11 a^{5} + 21 a^{4} + 4 a^{3} + 15 a^{2} + 7 a + 20\right)\cdot 23^{194} + \left(a^{14} + 5 a^{13} + 22 a^{12} + 12 a^{11} + 5 a^{10} + 7 a^{9} + 16 a^{8} + 7 a^{7} + 5 a^{6} + 11 a^{5} + 19 a^{4} + 8 a^{3} + 2 a^{2} + 6 a + 14\right)\cdot 23^{195} + \left(13 a^{14} + 15 a^{13} + 13 a^{12} + 9 a^{11} + 10 a^{10} + 7 a^{9} + 16 a^{8} + 18 a^{7} + 4 a^{6} + a^{5} + 19 a^{4} + 16 a^{3} + 20 a^{2} + 4 a + 19\right)\cdot 23^{196} + \left(22 a^{14} + 11 a^{13} + 10 a^{12} + 19 a^{11} + 17 a^{10} + 3 a^{9} + a^{8} + 12 a^{7} + 12 a^{6} + 11 a^{4} + 7 a^{3} + 16 a^{2} + 10 a + 22\right)\cdot 23^{197} + \left(a^{14} + 2 a^{13} + 11 a^{12} + 20 a^{11} + 3 a^{10} + 21 a^{9} + 14 a^{8} + 7 a^{7} + 5 a^{6} + 2 a^{5} + 9 a^{4} + 4 a^{3} + 21 a^{2} + 13 a + 4\right)\cdot 23^{198} + \left(12 a^{14} + 22 a^{13} + 22 a^{12} + 17 a^{11} + 2 a^{10} + 22 a^{9} + 19 a^{8} + 14 a^{7} + 8 a^{6} + 2 a^{5} + 19 a^{4} + 5 a^{3} + 16 a^{2} + 22 a + 11\right)\cdot 23^{199} + \left(15 a^{14} + 18 a^{13} + 14 a^{12} + 13 a^{11} + 16 a^{10} + 4 a^{9} + 19 a^{8} + 4 a^{7} + 16 a^{6} + 13 a^{5} + 18 a^{3} + 14 a^{2} + 11 a + 10\right)\cdot 23^{200} + \left(18 a^{14} + 17 a^{13} + 22 a^{12} + 16 a^{11} + 6 a^{10} + a^{9} + 11 a^{8} + 13 a^{7} + 6 a^{6} + 18 a^{5} + 7 a^{4} + 17 a^{3} + 18 a^{2} + 7 a + 2\right)\cdot 23^{201} + \left(7 a^{14} + 22 a^{12} + 17 a^{11} + 2 a^{10} + 11 a^{9} + 20 a^{8} + 18 a^{7} + 16 a^{6} + 15 a^{5} + 18 a^{4} + 15 a^{3} + 20 a^{2} + 18 a + 8\right)\cdot 23^{202} + \left(2 a^{14} + 16 a^{13} + 9 a^{12} + 14 a^{11} + 6 a^{10} + 9 a^{9} + 9 a^{8} + 19 a^{7} + 20 a^{6} + 10 a^{5} + 8 a^{4} + 7 a^{3} + 21 a^{2} + 21 a + 2\right)\cdot 23^{203} + \left(2 a^{14} + 4 a^{13} + 5 a^{12} + 5 a^{11} + 6 a^{10} + a^{9} + 17 a^{8} + 15 a^{7} + a^{6} + 17 a^{5} + 3 a^{4} + 5 a^{3} + 6 a^{2} + 3 a + 17\right)\cdot 23^{204} + \left(13 a^{14} + 8 a^{13} + 15 a^{12} + 17 a^{11} + 10 a^{10} + 22 a^{9} + 13 a^{8} + 7 a^{7} + 13 a^{6} + 18 a^{5} + 18 a^{4} + 11 a^{3} + 14 a^{2} + 9\right)\cdot 23^{205} + \left(12 a^{14} + 20 a^{13} + 11 a^{12} + 8 a^{11} + a^{10} + 7 a^{9} + 7 a^{8} + 21 a^{7} + 8 a^{6} + 9 a^{5} + 16 a^{4} + 6 a^{3} + 19 a^{2} + 6 a + 17\right)\cdot 23^{206} + \left(16 a^{14} + 3 a^{13} + 18 a^{12} + 11 a^{11} + 7 a^{10} + 19 a^{9} + 19 a^{8} + 7 a^{7} + 13 a^{6} + 3 a^{5} + 18 a^{4} + a^{3} + 19 a\right)\cdot 23^{207} + \left(8 a^{14} + 7 a^{13} + 3 a^{12} + 17 a^{11} + 12 a^{10} + 21 a^{9} + 17 a^{8} + 21 a^{7} + 11 a^{5} + 18 a^{4} + 19 a^{3} + 8 a^{2} + 9 a + 19\right)\cdot 23^{208} + \left(7 a^{14} + 10 a^{13} + 11 a^{12} + 10 a^{10} + 7 a^{9} + 2 a^{8} + 3 a^{6} + 15 a^{5} + 18 a^{4} + 8 a^{3} + 12 a^{2} + 8 a + 15\right)\cdot 23^{209} + \left(9 a^{14} + 5 a^{13} + a^{12} + 22 a^{11} + 12 a^{10} + 14 a^{9} + 4 a^{8} + 21 a^{7} + 6 a^{6} + 16 a^{5} + 9 a^{4} + 12 a^{3} + 9 a^{2} + 9 a + 12\right)\cdot 23^{210} + \left(4 a^{14} + 15 a^{13} + 22 a^{12} + 12 a^{11} + a^{10} + 5 a^{9} + 10 a^{8} + 16 a^{7} + 6 a^{6} + a^{5} + 19 a^{4} + 11 a^{3} + 2 a^{2} + 10 a + 5\right)\cdot 23^{211} + \left(4 a^{14} + 5 a^{13} + 9 a^{12} + 12 a^{11} + 22 a^{10} + 10 a^{9} + 3 a^{8} + 16 a^{7} + 14 a^{6} + 18 a^{5} + 20 a^{4} + 2 a^{3} + 3 a^{2} + 19 a + 7\right)\cdot 23^{212} + \left(4 a^{14} + 2 a^{13} + 4 a^{11} + 7 a^{10} + 20 a^{9} + 16 a^{8} + 21 a^{7} + 15 a^{6} + 2 a^{5} + 10 a^{4} + 7 a^{3} + 15 a + 20\right)\cdot 23^{213} + \left(15 a^{14} + 3 a^{13} + 8 a^{10} + 8 a^{8} + 2 a^{7} + 17 a^{6} + 10 a^{5} + 8 a^{4} + 6 a^{3} + 19 a^{2} + 13 a + 22\right)\cdot 23^{214} + \left(21 a^{14} + 6 a^{13} + 12 a^{12} + 13 a^{11} + 3 a^{10} + 15 a^{9} + 13 a^{8} + 13 a^{7} + 15 a^{5} + 11 a^{4} + 21 a^{3} + 16 a^{2} + 2 a + 4\right)\cdot 23^{215} + \left(3 a^{14} + 21 a^{13} + 17 a^{12} + 10 a^{11} + 19 a^{10} + 17 a^{9} + 11 a^{8} + 5 a^{7} + 15 a^{6} + 14 a^{4} + 16 a^{3} + 8 a^{2} + 16 a + 13\right)\cdot 23^{216} + \left(21 a^{14} + 7 a^{13} + 21 a^{12} + 2 a^{11} + 17 a^{10} + 2 a^{9} + 19 a^{8} + 2 a^{7} + a^{6} + 13 a^{5} + 19 a^{4} + 2 a^{3} + 8 a^{2} + a + 8\right)\cdot 23^{217} + \left(5 a^{14} + a^{13} + 8 a^{12} + 19 a^{11} + 20 a^{10} + 17 a^{9} + a^{8} + a^{7} + 22 a^{6} + 3 a^{5} + 19 a^{4} + 5 a^{3} + 7 a^{2} + 13 a + 7\right)\cdot 23^{218} + \left(a^{14} + 4 a^{13} + 4 a^{12} + 12 a^{11} + 11 a^{10} + a^{9} + 4 a^{8} + 3 a^{7} + 20 a^{6} + 9 a^{5} + 10 a^{4} + 4 a^{3} + 10 a^{2} + 19 a + 1\right)\cdot 23^{219} + \left(17 a^{14} + 6 a^{13} + 18 a^{12} + 17 a^{11} + 10 a^{10} + 20 a^{9} + 19 a^{8} + 5 a^{7} + 10 a^{6} + 22 a^{5} + 8 a^{3} + 20 a^{2} + 4 a + 5\right)\cdot 23^{220} + \left(8 a^{14} + 6 a^{13} + a^{11} + a^{10} + 21 a^{9} + 20 a^{8} + 3 a^{7} + 15 a^{6} + 21 a^{5} + 2 a^{4} + 21 a^{3} + 5 a^{2} + 8 a + 11\right)\cdot 23^{221} + \left(3 a^{14} + 14 a^{13} + 13 a^{12} + 12 a^{11} + 22 a^{10} + 13 a^{9} + 15 a^{8} + 8 a^{7} + 20 a^{6} + 3 a^{5} + 18 a^{4} + 11 a^{3} + 14 a + 11\right)\cdot 23^{222} + \left(12 a^{14} + 10 a^{13} + 9 a^{12} + 7 a^{11} + 9 a^{10} + 8 a^{9} + 22 a^{8} + 21 a^{7} + 14 a^{6} + 7 a^{5} + 4 a^{4} + 16 a^{3} + 18 a^{2} + 17 a + 15\right)\cdot 23^{223} + \left(19 a^{14} + 9 a^{13} + 18 a^{12} + 20 a^{11} + 13 a^{10} + 18 a^{9} + 4 a^{8} + 13 a^{7} + 22 a^{6} + 6 a^{5} + 14 a^{4} + 2 a^{3} + 19 a^{2} + a + 12\right)\cdot 23^{224} + \left(2 a^{14} + 18 a^{13} + 20 a^{12} + 17 a^{11} + 15 a^{9} + 11 a^{8} + 15 a^{7} + 14 a^{6} + a^{5} + 3 a^{4} + 12 a^{3} + 17 a^{2} + 22 a + 8\right)\cdot 23^{225} + \left(15 a^{14} + 17 a^{13} + 5 a^{12} + 21 a^{11} + 8 a^{10} + 14 a^{9} + 13 a^{8} + 13 a^{7} + 8 a^{6} + 13 a^{5} + 15 a^{4} + 7 a^{3} + 15 a\right)\cdot 23^{226} + \left(12 a^{14} + 14 a^{13} + 13 a^{12} + 10 a^{11} + 10 a^{10} + 10 a^{9} + 15 a^{8} + 17 a^{7} + 3 a^{6} + 8 a^{5} + 19 a^{4} + 18 a^{3} + 12 a^{2} + 17 a + 6\right)\cdot 23^{227} + \left(7 a^{14} + 2 a^{13} + 14 a^{12} + 2 a^{11} + 16 a^{10} + 10 a^{9} + 10 a^{8} + 17 a^{7} + 13 a^{6} + 3 a^{5} + 18 a^{4} + 9 a^{3} + 2 a + 20\right)\cdot 23^{228} + \left(7 a^{14} + 22 a^{13} + 14 a^{12} + 7 a^{11} + 21 a^{10} + 2 a^{9} + 7 a^{8} + 2 a^{7} + 2 a^{6} + 18 a^{5} + 13 a^{4} + 15 a^{3} + 9 a^{2} + 15 a + 11\right)\cdot 23^{229} + \left(3 a^{14} + 2 a^{13} + 20 a^{12} + 4 a^{11} + 20 a^{10} + a^{9} + 13 a^{8} + 2 a^{7} + 3 a^{6} + 21 a^{5} + 9 a^{4} + 5 a^{3} + 13 a^{2} + 10 a + 20\right)\cdot 23^{230} + \left(15 a^{14} + 21 a^{13} + 21 a^{12} + 15 a^{11} + 22 a^{10} + 4 a^{9} + 9 a^{8} + 6 a^{7} + 18 a^{6} + 19 a^{5} + 9 a^{4} + 12 a^{3} + 15 a + 19\right)\cdot 23^{231} + \left(21 a^{14} + 17 a^{13} + 3 a^{12} + 12 a^{11} + 13 a^{10} + 19 a^{9} + 21 a^{8} + 7 a^{7} + 8 a^{6} + 5 a^{5} + 17 a^{4} + 9 a^{3} + a^{2} + 3 a\right)\cdot 23^{232} + \left(a^{14} + 13 a^{12} + 7 a^{11} + 4 a^{10} + 7 a^{9} + 7 a^{8} + 12 a^{7} + 16 a^{6} + 13 a^{5} + 13 a^{4} + 9 a^{3} + 20 a^{2} + 13\right)\cdot 23^{233} + \left(2 a^{14} + 17 a^{13} + 13 a^{12} + 6 a^{11} + 5 a^{10} + 10 a^{9} + 9 a^{8} + 20 a^{7} + 16 a^{6} + 11 a^{4} + 13 a^{3} + 22 a^{2} + 19 a + 10\right)\cdot 23^{234} + \left(6 a^{14} + 3 a^{13} + 21 a^{11} + 17 a^{10} + 19 a^{9} + 11 a^{8} + 11 a^{7} + 16 a^{6} + 4 a^{5} + 7 a^{4} + 8 a^{3} + 19 a^{2} + 12 a + 21\right)\cdot 23^{235} + \left(17 a^{14} + 9 a^{13} + 4 a^{11} + 4 a^{10} + 8 a^{8} + 6 a^{7} + 2 a^{6} + 12 a^{5} + 4 a^{4} + 19 a^{3} + 21 a^{2} + 9 a + 13\right)\cdot 23^{236} + \left(6 a^{14} + 11 a^{13} + 9 a^{12} + 9 a^{11} + 18 a^{10} + 11 a^{9} + 20 a^{8} + 5 a^{7} + 14 a^{6} + 7 a^{5} + a^{4} + 14 a^{3} + 6 a^{2} + 8 a + 10\right)\cdot 23^{237} + \left(9 a^{14} + 13 a^{13} + 3 a^{12} + 17 a^{11} + 8 a^{10} + 2 a^{9} + 3 a^{8} + 21 a^{7} + a^{5} + 5 a^{4} + 20 a^{3} + 7 a^{2} + 5 a + 7\right)\cdot 23^{238} + \left(12 a^{14} + 22 a^{13} + 5 a^{12} + 3 a^{11} + 11 a^{10} + 20 a^{9} + 9 a^{8} + 3 a^{7} + 12 a^{6} + 2 a^{5} + 11 a^{4} + 21 a^{3} + 4 a^{2} + 9 a + 11\right)\cdot 23^{239} + \left(11 a^{14} + 9 a^{13} + 22 a^{12} + 14 a^{11} + 2 a^{10} + 21 a^{9} + 22 a^{8} + 4 a^{7} + 4 a^{6} + 5 a^{5} + 22 a^{3} + 4 a^{2} + 4 a + 13\right)\cdot 23^{240} + \left(4 a^{14} + 22 a^{13} + 5 a^{12} + 21 a^{11} + 6 a^{10} + 19 a^{9} + 19 a^{8} + 20 a^{7} + 11 a^{6} + 14 a^{5} + 16 a^{4} + 9 a^{3} + 11 a^{2} + 16 a + 21\right)\cdot 23^{241} + \left(16 a^{14} + 7 a^{13} + 21 a^{12} + 12 a^{11} + 22 a^{10} + 21 a^{9} + 14 a^{8} + 6 a^{7} + 3 a^{6} + 8 a^{5} + 13 a^{4} + 13 a^{3} + 22 a^{2} + 16 a + 9\right)\cdot 23^{242} + \left(11 a^{14} + 2 a^{13} + 16 a^{12} + 8 a^{11} + 18 a^{10} + a^{9} + 21 a^{8} + 2 a^{7} + 14 a^{6} + 2 a^{5} + 21 a^{4} + 10 a^{3} + 5 a^{2} + 11 a + 22\right)\cdot 23^{243} + \left(7 a^{14} + 11 a^{13} + 6 a^{12} + 8 a^{11} + a^{10} + 11 a^{9} + 7 a^{7} + 10 a^{6} + 8 a^{5} + 20 a^{4} + 16 a^{3} + 22 a^{2} + 14 a + 9\right)\cdot 23^{244} + \left(2 a^{14} + 20 a^{13} + 17 a^{12} + 17 a^{11} + 5 a^{10} + 14 a^{9} + 7 a^{8} + 4 a^{6} + 22 a^{4} + 6 a^{3} + 19 a^{2} + 8\right)\cdot 23^{245} + \left(a^{14} + 12 a^{13} + 12 a^{12} + 14 a^{11} + 5 a^{10} + 2 a^{9} + 6 a^{8} + 13 a^{7} + 9 a^{6} + 13 a^{5} + 21 a^{4} + 4 a^{2} + 17 a + 8\right)\cdot 23^{246} + \left(16 a^{14} + 14 a^{13} + 19 a^{12} + 8 a^{11} + 20 a^{9} + 5 a^{8} + 19 a^{7} + 9 a^{6} + 7 a^{5} + 18 a^{4} + 22 a^{3} + 13 a + 4\right)\cdot 23^{247} + \left(20 a^{14} + 6 a^{13} + 15 a^{12} + 17 a^{11} + 12 a^{10} + 13 a^{9} + 10 a^{8} + 9 a^{7} + 20 a^{6} + 3 a^{5} + 4 a^{4} + 15 a^{3} + 9 a^{2} + 5 a + 7\right)\cdot 23^{248} + \left(9 a^{14} + 2 a^{13} + 19 a^{12} + 22 a^{11} + 2 a^{10} + 21 a^{9} + 14 a^{8} + 5 a^{7} + 16 a^{6} + 4 a^{5} + 9 a^{4} + 8 a^{3} + 20 a^{2} + 12 a + 16\right)\cdot 23^{249} + \left(16 a^{14} + 8 a^{13} + 14 a^{12} + 6 a^{11} + 11 a^{10} + 11 a^{9} + 2 a^{8} + 7 a^{7} + 11 a^{6} + 21 a^{5} + 21 a^{4} + 7 a^{3} + 7 a^{2} + 6 a + 9\right)\cdot 23^{250} + \left(6 a^{14} + 13 a^{13} + 13 a^{12} + 10 a^{11} + 19 a^{10} + 15 a^{9} + 7 a^{8} + 13 a^{7} + 18 a^{6} + 10 a^{5} + 10 a^{4} + 17 a^{3} + 2 a + 8\right)\cdot 23^{251} + \left(17 a^{14} + 2 a^{13} + 8 a^{12} + 10 a^{11} + 16 a^{10} + 5 a^{9} + 9 a^{8} + 9 a^{7} + 10 a^{6} + 16 a^{5} + a^{4} + 22 a^{3} + 22 a^{2} + 16 a + 11\right)\cdot 23^{252} + \left(16 a^{14} + 19 a^{13} + 3 a^{12} + 16 a^{11} + 15 a^{10} + 2 a^{9} + 18 a^{8} + 13 a^{7} + 4 a^{6} + 4 a^{5} + 11 a^{4} + a^{3} + 11 a^{2} + 10 a + 19\right)\cdot 23^{253} + \left(13 a^{14} + 12 a^{13} + 19 a^{12} + 6 a^{11} + 16 a^{10} + 22 a^{9} + 18 a^{8} + 10 a^{7} + 3 a^{6} + a^{5} + 11 a^{4} + 8 a^{3} + 6 a^{2} + 18 a + 12\right)\cdot 23^{254} + \left(a^{14} + 18 a^{13} + 16 a^{12} + 22 a^{11} + 16 a^{10} + 11 a^{9} + 14 a^{8} + 3 a^{7} + 13 a^{6} + 5 a^{5} + 12 a^{4} + 16 a^{3} + 3 a^{2} + 18 a + 1\right)\cdot 23^{255} + \left(9 a^{14} + 8 a^{13} + 19 a^{12} + 4 a^{11} + 4 a^{10} + a^{9} + 21 a^{8} + 9 a^{7} + 5 a^{6} + 11 a^{5} + 7 a^{4} + 16 a^{3} + a^{2} + 19 a + 12\right)\cdot 23^{256} + \left(15 a^{14} + 9 a^{13} + a^{12} + 20 a^{11} + 10 a^{9} + 21 a^{8} + 8 a^{7} + 7 a^{6} + 7 a^{5} + 9 a^{4} + 11 a^{3} + 5 a^{2} + 15 a + 17\right)\cdot 23^{257} + \left(8 a^{14} + 5 a^{13} + 16 a^{12} + a^{11} + 15 a^{9} + 21 a^{7} + 4 a^{6} + 9 a^{5} + 19 a^{3} + 16 a^{2} + 6 a + 14\right)\cdot 23^{258} + \left(21 a^{14} + 8 a^{13} + 3 a^{12} + 13 a^{11} + 10 a^{10} + 12 a^{9} + 21 a^{8} + 10 a^{7} + 5 a^{6} + 16 a^{5} + 16 a^{4} + 3 a^{3} + 11 a^{2} + 2 a + 1\right)\cdot 23^{259} + \left(16 a^{14} + 6 a^{13} + 10 a^{12} + a^{11} + 2 a^{10} + 11 a^{9} + 2 a^{8} + 17 a^{7} + 19 a^{6} + 12 a^{5} + 9 a^{4} + 8 a^{3} + 2 a^{2} + 13 a + 13\right)\cdot 23^{260} + \left(18 a^{14} + 14 a^{13} + 16 a^{12} + 19 a^{10} + 12 a^{9} + 14 a^{8} + 22 a^{7} + 10 a^{6} + 16 a^{5} + 7 a^{4} + 14 a^{3} + a^{2} + 18 a + 7\right)\cdot 23^{261} + \left(22 a^{14} + 22 a^{13} + 11 a^{12} + 3 a^{11} + 20 a^{9} + 4 a^{8} + a^{7} + 13 a^{6} + 20 a^{5} + a^{4} + 7 a^{3} + 18 a^{2} + 22 a + 14\right)\cdot 23^{262} + \left(20 a^{14} + 11 a^{13} + 16 a^{11} + 3 a^{10} + 9 a^{9} + 18 a^{8} + 3 a^{7} + a^{6} + 13 a^{5} + 19 a^{4} + 18 a^{3} + 5 a^{2} + 14 a + 16\right)\cdot 23^{263} + \left(2 a^{14} + 16 a^{13} + 12 a^{12} + 6 a^{11} + 6 a^{10} + 15 a^{9} + 9 a^{8} + 16 a^{7} + a^{6} + 13 a^{5} + 4 a^{4} + 10 a^{3} + 3 a^{2} + 7 a + 20\right)\cdot 23^{264} + \left(20 a^{14} + 11 a^{13} + 13 a^{12} + 2 a^{11} + 22 a^{9} + 10 a^{8} + 14 a^{7} + 4 a^{6} + 14 a^{5} + 7 a^{4} + 4 a^{3} + 8 a^{2} + 5 a + 17\right)\cdot 23^{265} + \left(19 a^{14} + 14 a^{13} + 5 a^{11} + 11 a^{10} + 3 a^{9} + 20 a^{8} + 6 a^{7} + 7 a^{6} + a^{5} + 10 a^{4} + 3 a^{3} + 15 a^{2} + 9 a + 11\right)\cdot 23^{266} + \left(4 a^{14} + 7 a^{11} + 4 a^{10} + 8 a^{9} + 7 a^{8} + 3 a^{7} + 3 a^{6} + 20 a^{5} + 19 a^{4} + 19 a^{3} + 7 a^{2} + 3 a + 13\right)\cdot 23^{267} + \left(2 a^{14} + 7 a^{13} + 2 a^{12} + 2 a^{11} + 11 a^{10} + 5 a^{9} + 16 a^{8} + 6 a^{7} + 13 a^{6} + 10 a^{5} + 3 a^{4} + 19 a^{3} + 4 a + 5\right)\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 16 }$ $=$ $12 a^{14} + 18 a^{13} + 15 a^{12} + 7 a^{11} + 19 a^{10} + 11 a^{9} + 18 a^{8} + 22 a^{7} + 5 a^{5} + 9 a^{4} + 22 a^{3} + 15 a^{2} + 2 a + 12 + \left(8 a^{14} + 17 a^{13} + 12 a^{12} + 10 a^{11} + a^{10} + 3 a^{9} + 9 a^{8} + 6 a^{7} + 6 a^{6} + 5 a^{5} + 7 a^{4} + 5 a^{3} + 15 a^{2} + 20 a\right)\cdot 23 + \left(17 a^{14} + 12 a^{13} + 14 a^{12} + 17 a^{11} + a^{10} + 8 a^{9} + a^{8} + 14 a^{7} + 15 a^{6} + 18 a^{5} + 18 a^{4} + 8 a^{3} + 13 a^{2} + 16 a + 11\right)\cdot 23^{2} + \left(11 a^{14} + a^{13} + 17 a^{12} + 13 a^{11} + 2 a^{10} + 15 a^{9} + 15 a^{8} + 5 a^{7} + 18 a^{6} + 5 a^{5} + 7 a^{4} + 13 a^{3} + 21 a^{2} + 12 a + 9\right)\cdot 23^{3} + \left(6 a^{14} + 15 a^{13} + 8 a^{12} + 9 a^{11} + 3 a^{10} + 3 a^{9} + 4 a^{8} + 18 a^{7} + 18 a^{6} + 8 a^{5} + 8 a^{4} + a^{3} + 10 a^{2} + 7\right)\cdot 23^{4} + \left(20 a^{14} + 18 a^{13} + 17 a^{12} + 21 a^{11} + 18 a^{10} + 18 a^{9} + 18 a^{8} + 5 a^{7} + 10 a^{6} + 19 a^{5} + 21 a^{3} + 17 a^{2} + 18 a + 19\right)\cdot 23^{5} + \left(22 a^{14} + 8 a^{13} + 18 a^{12} + 22 a^{11} + 20 a^{10} + 8 a^{8} + 10 a^{7} + 12 a^{6} + 12 a^{5} + 16 a^{4} + 13 a^{3} + 16 a^{2} + 12 a + 8\right)\cdot 23^{6} + \left(15 a^{14} + 7 a^{13} + 20 a^{12} + 4 a^{10} + 11 a^{9} + 9 a^{8} + 2 a^{7} + 19 a^{6} + 8 a^{5} + 7 a^{4} + 17 a^{3} + 2 a^{2} + 9 a + 3\right)\cdot 23^{7} + \left(5 a^{14} + 19 a^{13} + 4 a^{12} + 12 a^{11} + 15 a^{9} + 19 a^{8} + 3 a^{7} + 2 a^{6} + 12 a^{5} + 9 a^{4} + 6 a^{3} + 20 a^{2} + 10 a + 16\right)\cdot 23^{8} + \left(18 a^{14} + 13 a^{13} + 14 a^{12} + 3 a^{11} + 17 a^{10} + 20 a^{9} + 21 a^{8} + 12 a^{7} + 6 a^{6} + 2 a^{5} + 22 a^{4} + 18 a^{3} + 2 a^{2} + 10 a + 16\right)\cdot 23^{9} + \left(5 a^{14} + 15 a^{13} + 21 a^{12} + 14 a^{11} + 3 a^{10} + 5 a^{9} + 22 a^{7} + 14 a^{6} + 3 a^{5} + 15 a^{4} + 14 a^{2} + 2 a\right)\cdot 23^{10} + \left(14 a^{14} + 9 a^{13} + 13 a^{12} + 5 a^{11} + 13 a^{10} + 15 a^{9} + 5 a^{8} + 19 a^{7} + 20 a^{6} + 18 a^{5} + 14 a^{4} + 7 a^{3} + 20 a^{2}\right)\cdot 23^{11} + \left(11 a^{14} + 14 a^{13} + 12 a^{12} + 5 a^{11} + 3 a^{9} + 17 a^{8} + 22 a^{7} + 4 a^{6} + 16 a^{5} + 11 a^{4} + 10 a^{3} + 4 a^{2} + 13 a + 21\right)\cdot 23^{12} + \left(11 a^{14} + 20 a^{13} + 12 a^{12} + 8 a^{11} + 10 a^{10} + 10 a^{9} + 21 a^{8} + 8 a^{7} + 8 a^{6} + 5 a^{5} + 20 a^{4} + 4 a^{2} + a + 7\right)\cdot 23^{13} + \left(15 a^{14} + 5 a^{13} + 12 a^{12} + 6 a^{11} + 4 a^{10} + 5 a^{9} + 14 a^{8} + 19 a^{7} + 17 a^{5} + 17 a^{4} + 9 a^{3} + 3 a^{2} + 3 a + 3\right)\cdot 23^{14} + \left(22 a^{14} + 9 a^{13} + 14 a^{12} + 8 a^{11} + 12 a^{10} + 12 a^{9} + 9 a^{8} + 14 a^{7} + 15 a^{6} + 10 a^{5} + 2 a^{4} + 20 a^{3} + 7 a^{2} + 20 a + 20\right)\cdot 23^{15} + \left(8 a^{14} + 5 a^{12} + 15 a^{11} + 2 a^{10} + 5 a^{9} + 16 a^{8} + 17 a^{7} + 4 a^{6} + 11 a^{5} + 5 a^{4} + 18 a^{3} + 18 a^{2} + 6 a + 13\right)\cdot 23^{16} + \left(8 a^{14} + 17 a^{13} + 16 a^{12} + 11 a^{11} + 14 a^{10} + 18 a^{9} + 4 a^{8} + 14 a^{7} + 19 a^{6} + 7 a^{5} + 2 a^{4} + 3 a^{3} + 18 a^{2} + 12 a + 21\right)\cdot 23^{17} + \left(3 a^{14} + 13 a^{13} + 9 a^{11} + 22 a^{10} + 8 a^{9} + 5 a^{8} + 2 a^{7} + 2 a^{6} + 19 a^{5} + 2 a^{4} + 7 a^{3} + 18 a^{2} + 12 a + 13\right)\cdot 23^{18} + \left(20 a^{14} + 13 a^{13} + 15 a^{12} + 22 a^{11} + 11 a^{10} + 20 a^{8} + a^{7} + 4 a^{6} + 8 a^{5} + 7 a^{4} + 19 a^{3} + 15 a^{2} + a\right)\cdot 23^{19} + \left(4 a^{14} + 8 a^{13} + 10 a^{12} + 5 a^{11} + 6 a^{10} + 18 a^{9} + 10 a^{8} + 5 a^{7} + 10 a^{6} + 9 a^{5} + 18 a^{4} + 14 a^{3} + 9 a^{2} + 18 a + 4\right)\cdot 23^{20} + \left(18 a^{14} + 7 a^{13} + 22 a^{12} + 19 a^{11} + 21 a^{9} + 10 a^{8} + 3 a^{7} + 3 a^{6} + 20 a^{5} + 3 a^{4} + a^{3} + 2 a^{2} + 7 a + 2\right)\cdot 23^{21} + \left(14 a^{14} + 14 a^{13} + a^{12} + 16 a^{11} + 3 a^{10} + 16 a^{9} + 19 a^{8} + 7 a^{7} + 8 a^{6} + 21 a^{5} + 9 a^{4} + 10 a^{3} + 19 a^{2} + 16 a + 7\right)\cdot 23^{22} + \left(11 a^{14} + 13 a^{13} + 14 a^{12} + 15 a^{11} + 8 a^{10} + 17 a^{9} + 9 a^{8} + 9 a^{7} + 10 a^{6} + 11 a^{5} + 19 a^{4} + 18 a^{3} + 8 a^{2} + 16 a + 10\right)\cdot 23^{23} + \left(4 a^{14} + 2 a^{13} + 16 a^{12} + 10 a^{11} + a^{10} + 3 a^{9} + 12 a^{8} + 4 a^{6} + 17 a^{5} + a^{4} + 4 a^{3} + 13 a^{2} + 10 a + 6\right)\cdot 23^{24} + \left(3 a^{14} + 14 a^{13} + 3 a^{12} + 5 a^{11} + 20 a^{10} + 5 a^{9} + 7 a^{7} + 6 a^{6} + 17 a^{5} + 19 a^{4} + 8 a^{3} + 4 a^{2} + 10 a + 4\right)\cdot 23^{25} + \left(13 a^{14} + 20 a^{13} + 7 a^{12} + 3 a^{11} + 3 a^{10} + 22 a^{9} + 9 a^{8} + 21 a^{7} + 9 a^{6} + 2 a^{5} + 13 a^{4} + 3 a^{3} + 16 a^{2} + 2 a + 14\right)\cdot 23^{26} + \left(19 a^{14} + 3 a^{13} + 19 a^{12} + 18 a^{11} + 14 a^{10} + 22 a^{9} + 18 a^{8} + 18 a^{7} + a^{6} + 4 a^{5} + 2 a^{4} + 17 a^{2} + 2 a + 16\right)\cdot 23^{27} + \left(2 a^{14} + 22 a^{13} + 10 a^{12} + 16 a^{11} + 20 a^{10} + 6 a^{9} + 15 a^{8} + a^{7} + 9 a^{6} + 5 a^{5} + 19 a^{4} + 9 a^{3} + 6 a^{2} + 4 a + 15\right)\cdot 23^{28} + \left(18 a^{14} + 4 a^{13} + 6 a^{12} + 18 a^{11} + 12 a^{10} + a^{9} + a^{8} + 13 a^{7} + 20 a^{6} + a^{4} + 8 a^{2} + 6 a + 4\right)\cdot 23^{29} + \left(14 a^{14} + 2 a^{12} + 6 a^{11} + 8 a^{10} + 3 a^{8} + 10 a^{7} + 21 a^{6} + 22 a^{5} + 11 a^{4} + 3 a^{3} + 5 a^{2} + 21 a + 18\right)\cdot 23^{30} + \left(18 a^{14} + 3 a^{13} + 9 a^{12} + a^{11} + 7 a^{10} + 8 a^{9} + 22 a^{8} + 8 a^{7} + 10 a^{6} + 12 a^{5} + 6 a^{4} + 12 a^{3} + 6 a^{2} + 8 a + 19\right)\cdot 23^{31} + \left(15 a^{14} + 2 a^{13} + 22 a^{12} + 2 a^{11} + 14 a^{10} + 18 a^{9} + 2 a^{8} + 4 a^{7} + 15 a^{6} + 2 a^{5} + 21 a^{4} + 2 a^{3} + 8 a^{2} + 7 a + 2\right)\cdot 23^{32} + \left(3 a^{14} + 4 a^{13} + 3 a^{12} + a^{11} + 2 a^{10} + a^{9} + 7 a^{8} + 19 a^{7} + 21 a^{6} + 9 a^{5} + a^{4} + 2 a^{3} + 15 a^{2} + 11 a + 22\right)\cdot 23^{33} + \left(21 a^{14} + 4 a^{13} + 3 a^{12} + a^{11} + 5 a^{10} + 5 a^{9} + 15 a^{8} + 22 a^{7} + 15 a^{6} + 8 a^{5} + 17 a^{4} + 22 a^{3} + 19 a^{2} + 8 a + 8\right)\cdot 23^{34} + \left(4 a^{14} + 12 a^{13} + 6 a^{12} + 4 a^{11} + 22 a^{10} + 20 a^{9} + 7 a^{8} + 15 a^{7} + 5 a^{6} + 3 a^{5} + a^{4} + 6 a^{3} + 7 a^{2} + 16 a + 15\right)\cdot 23^{35} + \left(21 a^{14} + 8 a^{13} + a^{12} + 11 a^{11} + 4 a^{10} + 17 a^{9} + 10 a^{8} + 13 a^{6} + 3 a^{5} + 6 a^{4} + 22 a^{3} + 8 a^{2} + 20 a + 18\right)\cdot 23^{36} + \left(17 a^{14} + 7 a^{13} + 21 a^{12} + 3 a^{11} + 21 a^{10} + 21 a^{9} + 15 a^{8} + 10 a^{7} + 7 a^{6} + 21 a^{5} + 22 a^{4} + 18 a^{3} + 15 a^{2} + 9 a + 9\right)\cdot 23^{37} + \left(16 a^{14} + 2 a^{13} + a^{12} + 8 a^{11} + 15 a^{10} + a^{9} + 16 a^{8} + 21 a^{7} + 3 a^{6} + 3 a^{5} + a^{4} + 17 a^{3} + 3 a^{2} + 2 a + 16\right)\cdot 23^{38} + \left(11 a^{14} + 4 a^{13} + 9 a^{12} + 4 a^{11} + 8 a^{10} + 13 a^{9} + 15 a^{8} + 11 a^{7} + 4 a^{6} + 15 a^{5} + 7 a^{4} + 19 a^{3} + a^{2} + 10 a + 8\right)\cdot 23^{39} + \left(18 a^{14} + a^{13} + 19 a^{12} + 10 a^{10} + 20 a^{9} + 21 a^{8} + 22 a^{7} + 6 a^{5} + 12 a^{4} + 17 a^{3} + 20 a^{2} + 7 a + 6\right)\cdot 23^{40} + \left(10 a^{14} + 21 a^{13} + 8 a^{12} + 17 a^{11} + 19 a^{10} + 11 a^{9} + 17 a^{8} + 4 a^{7} + 11 a^{5} + 12 a^{4} + 9 a^{3} + 14 a^{2} + 5 a + 5\right)\cdot 23^{41} + \left(5 a^{14} + 12 a^{13} + 12 a^{12} + 2 a^{10} + 9 a^{9} + 6 a^{8} + 15 a^{7} + 12 a^{6} + 22 a^{5} + 8 a^{4} + 16 a^{3} + 2 a^{2} + 3 a + 19\right)\cdot 23^{42} + \left(16 a^{14} + 21 a^{13} + 22 a^{10} + 2 a^{9} + 21 a^{8} + a^{7} + 14 a^{5} + 9 a^{4} + 19 a^{3} + 13 a^{2} + a + 18\right)\cdot 23^{43} + \left(16 a^{14} + 21 a^{13} + 19 a^{12} + 15 a^{11} + 9 a^{10} + 6 a^{9} + 21 a^{8} + a^{6} + 19 a^{5} + 9 a^{4} + 4 a^{3} + a^{2} + 6 a + 11\right)\cdot 23^{44} + \left(18 a^{14} + 14 a^{13} + 3 a^{12} + 22 a^{11} + 12 a^{10} + 17 a^{9} + 18 a^{8} + 21 a^{7} + 6 a^{6} + 22 a^{5} + 9 a^{4} + 9 a^{3} + 6 a^{2} + 16 a + 17\right)\cdot 23^{45} + \left(17 a^{14} + 3 a^{13} + 10 a^{12} + 17 a^{11} + 5 a^{10} + 7 a^{9} + 17 a^{8} + 4 a^{7} + 2 a^{6} + 22 a^{5} + 21 a^{4} + 5 a^{3} + 6 a^{2} + 13 a + 8\right)\cdot 23^{46} + \left(5 a^{14} + 14 a^{13} + 12 a^{12} + 5 a^{11} + 20 a^{9} + 21 a^{8} + 21 a^{7} + 6 a^{6} + 11 a^{5} + 19 a^{4} + 13 a^{3} + 2 a^{2} + 12 a + 4\right)\cdot 23^{47} + \left(13 a^{14} + 5 a^{13} + 17 a^{12} + 6 a^{11} + 21 a^{10} + 13 a^{9} + 7 a^{8} + 10 a^{7} + 9 a^{6} + 18 a^{5} + 20 a^{4} + 6 a^{3} + 11 a^{2} + 6 a + 21\right)\cdot 23^{48} + \left(20 a^{14} + 6 a^{13} + 12 a^{12} + 13 a^{11} + 9 a^{7} + 4 a^{6} + 2 a^{5} + 2 a^{4} + 3 a^{3} + 3 a^{2} + 18 a + 17\right)\cdot 23^{49} + \left(11 a^{14} + 5 a^{13} + 8 a^{12} + 3 a^{11} + 3 a^{10} + 7 a^{9} + 17 a^{8} + 18 a^{7} + 12 a^{6} + 3 a^{5} + 22 a^{4} + 15 a^{3} + 4 a^{2} + 18 a + 17\right)\cdot 23^{50} + \left(9 a^{14} + 9 a^{13} + 2 a^{12} + 13 a^{11} + 21 a^{10} + 15 a^{9} + 6 a^{8} + 21 a^{7} + 5 a^{6} + 22 a^{5} + 21 a^{4} + 9 a^{3} + 4 a^{2} + 10 a + 3\right)\cdot 23^{51} + \left(18 a^{14} + 22 a^{13} + 21 a^{12} + 11 a^{11} + 8 a^{10} + 4 a^{9} + 8 a^{8} + 16 a^{7} + 18 a^{6} + 19 a^{5} + 14 a^{4} + 2 a^{3} + 11 a^{2} + 9 a + 12\right)\cdot 23^{52} + \left(16 a^{14} + 21 a^{13} + 17 a^{12} + 6 a^{11} + 9 a^{10} + 17 a^{9} + 7 a^{8} + 7 a^{7} + 2 a^{6} + 14 a^{5} + 20 a^{3} + 20 a^{2} + a + 4\right)\cdot 23^{53} + \left(20 a^{13} + 18 a^{12} + 22 a^{11} + 16 a^{10} + 17 a^{9} + 17 a^{8} + 22 a^{7} + 14 a^{6} + 22 a^{5} + 15 a^{4} + 14 a^{3} + 17 a^{2} + 13 a + 7\right)\cdot 23^{54} + \left(15 a^{14} + 18 a^{13} + 16 a^{12} + 7 a^{11} + 10 a^{10} + 7 a^{9} + 8 a^{8} + a^{7} + 9 a^{6} + 8 a^{5} + 2 a^{4} + 11 a^{3} + 21 a^{2} + 7 a + 14\right)\cdot 23^{55} + \left(6 a^{14} + 6 a^{13} + 22 a^{12} + 3 a^{11} + 11 a^{10} + 21 a^{9} + 8 a^{8} + 16 a^{7} + 18 a^{6} + 18 a^{5} + 21 a^{4} + 2 a^{2} + 5 a + 2\right)\cdot 23^{56} + \left(20 a^{13} + 21 a^{12} + 18 a^{11} + 8 a^{10} + 22 a^{9} + 21 a^{8} + 16 a^{7} + 8 a^{6} + 22 a^{5} + 4 a^{4} + 10 a^{3} + 16 a^{2} + 8 a + 3\right)\cdot 23^{57} + \left(20 a^{14} + 17 a^{13} + 12 a^{12} + 15 a^{11} + 11 a^{10} + 20 a^{9} + 8 a^{8} + 14 a^{7} + 12 a^{5} + 7 a^{3} + 15 a^{2} + 19 a + 19\right)\cdot 23^{58} + \left(14 a^{14} + 18 a^{13} + 18 a^{12} + 15 a^{11} + 5 a^{10} + 11 a^{9} + 21 a^{8} + 18 a^{7} + 11 a^{6} + 21 a^{5} + 22 a^{4} + 13 a^{3} + 7 a^{2} + 4 a + 17\right)\cdot 23^{59} + \left(10 a^{14} + 13 a^{13} + 5 a^{12} + 12 a^{11} + 3 a^{10} + 10 a^{9} + 10 a^{8} + 6 a^{7} + 16 a^{6} + 7 a^{5} + 7 a^{4} + 21 a^{3} + 9 a^{2} + 15 a + 22\right)\cdot 23^{60} + \left(13 a^{14} + 11 a^{13} + 6 a^{12} + 13 a^{11} + 19 a^{10} + 21 a^{9} + 5 a^{8} + 11 a^{7} + 6 a^{6} + 19 a^{4} + 10 a^{3} + 18 a^{2} + 17 a + 11\right)\cdot 23^{61} + \left(16 a^{14} + 10 a^{13} + 5 a^{12} + 16 a^{11} + a^{10} + 21 a^{9} + 8 a^{8} + 14 a^{7} + 14 a^{6} + 4 a^{4} + 8 a^{3} + 16 a^{2} + 5 a + 20\right)\cdot 23^{62} + \left(8 a^{14} + 5 a^{13} + 16 a^{11} + 11 a^{9} + 20 a^{8} + 13 a^{7} + 16 a^{6} + 8 a^{5} + 7 a^{4} + 6 a^{3} + 4 a^{2} + 4 a + 11\right)\cdot 23^{63} + \left(2 a^{14} + 13 a^{13} + 9 a^{12} + 9 a^{11} + 5 a^{10} + a^{9} + 11 a^{8} + 9 a^{7} + 2 a^{6} + 9 a^{5} + 14 a^{4} + a^{3} + 16 a^{2} + 19 a + 11\right)\cdot 23^{64} + \left(20 a^{14} + 22 a^{13} + 10 a^{12} + 16 a^{11} + 22 a^{10} + 3 a^{9} + 14 a^{8} + 20 a^{7} + 20 a^{6} + 18 a^{5} + 18 a^{4} + 16 a^{3} + 15 a^{2} + 4 a + 12\right)\cdot 23^{65} + \left(12 a^{14} + 18 a^{12} + a^{11} + 7 a^{10} + 12 a^{9} + 10 a^{8} + 15 a^{7} + 15 a^{6} + 3 a^{5} + 20 a^{4} + 16 a^{3} + 11 a^{2} + 17 a + 19\right)\cdot 23^{66} + \left(14 a^{14} + 10 a^{13} + 9 a^{12} + 12 a^{11} + 4 a^{10} + 19 a^{9} + 9 a^{8} + a^{7} + 17 a^{6} + 15 a^{5} + 12 a^{3} + 5 a^{2} + 9 a + 7\right)\cdot 23^{67} + \left(9 a^{14} + 7 a^{13} + 7 a^{12} + 14 a^{11} + 2 a^{10} + 12 a^{9} + 12 a^{8} + 10 a^{7} + 6 a^{6} + 15 a^{5} + 14 a^{4} + 7 a^{3} + 4 a^{2} + a + 2\right)\cdot 23^{68} + \left(15 a^{14} + 14 a^{13} + 9 a^{12} + 3 a^{11} + a^{10} + 15 a^{9} + 7 a^{8} + 19 a^{7} + 14 a^{6} + 6 a^{5} + 15 a^{4} + 18 a^{3} + 21 a^{2} + 11 a + 11\right)\cdot 23^{69} + \left(20 a^{14} + 17 a^{13} + 15 a^{12} + 8 a^{11} + 7 a^{10} + 20 a^{9} + 8 a^{8} + 2 a^{7} + a^{6} + 20 a^{5} + 2 a^{4} + 10 a^{3} + 19 a^{2} + 17 a + 20\right)\cdot 23^{70} + \left(6 a^{14} + 15 a^{13} + 4 a^{12} + 14 a^{11} + 10 a^{10} + 14 a^{9} + 11 a^{8} + 22 a^{7} + 13 a^{6} + 14 a^{5} + 5 a^{4} + 8 a^{3} + 19 a^{2} + 9 a\right)\cdot 23^{71} + \left(21 a^{14} + 9 a^{13} + 14 a^{12} + 13 a^{11} + a^{10} + 21 a^{9} + 4 a^{8} + 14 a^{7} + 14 a^{6} + 16 a^{5} + 6 a^{4} + 10 a^{3} + 3 a^{2} + 2 a + 22\right)\cdot 23^{72} + \left(20 a^{14} + 14 a^{13} + 3 a^{12} + 17 a^{11} + 16 a^{10} + 2 a^{9} + 6 a^{8} + 15 a^{7} + 20 a^{6} + 17 a^{5} + 3 a^{4} + 12 a^{3} + 21 a^{2} + 22 a + 3\right)\cdot 23^{73} + \left(2 a^{14} + 2 a^{13} + 7 a^{12} + 22 a^{11} + a^{10} + 13 a^{9} + 13 a^{8} + 18 a^{7} + 21 a^{6} + 15 a^{5} + 6 a^{4} + 6 a^{3} + 20 a^{2} + 9 a + 13\right)\cdot 23^{74} + \left(11 a^{14} + 8 a^{13} + 10 a^{12} + 10 a^{11} + 8 a^{10} + 17 a^{9} + 8 a^{8} + 12 a^{7} + 3 a^{6} + 5 a^{5} + 15 a^{3} + 13 a^{2} + 10 a + 22\right)\cdot 23^{75} + \left(5 a^{14} + 2 a^{13} + 19 a^{12} + 8 a^{11} + a^{10} + 5 a^{9} + 6 a^{8} + 7 a^{7} + 15 a^{6} + 6 a^{5} + 17 a^{4} + 22 a^{3} + 19 a^{2} + 22 a + 2\right)\cdot 23^{76} + \left(19 a^{14} + 11 a^{13} + 16 a^{11} + 7 a^{10} + 18 a^{9} + 7 a^{8} + 16 a^{7} + 19 a^{6} + 7 a^{5} + 19 a^{4} + 22 a^{3} + 5 a^{2} + 10 a + 17\right)\cdot 23^{77} + \left(11 a^{14} + 8 a^{13} + 18 a^{12} + 5 a^{11} + 3 a^{10} + 14 a^{9} + 7 a^{8} + 8 a^{7} + 19 a^{6} + a^{5} + 11 a^{4} + 12 a^{3} + 14 a^{2} + 14 a + 15\right)\cdot 23^{78} + \left(9 a^{14} + 20 a^{13} + 22 a^{12} + 15 a^{11} + 4 a^{10} + 6 a^{9} + 7 a^{8} + 15 a^{7} + 22 a^{6} + 16 a^{5} + 8 a^{4} + 10 a^{3} + 21 a^{2} + 6 a + 4\right)\cdot 23^{79} + \left(6 a^{14} + a^{13} + 6 a^{11} + 2 a^{10} + 7 a^{9} + 22 a^{8} + 14 a^{7} + 2 a^{6} + 4 a^{5} + 4 a^{4} + 6 a^{3} + 20 a^{2} + 17 a + 22\right)\cdot 23^{80} + \left(10 a^{14} + 7 a^{13} + 3 a^{12} + 3 a^{10} + 16 a^{9} + 3 a^{8} + 19 a^{7} + 4 a^{6} + 4 a^{5} + 3 a^{3} + 15 a^{2} + 12 a + 1\right)\cdot 23^{81} + \left(a^{14} + 12 a^{13} + 8 a^{12} + 4 a^{11} + 14 a^{10} + 12 a^{8} + 14 a^{7} + 10 a^{6} + 15 a^{5} + 16 a^{4} + 13 a^{3} + a^{2} + 18 a + 2\right)\cdot 23^{82} + \left(7 a^{14} + 15 a^{13} + 11 a^{12} + 4 a^{11} + 8 a^{10} + 19 a^{9} + 21 a^{8} + 13 a^{7} + 8 a^{6} + a^{5} + 13 a^{4} + 13 a^{3} + 5 a^{2} + 7 a + 7\right)\cdot 23^{83} + \left(10 a^{14} + 12 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4\right)\cdot 23^{95} + \left(12 a^{14} + 21 a^{12} + 16 a^{11} + 21 a^{10} + 15 a^{9} + 10 a^{8} + 22 a^{7} + 17 a^{6} + 18 a^{3} + 9 a^{2} + 10\right)\cdot 23^{96} + \left(8 a^{14} + 8 a^{13} + 14 a^{12} + a^{11} + 9 a^{10} + 18 a^{9} + 18 a^{7} + 2 a^{6} + 17 a^{5} + 17 a^{3} + 4 a^{2} + 10 a + 9\right)\cdot 23^{97} + \left(15 a^{14} + 19 a^{13} + 20 a^{12} + a^{11} + 18 a^{10} + 20 a^{9} + a^{8} + 4 a^{7} + 20 a^{6} + 2 a^{5} + 21 a^{4} + 9 a^{3} + 19 a^{2} + 10 a + 10\right)\cdot 23^{98} + \left(12 a^{14} + 17 a^{13} + 16 a^{12} + 4 a^{11} + a^{10} + 11 a^{9} + 4 a^{8} + 13 a^{7} + 12 a^{6} + 15 a^{5} + 5 a^{4} + 11 a^{3} + 18 a^{2} + 17 a + 18\right)\cdot 23^{99} + \left(6 a^{14} + 18 a^{13} + 20 a^{12} + 20 a^{11} + 6 a^{10} + 9 a^{9} + 20 a^{8} + a^{7} + 2 a^{6} + a^{5} + 2 a^{4} + 6 a^{3} + 22 a^{2} + 3 a + 19\right)\cdot 23^{100} + \left(11 a^{14} + 7 a^{13} + 16 a^{12} + 22 a^{11} + 10 a^{10} + 10 a^{9} + 20 a^{8} + 16 a^{7} + 2 a^{6} + 13 a^{5} + 13 a^{4} + 18 a^{3} + 2 a^{2} + 17 a + 14\right)\cdot 23^{101} + \left(13 a^{14} + 13 a^{13} + 13 a^{12} + 8 a^{11} + 4 a^{10} + a^{9} + 19 a^{8} + 12 a^{7} + 2 a^{6} + 8 a^{5} + a^{4} + 2 a^{3} + 10 a^{2} + 5 a + 18\right)\cdot 23^{102} + \left(5 a^{14} + 11 a^{13} + 15 a^{12} + 13 a^{11} + 7 a^{9} + 18 a^{8} + 5 a^{7} + 5 a^{6} + 8 a^{5} + 4 a^{4} + 19 a^{3} + 20 a + 4\right)\cdot 23^{103} + \left(8 a^{14} + 16 a^{13} + 3 a^{12} + 12 a^{11} + 21 a^{10} + 2 a^{9} + 15 a^{8} + 19 a^{7} + 13 a^{6} + 21 a^{5} + 6 a^{4} + 20 a^{3} + 3 a^{2} + 3 a + 11\right)\cdot 23^{104} + \left(17 a^{14} + 12 a^{13} + 15 a^{12} + 7 a^{11} + 11 a^{10} + 14 a^{9} + 17 a^{8} + 22 a^{7} + 4 a^{6} + 15 a^{5} + 17 a^{4} + 11 a^{3} + 19 a^{2} + 3 a + 17\right)\cdot 23^{105} + \left(9 a^{14} + a^{13} + 19 a^{12} + 21 a^{11} + 19 a^{10} + 4 a^{9} + 7 a^{8} + 12 a^{7} + 17 a^{6} + a^{5} + 8 a^{4} + 19 a^{3} + 14 a^{2} + 22 a + 13\right)\cdot 23^{106} + \left(18 a^{14} + 4 a^{13} + 2 a^{11} + 15 a^{10} + 15 a^{9} + 2 a^{8} + 10 a^{6} + 22 a^{5} + 5 a^{4} + 20 a^{3} + 21 a^{2} + 3 a + 9\right)\cdot 23^{107} + \left(12 a^{14} + 15 a^{13} + 17 a^{12} + 10 a^{11} + 14 a^{10} + 10 a^{9} + 2 a^{8} + 6 a^{7} + 2 a^{6} + 20 a^{5} + 14 a^{4} + 8 a^{3} + 2 a^{2} + 2 a + 10\right)\cdot 23^{108} + \left(4 a^{14} + 21 a^{13} + 18 a^{12} + a^{11} + 11 a^{10} + 5 a^{9} + 6 a^{8} + 19 a^{7} + 20 a^{6} + 17 a^{5} + 4 a^{4} + a^{3} + a + 10\right)\cdot 23^{109} + \left(9 a^{14} + 4 a^{13} + 22 a^{12} + 14 a^{11} + 17 a^{10} + 6 a^{9} + 18 a^{8} + 22 a^{7} + 11 a^{6} + 7 a^{5} + 19 a^{4} + 5 a^{3} + 6 a + 16\right)\cdot 23^{110} + \left(9 a^{14} + 5 a^{13} + 6 a^{12} + 12 a^{11} + 16 a^{10} + 11 a^{9} + 6 a^{8} + 17 a^{7} + 12 a^{6} + 14 a^{5} + 10 a^{4} + 15 a^{3} + 3 a^{2} + 16 a + 3\right)\cdot 23^{111} + \left(9 a^{14} + a^{13} + a^{12} + 15 a^{11} + 3 a^{10} + a^{9} + 5 a^{8} + 12 a^{7} + 18 a^{6} + 17 a^{5} + a^{4} + 20 a^{3} + 22 a^{2} + 3 a + 16\right)\cdot 23^{112} + \left(14 a^{14} + 10 a^{13} + 12 a^{12} + 13 a^{11} + 3 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\left(18 a^{14} + 8 a^{13} + 15 a^{12} + 9 a^{11} + 17 a^{10} + 3 a^{9} + 12 a^{8} + 12 a^{6} + 14 a^{5} + 22 a^{4} + 8 a^{3} + 20 a^{2} + 8 a + 18\right)\cdot 23^{119} + \left(11 a^{14} + 14 a^{13} + 19 a^{12} + 2 a^{11} + 11 a^{10} + 10 a^{9} + 11 a^{8} + 12 a^{7} + 9 a^{6} + 12 a^{5} + 13 a^{4} + 13 a^{3} + 20 a^{2} + 8 a + 8\right)\cdot 23^{120} + \left(11 a^{14} + 16 a^{13} + 12 a^{12} + 22 a^{11} + 20 a^{10} + 20 a^{9} + 18 a^{8} + 12 a^{7} + 7 a^{6} + 16 a^{5} + 15 a^{4} + 12 a^{3} + 12 a^{2} + 14 a + 5\right)\cdot 23^{121} + \left(4 a^{14} + 11 a^{13} + 19 a^{12} + 14 a^{11} + 20 a^{10} + 15 a^{9} + 2 a^{8} + 3 a^{7} + 7 a^{6} + 8 a^{5} + 14 a^{4} + 14 a^{3} + 19 a^{2} + 8 a + 2\right)\cdot 23^{122} + \left(7 a^{14} + 5 a^{13} + 12 a^{12} + a^{11} + 21 a^{10} + 16 a^{9} + 9 a^{8} + 16 a^{7} + 4 a^{6} + 12 a^{5} + 18 a^{4} + 20 a^{3} + 17 a^{2} + 17 a + 18\right)\cdot 23^{123} + \left(5 a^{14} + 6 a^{13} + 19 a^{12} + 4 a^{10} + 7 a^{9} + 8 a^{8} + 15 a^{7} + 7 a^{6} + 13 a^{5} + 22 a^{4} + 3 a^{3} + 17 a^{2} + 22 a + 9\right)\cdot 23^{124} + \left(7 a^{14} + 13 a^{13} + 13 a^{12} + 16 a^{11} + 22 a^{10} + 6 a^{8} + 14 a^{7} + 12 a^{6} + 2 a^{5} + 18 a^{3} + 12 a^{2} + 11 a + 20\right)\cdot 23^{125} + \left(4 a^{14} + 14 a^{13} + a^{12} + 3 a^{11} + 22 a^{10} + 6 a^{9} + 16 a^{8} + 10 a^{7} + 18 a^{6} + a^{5} + 11 a^{4} + a^{3} + 21 a^{2} + 14 a + 19\right)\cdot 23^{126} + \left(19 a^{14} + 4 a^{13} + 13 a^{12} + 14 a^{11} + 12 a^{10} + 6 a^{9} + 2 a^{8} + 6 a^{7} + 14 a^{6} + 4 a^{5} + 20 a^{4} + 20 a^{3} + 14 a^{2} + 12 a + 3\right)\cdot 23^{127} + \left(7 a^{14} + 17 a^{13} + 2 a^{12} + 15 a^{11} + 3 a^{10} + 8 a^{9} + 12 a^{8} + 16 a^{7} + 19 a^{6} + 2 a^{5} + 14 a^{4} + 18 a^{3} + 6 a^{2} + 9 a + 10\right)\cdot 23^{128} + \left(6 a^{14} + 14 a^{13} + 3 a^{12} + 19 a^{11} + 18 a^{10} + 16 a^{9} + 11 a^{8} + 13 a^{7} + 19 a^{6} + 15 a^{5} + 21 a^{4} + 11 a^{3} + 8 a^{2} + 19 a + 13\right)\cdot 23^{129} + \left(a^{14} + 22 a^{13} + 17 a^{12} + 5 a^{11} + 19 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a^{9} + 7 a^{8} + 5 a^{7} + 11 a^{6} + 5 a^{5} + 6 a^{4} + 4 a^{3} + 15 a^{2} + a + 3\right)\cdot 23^{136} + \left(14 a^{14} + 14 a^{13} + 18 a^{12} + 2 a^{11} + 14 a^{10} + 7 a^{9} + 4 a^{8} + 18 a^{7} + 11 a^{6} + 4 a^{5} + 6 a^{4} + 7 a^{3} + 19 a^{2} + a + 18\right)\cdot 23^{137} + \left(2 a^{14} + 4 a^{13} + 10 a^{12} + 19 a^{11} + 13 a^{10} + 11 a^{9} + 7 a^{8} + 19 a^{7} + 3 a^{5} + a^{4} + 12 a^{3} + a^{2} + 9 a + 16\right)\cdot 23^{138} + \left(22 a^{14} + 10 a^{12} + 9 a^{11} + 2 a^{10} + 19 a^{9} + 21 a^{8} + 14 a^{7} + 3 a^{6} + 2 a^{5} + a^{4} + 22 a^{3} + 13 a^{2} + 8 a + 15\right)\cdot 23^{139} + \left(22 a^{13} + 10 a^{12} + 18 a^{11} + a^{10} + 19 a^{9} + 10 a^{8} + 7 a^{7} + 19 a^{6} + 17 a^{5} + 19 a^{4} + 16 a^{3} + 17 a^{2} + 10 a + 16\right)\cdot 23^{140} + \left(16 a^{14} + 4 a^{13} + 22 a^{12} + 2 a^{11} + 22 a^{10} + 22 a^{9} + 10 a^{8} + 10 a^{7} + 9 a^{6} + 17 a^{5} + 13 a^{4} + 12 a^{3} + 20 a^{2} + 15\right)\cdot 23^{141} + \left(15 a^{14} + 17 a^{13} + 6 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13\right)\cdot 23^{147} + \left(21 a^{14} + 17 a^{13} + 12 a^{12} + 9 a^{11} + 8 a^{10} + 18 a^{9} + 21 a^{8} + 20 a^{7} + 3 a^{6} + 19 a^{4} + 9 a^{3} + 6 a^{2} + a + 9\right)\cdot 23^{148} + \left(2 a^{14} + 3 a^{13} + 14 a^{12} + 18 a^{11} + 9 a^{10} + 14 a^{9} + 4 a^{8} + 3 a^{7} + 3 a^{6} + 5 a^{5} + 12 a^{4} + 9 a^{3} + 18 a^{2} + 9 a + 22\right)\cdot 23^{149} + \left(18 a^{14} + 12 a^{13} + 21 a^{12} + 13 a^{11} + 21 a^{10} + 20 a^{9} + 14 a^{8} + 6 a^{7} + 4 a^{6} + 4 a^{5} + 3 a^{4} + 18 a^{3} + 20 a^{2} + 18 a + 1\right)\cdot 23^{150} + \left(19 a^{14} + 13 a^{13} + 3 a^{12} + 17 a^{11} + 12 a^{10} + 12 a^{8} + 14 a^{7} + 8 a^{6} + 16 a^{5} + 9 a^{4} + 17 a^{3} + 6 a^{2} + 18 a + 9\right)\cdot 23^{151} + \left(8 a^{14} + 11 a^{13} + 10 a^{12} + 10 a^{11} + 22 a^{10} + 22 a^{9} + 12 a^{8} + 15 a^{7} + a^{6} + 6 a^{5} + 21 a^{4} + 12 a^{3} + 16 a^{2} + 7 a + 15\right)\cdot 23^{152} + \left(8 a^{14} + 7 a^{13} + 3 a^{12} + 15 a^{11} + a^{10} + 9 a^{9} + 7 a^{8} + 9 a^{7} + 19 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\left(13 a^{14} + 19 a^{13} + 19 a^{12} + 18 a^{11} + 9 a^{10} + 8 a^{9} + 2 a^{8} + 7 a^{7} + 8 a^{6} + 15 a^{5} + 3 a^{4} + a^{3} + 6 a^{2} + 12 a + 7\right)\cdot 23^{165} + \left(11 a^{14} + 14 a^{13} + a^{12} + 14 a^{11} + 8 a^{10} + 2 a^{9} + a^{8} + 21 a^{7} + a^{6} + 16 a^{5} + a^{4} + 5 a^{3} + 21 a + 9\right)\cdot 23^{166} + \left(8 a^{14} + 13 a^{13} + 15 a^{12} + 8 a^{11} + 5 a^{10} + 9 a^{9} + 18 a^{8} + 5 a^{7} + 17 a^{6} + 7 a^{5} + 20 a^{4} + a^{3} + 7 a^{2} + 6 a + 17\right)\cdot 23^{167} + \left(11 a^{14} + 20 a^{13} + 14 a^{12} + 14 a^{11} + 21 a^{10} + 22 a^{9} + 9 a^{8} + 2 a^{7} + 18 a^{6} + a^{5} + 4 a^{4} + 3 a^{3} + 17 a^{2} + 19 a + 8\right)\cdot 23^{168} + \left(11 a^{14} + 15 a^{13} + 18 a^{12} + 2 a^{11} + 2 a^{10} + 22 a^{8} + 17 a^{7} + 17 a^{6} + 8 a^{5} + 22 a^{4} + 3 a^{3} + 18 a^{2} + 19 a + 5\right)\cdot 23^{169} + \left(17 a^{14} + 4 a^{13} + 14 a^{12} + 6 a^{11} + 11 a^{10} + 22 a^{9} + 16 a^{8} + 2 a^{7} + 5 a^{6} + 16 a^{5} + 11 a^{4} + 22 a^{3} + 13 a^{2} + 11 a + 14\right)\cdot 23^{170} + \left(16 a^{14} + 16 a^{13} + 6 a^{12} + 4 a^{11} + 4 a^{10} + 12 a^{9} + 20 a^{8} + 20 a^{7} + 20 a^{6} + 20 a^{5} + 16 a^{4} + 10 a^{3} + 13 a^{2} + 16 a + 19\right)\cdot 23^{171} + \left(10 a^{14} + 22 a^{13} + 10 a^{12} + a^{11} + 4 a^{10} + 21 a^{9} + 7 a^{8} + 12 a^{7} + 5 a^{6} + 9 a^{5} + 10 a^{4} + 12 a + 15\right)\cdot 23^{172} + \left(8 a^{14} + 4 a^{13} + 16 a^{12} + 20 a^{11} + 13 a^{10} + 2 a^{9} + 7 a^{8} + 19 a^{7} + 15 a^{6} + 17 a^{5} + 4 a^{4} + 21 a^{3} + 19 a^{2} + 11 a + 11\right)\cdot 23^{173} + \left(15 a^{14} + 9 a^{13} + 4 a^{12} + 9 a^{11} + 2 a^{10} + 20 a^{9} + 6 a^{8} + 13 a^{7} + a^{6} + 4 a^{5} + 8 a^{4} + 22 a^{3} + 12 a^{2} + 7 a + 9\right)\cdot 23^{174} + \left(10 a^{14} + 4 a^{13} + 11 a^{12} + 16 a^{11} + 13 a^{10} + 17 a^{9} + 3 a^{8} + 5 a^{7} + 6 a^{6} + 16 a^{5} + 21 a^{4} + 2 a^{3} + 15 a^{2} + 20 a + 10\right)\cdot 23^{175} + \left(3 a^{14} + 21 a^{13} + 20 a^{12} + 18 a^{11} + 5 a^{10} + 18 a^{9} + 14 a^{8} + 2 a^{7} + 13 a^{6} + 21 a^{5} + 2 a^{4} + 4 a^{3} + 21 a^{2} + 9 a + 10\right)\cdot 23^{176} + \left(4 a^{14} + 5 a^{13} + 17 a^{12} + 13 a^{11} + 15 a^{10} + 9 a^{9} + 20 a^{8} + 3 a^{7} + 17 a^{6} + 18 a^{5} + 2 a^{4} + 5 a^{3} + 12 a^{2} + 14 a + 10\right)\cdot 23^{177} + \left(17 a^{14} + 21 a^{13} + 16 a^{12} + a^{11} + 5 a^{10} + 17 a^{9} + 11 a^{8} + 21 a^{7} + 15 a^{6} + 15 a^{5} + 16 a^{4} + 17 a^{3} + 19 a^{2} + 4 a + 18\right)\cdot 23^{178} + \left(15 a^{14} + 15 a^{13} + 22 a^{12} + 22 a^{11} + 5 a^{10} + 13 a^{9} + 8 a^{8} + 9 a^{7} + 18 a^{6} + 10 a^{5} + 9 a^{4} + 12 a^{3} + a^{2} + 11 a + 4\right)\cdot 23^{179} + \left(16 a^{14} + 15 a^{13} + 6 a^{12} + 17 a^{11} + 20 a^{10} + 11 a^{9} + 9 a^{8} + 14 a^{7} + 12 a^{6} + 19 a^{5} + 22 a^{4} + 3 a^{3} + 8 a^{2} + 21 a + 17\right)\cdot 23^{180} + \left(4 a^{14} + 14 a^{13} + 14 a^{12} + 18 a^{11} + 15 a^{10} + 15 a^{9} + 14 a^{8} + 5 a^{7} + 6 a^{6} + 17 a^{5} + 7 a^{4} + 12 a^{2} + a + 9\right)\cdot 23^{181} + \left(8 a^{14} + 3 a^{13} + a^{12} + 22 a^{11} + 8 a^{10} + 2 a^{8} + 13 a^{7} + 5 a^{6} + 4 a^{5} + 20 a^{4} + 15 a^{3} + 2 a^{2} + 8 a + 18\right)\cdot 23^{182} + \left(8 a^{14} + 16 a^{13} + 16 a^{12} + 19 a^{11} + 18 a^{10} + 17 a^{9} + 14 a^{8} + 14 a^{7} + 8 a^{6} + 11 a^{5} + 22 a^{4} + 21 a^{3} + 11 a^{2} + 8 a + 20\right)\cdot 23^{183} + \left(18 a^{14} + 17 a^{13} + 11 a^{12} + 12 a^{11} + 2 a^{10} + 18 a^{9} + 17 a^{8} + 8 a^{7} + 2 a^{6} + 17 a^{5} + 9 a^{4} + 12 a^{3} + 3 a^{2} + 18 a + 18\right)\cdot 23^{184} + \left(19 a^{14} + 15 a^{13} + 19 a^{12} + 18 a^{11} + 17 a^{9} + 19 a^{8} + 9 a^{7} + 9 a^{6} + 10 a^{5} + 3 a^{4} + 20 a^{3} + 21 a^{2} + 3 a + 18\right)\cdot 23^{185} + \left(20 a^{14} + 22 a^{13} + 20 a^{12} + 10 a^{11} + 18 a^{10} + 15 a^{9} + 5 a^{8} + 5 a^{7} + a^{6} + 22 a^{5} + 2 a^{4} + 10 a^{3} + 7 a^{2} + 8 a + 1\right)\cdot 23^{186} + \left(14 a^{14} + 11 a^{13} + 13 a^{12} + 6 a^{11} + 14 a^{10} + 7 a^{9} + 9 a^{8} + 13 a^{7} + 9 a^{6} + 12 a^{5} + 3 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\left(3 a^{14} + 21 a^{12} + 11 a^{11} + 4 a^{10} + 14 a^{9} + 9 a^{8} + 11 a^{7} + 9 a^{6} + 15 a^{5} + 2 a^{4} + 9 a^{3} + 14 a^{2} + 2\right)\cdot 23^{199} + \left(4 a^{14} + 19 a^{13} + 18 a^{12} + 2 a^{10} + 4 a^{9} + 19 a^{8} + 8 a^{7} + 20 a^{6} + 2 a^{5} + 21 a^{4} + 6 a^{3} + 5 a^{2} + 18 a + 20\right)\cdot 23^{200} + \left(15 a^{14} + 18 a^{13} + 10 a^{12} + 15 a^{11} + 12 a^{9} + 8 a^{8} + 21 a^{7} + a^{6} + 5 a^{5} + 22 a^{4} + 15 a^{3} + 20 a^{2} + 21 a + 13\right)\cdot 23^{201} + \left(20 a^{14} + 15 a^{12} + 7 a^{11} + 12 a^{9} + 9 a^{8} + 8 a^{7} + 21 a^{6} + 13 a^{5} + 18 a^{4} + 8 a^{2} + 7 a + 10\right)\cdot 23^{202} + \left(10 a^{13} + 16 a^{12} + 7 a^{11} + 13 a^{10} + 2 a^{9} + 16 a^{8} + 6 a^{7} + 13 a^{6} + 17 a^{5} + 5 a^{3} + 14 a^{2} + 14 a + 5\right)\cdot 23^{203} + \left(8 a^{14} + 19 a^{13} + 21 a^{11} + 21 a^{10} + 21 a^{9} + 3 a^{8} + 17 a^{7} + 13 a^{5} + a^{4} + 22 a^{3} + 8 a^{2} + 3 a + 22\right)\cdot 23^{204} + \left(19 a^{14} + 21 a^{13} + 14 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23^{210} + \left(20 a^{14} + 22 a^{13} + 10 a^{12} + 10 a^{11} + 4 a^{10} + 7 a^{9} + 11 a^{7} + 6 a^{6} + 4 a^{5} + 11 a^{4} + 15 a^{3} + 10 a^{2} + 5 a + 20\right)\cdot 23^{211} + \left(4 a^{14} + 17 a^{13} + 20 a^{12} + 4 a^{11} + 22 a^{10} + 14 a^{9} + 2 a^{8} + 9 a^{7} + 14 a^{6} + 15 a^{5} + 13 a^{4} + 12 a^{3} + 15 a^{2} + 21 a + 20\right)\cdot 23^{212} + \left(20 a^{14} + 11 a^{13} + 7 a^{12} + 19 a^{11} + 4 a^{10} + 19 a^{9} + 15 a^{8} + 21 a^{7} + 7 a^{6} + 21 a^{5} + 5 a^{4} + 13 a^{3} + 12 a^{2} + 10 a + 20\right)\cdot 23^{213} + \left(21 a^{14} + a^{13} + 17 a^{12} + 20 a^{11} + 11 a^{9} + 12 a^{8} + 6 a^{6} + 7 a^{5} + 16 a^{4} + 5 a^{2} + 13 a + 14\right)\cdot 23^{214} + \left(18 a^{14} + 15 a^{13} + 21 a^{12} + 19 a^{11} + 22 a^{10} + 16 a^{9} + 18 a^{8} + 15 a^{7} + 9 a^{5} + a^{4} + 3 a^{2} + 12 a + 15\right)\cdot 23^{215} + \left(17 a^{14} + 12 a^{13} + 18 a^{12} + 10 a^{11} + 8 a^{10} + 18 a^{9} + 9 a^{8} + 9 a^{7} + 5 a^{6} + 3 a^{5} + 14 a^{4} + 15 a^{3} + 4 a^{2} + 7 a + 16\right)\cdot 23^{216} + \left(5 a^{14} + 16 a^{13} + 16 a^{12} + 11 a^{11} + 12 a^{9} + 12 a^{8} + 14 a^{7} + 15 a^{6} + 12 a^{5} + 10 a^{4} + 3 a^{3} + 20 a^{2} + 10 a + 8\right)\cdot 23^{217} + \left(19 a^{14} + 3 a^{13} + 5 a^{12} + 12 a^{11} + 14 a^{10} + 19 a^{9} + 3 a^{8} + 4 a^{7} + 18 a^{6} + 15 a^{5} + 18 a^{4} + 4 a^{3} + 11 a^{2} + 3 a + 6\right)\cdot 23^{218} + \left(16 a^{14} + 6 a^{13} + 8 a^{12} + 22 a^{11} + 4 a^{10} + 13 a^{9} + 17 a^{8} + 10 a^{7} + 3 a^{6} + 2 a^{5} + 3 a^{4} + 5 a^{3} + 13 a^{2} + 10 a + 4\right)\cdot 23^{219} + \left(16 a^{14} + 2 a^{13} + 19 a^{12} + 21 a^{10} + 21 a^{9} + 6 a^{8} + 3 a^{7} + 3 a^{6} + 22 a^{5} + a^{4} + 9 a^{3} + 14 a^{2} + 13 a + 17\right)\cdot 23^{220} + \left(13 a^{14} + 2 a^{13} + 13 a^{12} + 6 a^{11} + 10 a^{10} + 14 a^{9} + 17 a^{8} + 17 a^{7} + 21 a^{6} + 6 a^{5} + 5 a^{4} + 7 a^{3} + 2 a^{2} + a + 16\right)\cdot 23^{221} + \left(10 a^{14} + 10 a^{13} + 6 a^{12} + 21 a^{11} + 16 a^{10} + 14 a^{9} + 14 a^{8} + 12 a^{7} + 15 a^{6} + 20 a^{5} + 2 a^{4} + 9 a^{3} + 11 a^{2} + 2 a + 22\right)\cdot 23^{222} + \left(16 a^{13} + 14 a^{12} + 18 a^{11} + 13 a^{10} + 13 a^{9} + 4 a^{8} + 21 a^{7} + 15 a^{6} + 6 a^{5} + 7 a^{4} + 7 a^{3} + 13 a^{2} + 20 a + 19\right)\cdot 23^{223} + \left(21 a^{14} + 16 a^{13} + 21 a^{12} + 13 a^{11} + 20 a^{10} + 14 a^{9} + 7 a^{8} + 22 a^{7} + 13 a^{6} + 14 a^{5} + 8 a^{4} + 6 a^{3} + 7 a^{2} + 6 a + 17\right)\cdot 23^{224} + \left(17 a^{14} + 22 a^{13} + a^{12} + 22 a^{11} + 19 a^{10} + 10 a^{9} + 20 a^{8} + 20 a^{7} + 17 a^{6} + 13 a^{5} + 4 a^{4} + 13 a^{3} + 5 a^{2} + 5 a + 4\right)\cdot 23^{225} + \left(20 a^{14} + 2 a^{12} + 6 a^{11} + 14 a^{10} + 6 a^{9} + 18 a^{8} + 10 a^{7} + 17 a^{6} + 6 a^{5} + 15 a^{3} + 11 a^{2} + 5 a + 12\right)\cdot 23^{226} + \left(3 a^{14} + 18 a^{13} + 7 a^{12} + 12 a^{11} + 2 a^{10} + 4 a^{9} + 6 a^{8} + 2 a^{7} + 19 a^{6} + 10 a^{5} + 22 a^{4} + a^{3} + 17 a^{2} + 20 a + 4\right)\cdot 23^{227} + \left(10 a^{14} + 5 a^{13} + 7 a^{12} + 9 a^{11} + 9 a^{10} + 14 a^{9} + 15 a^{8} + 7 a^{7} + 2 a^{6} + 14 a^{5} + a^{4} + 12 a^{3} + 9 a^{2} + 7 a + 14\right)\cdot 23^{228} + \left(9 a^{14} + 8 a^{12} + 21 a^{11} + 9 a^{10} + 6 a^{9} + 7 a^{8} + 18 a^{7} + 4 a^{6} + 16 a^{5} + 18 a^{4} + a^{3} + 14 a^{2} + 15 a + 19\right)\cdot 23^{229} + \left(21 a^{14} + 22 a^{13} + 3 a^{12} + 13 a^{11} + 17 a^{10} + 17 a^{9} + 6 a^{8} + 4 a^{7} + 13 a^{6} + 4 a^{5} + 19 a^{4} + 15 a^{3} + 13 a^{2} + 4 a + 21\right)\cdot 23^{230} + \left(14 a^{14} + 20 a^{13} + 3 a^{12} + 6 a^{11} + 14 a^{10} + 7 a^{9} + 13 a^{8} + 22 a^{6} + a^{5} + 9 a^{4} + 13 a^{3} + 11 a^{2} + 16 a + 5\right)\cdot 23^{231} + \left(11 a^{14} + 10 a^{13} + 12 a^{12} + 14 a^{11} + 10 a^{10} + 13 a^{9} + 16 a^{8} + 18 a^{7} + 17 a^{6} + 19 a^{5} + 15 a^{4} + 22 a^{3} + 13 a^{2} + 10 a + 9\right)\cdot 23^{232} + \left(a^{14} + 6 a^{13} + 9 a^{12} + 17 a^{11} + 13 a^{10} + 19 a^{9} + 4 a^{8} + 14 a^{7} + 8 a^{6} + 5 a^{5} + 11 a^{4} + 5 a^{3} + 8 a^{2} + 3 a + 14\right)\cdot 23^{233} + \left(19 a^{14} + 16 a^{13} + 16 a^{12} + 12 a^{11} + 10 a^{10} + 15 a^{9} + 8 a^{8} + 19 a^{7} + 5 a^{6} + 8 a^{5} + 10 a^{4} + 21 a^{3} + 5 a + 14\right)\cdot 23^{234} + \left(7 a^{14} + 20 a^{12} + 13 a^{11} + 10 a^{10} + 22 a^{9} + 20 a^{8} + 17 a^{7} + 16 a^{6} + 22 a^{5} + 11 a^{4} + 7 a^{3} + 7 a^{2} + 12 a + 18\right)\cdot 23^{235} + \left(21 a^{14} + 14 a^{13} + 10 a^{12} + 20 a^{11} + 20 a^{10} + 19 a^{9} + 17 a^{8} + 13 a^{7} + 20 a^{6} + 6 a^{5} + 20 a^{4} + 12 a^{3} + 10 a^{2} + a + 10\right)\cdot 23^{236} + \left(11 a^{14} + 13 a^{13} + 4 a^{12} + 7 a^{11} + 18 a^{10} + 7 a^{9} + 21 a^{8} + 16 a^{7} + 5 a^{6} + 19 a^{5} + 3 a^{4} + 11 a^{3} + 9 a^{2} + 3 a + 5\right)\cdot 23^{237} + \left(20 a^{14} + 8 a^{13} + 12 a^{12} + 21 a^{11} + 21 a^{10} + 17 a^{9} + 6 a^{8} + 20 a^{7} + 5 a^{6} + 22 a^{5} + 9 a^{4} + 7 a^{3} + 3 a^{2} + 2 a + 9\right)\cdot 23^{238} + \left(20 a^{14} + 10 a^{13} + 10 a^{12} + 8 a^{11} + 20 a^{10} + 17 a^{9} + 7 a^{8} + 11 a^{7} + 13 a^{5} + 21 a^{4} + 22 a^{3} + 6 a^{2} + 20 a + 4\right)\cdot 23^{239} + \left(22 a^{14} + 7 a^{13} + 21 a^{12} + 17 a^{11} + 22 a^{10} + 22 a^{9} + 12 a^{8} + a^{7} + 2 a^{6} + 16 a^{5} + 5 a^{4} + 5 a^{3} + 10 a^{2} + 8 a + 13\right)\cdot 23^{240} + \left(5 a^{14} + 7 a^{13} + 18 a^{12} + 16 a^{11} + 10 a^{10} + 9 a^{9} + 5 a^{8} + 22 a^{7} + 12 a^{6} + 22 a^{5} + 21 a^{4} + 4 a^{3} + 13 a^{2} + 10\right)\cdot 23^{241} + \left(4 a^{14} + 19 a^{13} + 16 a^{12} + 13 a^{11} + 18 a^{10} + 2 a^{9} + a^{8} + 8 a^{7} + 16 a^{6} + 9 a^{5} + 6 a^{4} + 11 a^{3} + 8 a + 18\right)\cdot 23^{242} + \left(13 a^{14} + a^{13} + 15 a^{12} + 2 a^{11} + 18 a^{10} + 21 a^{9} + 14 a^{8} + 9 a^{7} + 15 a^{6} + 18 a^{5} + 15 a^{4} + 16 a^{2} + 19 a + 22\right)\cdot 23^{243} + \left(15 a^{14} + 5 a^{13} + 2 a^{12} + 8 a^{11} + 8 a^{10} + 11 a^{9} + 6 a^{8} + 20 a^{7} + 14 a^{6} + 13 a^{5} + 19 a^{4} + 11 a^{3} + 11 a^{2} + 9 a + 13\right)\cdot 23^{244} + \left(12 a^{14} + 15 a^{13} + 20 a^{12} + 17 a^{11} + 19 a^{10} + 10 a^{9} + 19 a^{8} + 2 a^{7} + 2 a^{6} + 9 a^{5} + 22 a^{4} + 2 a^{3} + 16 a^{2} + 7 a + 10\right)\cdot 23^{245} + \left(13 a^{14} + 10 a^{13} + 4 a^{12} + 14 a^{11} + 7 a^{10} + 11 a^{9} + 9 a^{8} + 10 a^{7} + 6 a^{6} + 18 a^{5} + 17 a^{4} + 6 a^{3} + 8 a^{2} + 15\right)\cdot 23^{246} + \left(15 a^{14} + 9 a^{13} + 2 a^{12} + 19 a^{11} + 16 a^{10} + 6 a^{9} + 6 a^{7} + 21 a^{6} + 8 a^{5} + 11 a^{4} + 4 a^{3} + 13 a^{2} + 7 a + 17\right)\cdot 23^{247} + \left(22 a^{14} + 17 a^{13} + 13 a^{12} + 15 a^{11} + 10 a^{10} + 2 a^{9} + 8 a^{8} + 9 a^{6} + 12 a^{5} + 8 a^{4} + 19 a^{3} + 9 a^{2} + 16 a + 17\right)\cdot 23^{248} + \left(a^{14} + 10 a^{13} + 14 a^{12} + 18 a^{11} + 6 a^{10} + 6 a^{9} + 13 a^{7} + a^{6} + 9 a^{5} + 17 a^{3} + 22 a + 2\right)\cdot 23^{249} + \left(10 a^{14} + 22 a^{13} + 17 a^{12} + 12 a^{11} + 5 a^{10} + 15 a^{9} + 6 a^{8} + 9 a^{7} + 8 a^{6} + 11 a^{5} + 19 a^{4} + 18 a^{3} + 8 a^{2} + 19 a + 18\right)\cdot 23^{250} + \left(22 a^{14} + 14 a^{13} + 14 a^{12} + 5 a^{11} + 22 a^{10} + 11 a^{9} + 17 a^{8} + 14 a^{7} + 14 a^{6} + 20 a^{5} + 9 a^{4} + 21 a^{3} + 8 a^{2} + 3 a + 16\right)\cdot 23^{251} + \left(16 a^{14} + 9 a^{13} + 14 a^{12} + a^{11} + 5 a^{10} + 16 a^{9} + 12 a^{7} + 9 a^{5} + 7 a^{4} + 12 a^{3} + 5 a^{2} + 2 a + 16\right)\cdot 23^{252} + \left(16 a^{14} + 19 a^{13} + a^{12} + a^{11} + 6 a^{10} + 4 a^{9} + 6 a^{8} + 22 a^{7} + 4 a^{6} + 15 a^{5} + 15 a^{4} + 22 a^{3} + a^{2} + 12 a + 1\right)\cdot 23^{253} + \left(8 a^{14} + 20 a^{13} + 2 a^{12} + 5 a^{10} + 14 a^{9} + 22 a^{8} + 9 a^{7} + 18 a^{6} + 4 a^{4} + 11 a^{3} + 3 a^{2} + 14 a + 22\right)\cdot 23^{254} + \left(13 a^{14} + 3 a^{13} + 17 a^{12} + 21 a^{11} + 18 a^{10} + 7 a^{9} + 18 a^{8} + 10 a^{7} + 2 a^{5} + 8 a^{4} + 4 a^{3} + 7 a^{2} + 15 a + 2\right)\cdot 23^{255} + \left(21 a^{14} + 5 a^{13} + 11 a^{12} + 11 a^{11} + 5 a^{10} + 14 a^{9} + 11 a^{8} + 2 a^{7} + 16 a^{6} + 20 a^{5} + 15 a^{4} + 7 a^{3} + 19 a^{2} + 18 a + 10\right)\cdot 23^{256} + \left(15 a^{14} + 13 a^{13} + 21 a^{12} + 9 a^{11} + 12 a^{10} + 17 a^{9} + 2 a^{8} + 12 a^{7} + a^{6} + 13 a^{5} + 7 a^{4} + 9 a^{2} + 5 a + 1\right)\cdot 23^{257} + \left(7 a^{14} + 21 a^{13} + 21 a^{12} + 14 a^{11} + 9 a^{10} + 2 a^{9} + 9 a^{8} + a^{7} + a^{6} + 7 a^{5} + 22 a^{4} + 6 a^{3} + 8 a^{2} + 13 a + 13\right)\cdot 23^{258} + \left(a^{14} + 22 a^{13} + 12 a^{12} + 20 a^{11} + 11 a^{10} + 10 a^{9} + 21 a^{8} + 22 a^{7} + 10 a^{6} + 20 a^{5} + 5 a^{4} + 20 a^{3} + 14 a^{2} + 10 a + 15\right)\cdot 23^{259} + \left(2 a^{14} + 12 a^{13} + 11 a^{12} + 10 a^{11} + 18 a^{10} + 6 a^{9} + 13 a^{8} + 16 a^{7} + 10 a^{6} + 18 a^{5} + 22 a^{4} + 6 a^{2} + 11 a + 15\right)\cdot 23^{260} + \left(20 a^{14} + 9 a^{13} + 19 a^{12} + 4 a^{10} + 17 a^{9} + 18 a^{8} + 17 a^{7} + 13 a^{6} + 22 a^{5} + 18 a^{3} + 14 a^{2} + 8 a + 3\right)\cdot 23^{261} + \left(7 a^{14} + 9 a^{13} + 22 a^{12} + 8 a^{11} + 16 a^{10} + 15 a^{9} + 18 a^{7} + 13 a^{6} + 12 a^{5} + 7 a^{4} + 22 a^{3} + 3 a^{2} + 3 a + 2\right)\cdot 23^{262} + \left(8 a^{14} + 2 a^{13} + 7 a^{12} + 9 a^{11} + 10 a^{10} + 8 a^{9} + 9 a^{8} + 3 a^{7} + 15 a^{6} + 17 a^{5} + 4 a^{4} + 6 a^{3} + 20 a^{2} + 19 a + 10\right)\cdot 23^{263} + \left(17 a^{14} + 17 a^{13} + 3 a^{12} + 12 a^{11} + 19 a^{10} + 5 a^{9} + 16 a^{8} + 8 a^{7} + 9 a^{6} + 16 a^{5} + 6 a^{4} + 3 a^{3} + 22 a + 7\right)\cdot 23^{264} + \left(19 a^{14} + 5 a^{13} + 6 a^{12} + 19 a^{11} + 19 a^{10} + a^{9} + 3 a^{8} + 6 a^{7} + 16 a^{6} + 7 a^{5} + 19 a^{4} + 7 a^{3} + 8 a^{2} + 4 a + 11\right)\cdot 23^{265} + \left(12 a^{13} + 8 a^{12} + 6 a^{11} + 15 a^{10} + 5 a^{9} + 15 a^{8} + 5 a^{7} + 14 a^{6} + 22 a^{5} + 3 a^{4} + 6 a^{3} + 19 a^{2} + 13 a + 19\right)\cdot 23^{266} + \left(18 a^{14} + 20 a^{13} + 3 a^{12} + 4 a^{11} + 17 a^{10} + 14 a^{9} + 8 a^{8} + 15 a^{7} + 7 a^{6} + 11 a^{5} + 15 a^{4} + 22 a^{3} + 12 a^{2} + 19 a + 21\right)\cdot 23^{267} + \left(13 a^{14} + 7 a^{13} + 9 a^{12} + 21 a^{11} + 6 a^{10} + 16 a^{9} + 5 a^{8} + 15 a^{7} + 5 a^{6} + 4 a^{5} + 14 a^{4} + 17 a^{3} + 5 a^{2} + 5 a + 4\right)\cdot 23^{268} +O\left(23^{ 269 }\right)$ $r_{ 17 }$ $=$ $13 a^{14} + 2 a^{13} + 4 a^{12} + 6 a^{11} + 13 a^{10} + 14 a^{9} + 21 a^{8} + 2 a^{7} + 6 a^{6} + 14 a^{5} + 17 a^{3} + 19 a^{2} + 17 a + 14 + \left(22 a^{14} + 16 a^{13} + 2 a^{11} + 21 a^{10} + 11 a^{9} + 15 a^{8} + 4 a^{7} + 2 a^{6} + 7 a^{5} + 15 a^{4} + 15 a^{3} + 7 a^{2} + 9 a + 12\right)\cdot 23 + \left(18 a^{14} + 7 a^{13} + 18 a^{12} + 12 a^{11} + 19 a^{10} + 19 a^{9} + 6 a^{8} + 17 a^{7} + 3 a^{6} + 6 a^{5} + 12 a^{4} + 16 a^{3} + 2 a^{2} + 20 a + 18\right)\cdot 23^{2} + \left(a^{14} + 15 a^{13} + 3 a^{12} + 8 a^{11} + 19 a^{10} + 19 a^{9} + 21 a^{8} + 7 a^{7} + 9 a^{6} + 21 a^{5} + 4 a^{4} + 21 a^{3} + 2 a^{2} + 2 a + 2\right)\cdot 23^{3} + \left(11 a^{14} + 18 a^{13} + 17 a^{12} + 3 a^{11} + a^{9} + 16 a^{8} + 2 a^{7} + 8 a^{6} + 2 a^{5} + 5 a^{4} + 18 a^{3} + 2 a^{2} + 19 a + 1\right)\cdot 23^{4} + \left(2 a^{14} + 20 a^{13} + 12 a^{12} + 18 a^{11} + 12 a^{10} + 15 a^{9} + 18 a^{8} + 18 a^{7} + 20 a^{6} + 5 a^{4} + a^{3} + 16 a^{2} + 17 a + 8\right)\cdot 23^{5} + \left(14 a^{14} + 17 a^{13} + 3 a^{12} + 13 a^{11} + 13 a^{10} + 4 a^{9} + 5 a^{8} + 20 a^{6} + 22 a^{5} + 3 a^{4} + 15 a^{3} + 3 a^{2} + 12 a + 11\right)\cdot 23^{6} + \left(2 a^{14} + 5 a^{13} + 21 a^{12} + 12 a^{11} + 20 a^{10} + 8 a^{9} + 3 a^{8} + 4 a^{7} + 11 a^{6} + 13 a^{5} + 21 a^{4} + 3 a^{3} + 10 a + 17\right)\cdot 23^{7} + \left(20 a^{14} + 11 a^{13} + 13 a^{12} + 5 a^{11} + 14 a^{10} + 12 a^{9} + 9 a^{8} + 13 a^{7} + 15 a^{5} + 21 a^{3} + 6 a^{2} + 20 a + 16\right)\cdot 23^{8} + \left(a^{14} + 14 a^{13} + 11 a^{12} + 19 a^{11} + 5 a^{10} + 16 a^{9} + 8 a^{7} + 13 a^{6} + 13 a^{5} + 18 a^{4} + 6 a^{2} + 13 a + 22\right)\cdot 23^{9} + \left(15 a^{14} + 10 a^{13} + 15 a^{12} + 18 a^{11} + 17 a^{10} + 21 a^{9} + 18 a^{7} + 15 a^{6} + a^{5} + 6 a^{4} + 22 a^{3} + 2 a^{2} + 17 a + 11\right)\cdot 23^{10} + \left(22 a^{13} + 15 a^{11} + 20 a^{10} + 22 a^{9} + 9 a^{8} + 2 a^{7} + 10 a^{6} + 3 a^{5} + 16 a^{4} + 21 a^{3} + 22 a^{2} + 15 a + 4\right)\cdot 23^{11} + \left(12 a^{14} + 6 a^{13} + 9 a^{12} + 14 a^{11} + 5 a^{10} + 18 a^{8} + a^{7} + 12 a^{6} + a^{5} + 7 a^{4} + 14 a^{3} + 20 a\right)\cdot 23^{12} + \left(10 a^{14} + 22 a^{13} + 19 a^{12} + 15 a^{11} + 2 a^{10} + 14 a^{9} + 3 a^{8} + 7 a^{7} + 22 a^{6} + 20 a^{4} + 3 a^{3} + 9 a^{2} + 7 a + 22\right)\cdot 23^{13} + \left(12 a^{14} + a^{13} + 15 a^{12} + 10 a^{11} + 9 a^{10} + 13 a^{9} + 14 a^{8} + 17 a^{7} + 18 a^{6} + 18 a^{5} + a^{4} + 9 a^{3} + 22 a^{2} + 15 a + 15\right)\cdot 23^{14} + \left(17 a^{14} + 22 a^{13} + 14 a^{12} + 6 a^{11} + 21 a^{10} + 13 a^{9} + 9 a^{8} + 15 a^{7} + 3 a^{6} + 10 a^{4} + 16 a^{3} + 14 a^{2} + 5 a + 3\right)\cdot 23^{15} + \left(3 a^{14} + 18 a^{13} + 11 a^{12} + 15 a^{11} + 12 a^{10} + 13 a^{9} + a^{8} + 16 a^{7} + 14 a^{6} + 12 a^{5} + 7 a^{4} + 9 a^{3} + 3 a^{2} + 5\right)\cdot 23^{16} + \left(8 a^{14} + 3 a^{13} + 12 a^{12} + 22 a^{11} + 14 a^{10} + 8 a^{9} + 15 a^{8} + 16 a^{7} + 6 a^{6} + 4 a^{5} + 20 a^{4} + 16 a^{3} + 9 a^{2} + 14 a + 4\right)\cdot 23^{17} + \left(7 a^{14} + 18 a^{13} + 21 a^{12} + 9 a^{11} + 8 a^{10} + 11 a^{9} + 14 a^{8} + 14 a^{7} + 9 a^{6} + 6 a^{5} + 18 a^{4} + 15 a^{3} + 22 a^{2} + a + 9\right)\cdot 23^{18} + \left(17 a^{14} + 9 a^{13} + 5 a^{12} + 11 a^{11} + 3 a^{10} + 17 a^{9} + 22 a^{8} + 3 a^{7} + 18 a^{6} + 15 a^{5} + 17 a^{4} + 10 a^{3} + 18 a^{2} + 12 a + 17\right)\cdot 23^{19} + \left(13 a^{14} + 11 a^{13} + a^{12} + 9 a^{11} + 20 a^{10} + 22 a^{9} + 3 a^{8} + 12 a^{7} + 6 a^{6} + 8 a^{5} + 5 a^{4} + 19 a^{3} + 14 a^{2} + 7 a + 9\right)\cdot 23^{20} + \left(20 a^{14} + 19 a^{13} + 20 a^{12} + 14 a^{11} + a^{10} + 11 a^{9} + 11 a^{8} + 16 a^{7} + 5 a^{6} + 21 a^{5} + 15 a^{4} + 9 a^{3} + 21 a^{2} + 18 a + 20\right)\cdot 23^{21} + \left(21 a^{14} + 11 a^{13} + 18 a^{10} + 15 a^{9} + 8 a^{8} + 21 a^{7} + 16 a^{6} + 5 a^{5} + 5 a^{4} + 7 a^{3} + a^{2} + 8 a + 6\right)\cdot 23^{22} + \left(21 a^{14} + 3 a^{13} + 16 a^{12} + 12 a^{11} + 18 a^{10} + 15 a^{9} + 5 a^{8} + 21 a^{7} + 2 a^{6} + 4 a^{5} + 5 a^{4} + 18 a^{3} + 6 a^{2} + 17 a + 10\right)\cdot 23^{23} + \left(7 a^{14} + 9 a^{13} + 6 a^{12} + 18 a^{11} + a^{10} + 7 a^{9} + 3 a^{8} + 21 a^{7} + 3 a^{6} + 11 a^{5} + 15 a^{4} + 21 a^{3} + 3 a^{2} + 13 a\right)\cdot 23^{24} + \left(9 a^{14} + 7 a^{13} + 21 a^{12} + 15 a^{11} + 10 a^{10} + 9 a^{9} + 18 a^{8} + 3 a^{7} + 3 a^{6} + 3 a^{5} + 18 a^{4} + 12 a^{3} + 8 a^{2} + 16 a + 12\right)\cdot 23^{25} + \left(5 a^{14} + 3 a^{13} + a^{12} + 8 a^{11} + 12 a^{10} + 21 a^{9} + 10 a^{8} + 6 a^{7} + 3 a^{6} + 9 a^{5} + a^{4} + 14 a^{3} + 9 a^{2} + 8 a + 16\right)\cdot 23^{26} + \left(19 a^{14} + 4 a^{13} + 14 a^{12} + 18 a^{11} + 20 a^{10} + 3 a^{9} + 5 a^{8} + a^{7} + 22 a^{6} + 14 a^{5} + 19 a^{4} + 5 a^{3} + 8 a^{2} + 12 a + 17\right)\cdot 23^{27} + \left(a^{14} + 19 a^{13} + 15 a^{12} + 12 a^{11} + 17 a^{10} + 9 a^{9} + a^{8} + 19 a^{7} + 2 a^{5} + 4 a^{4} + 12 a^{2} + 11 a + 12\right)\cdot 23^{28} + \left(22 a^{14} + 15 a^{13} + 11 a^{12} + 11 a^{11} + 6 a^{10} + 9 a^{9} + a^{8} + 9 a^{7} + 8 a^{6} + 7 a^{5} + 3 a^{4} + 10 a^{3} + 10 a^{2} + 2 a + 17\right)\cdot 23^{29} + \left(21 a^{14} + 21 a^{12} + 10 a^{11} + 17 a^{10} + 4 a^{9} + 5 a^{8} + 4 a^{7} + 21 a^{6} + 9 a^{5} + 17 a^{4} + 8 a^{3} + 17 a^{2} + 17 a + 22\right)\cdot 23^{30} + \left(22 a^{13} + 2 a^{12} + 21 a^{11} + 13 a^{10} + 7 a^{9} + 7 a^{8} + 11 a^{7} + 8 a^{5} + 12 a^{4} + 14 a^{3} + 11 a^{2} + 12 a + 3\right)\cdot 23^{31} + \left(11 a^{14} + 9 a^{13} + 15 a^{12} + 22 a^{11} + 11 a^{10} + 16 a^{9} + 18 a^{8} + 16 a^{7} + 7 a^{6} + 16 a^{5} + 5 a^{4} + 15 a^{3} + 20 a^{2} + 16 a + 20\right)\cdot 23^{32} + \left(8 a^{14} + 8 a^{13} + 9 a^{12} + 16 a^{11} + 19 a^{10} + 19 a^{9} + 3 a^{8} + 4 a^{7} + 18 a^{6} + 7 a^{5} + 10 a^{4} + a^{3} + 19 a^{2} + 15 a + 17\right)\cdot 23^{33} + \left(13 a^{14} + 20 a^{13} + 7 a^{12} + 13 a^{11} + a^{10} + 19 a^{9} + 12 a^{8} + 7 a^{7} + 14 a^{6} + 4 a^{5} + 7 a^{4} + 4 a^{3} + 17 a^{2} + 9 a + 8\right)\cdot 23^{34} + \left(20 a^{14} + 19 a^{13} + 8 a^{12} + 10 a^{11} + 10 a^{10} + 2 a^{8} + 7 a^{7} + 9 a^{6} + 16 a^{5} + 10 a^{4} + 7 a^{3} + 21 a^{2} + 4 a + 18\right)\cdot 23^{35} + \left(12 a^{14} + 3 a^{13} + 9 a^{12} + 2 a^{11} + 4 a^{10} + 17 a^{9} + 15 a^{8} + 20 a^{7} + 10 a^{6} + 22 a^{5} + 13 a^{4} + 13 a^{3} + 4 a^{2} + 7 a + 20\right)\cdot 23^{36} + \left(15 a^{13} + 18 a^{12} + 9 a^{11} + 15 a^{10} + 20 a^{9} + 3 a^{8} + 7 a^{7} + 19 a^{6} + a^{5} + 21 a^{4} + 3 a^{3} + 22 a^{2} + 13 a + 7\right)\cdot 23^{37} + \left(21 a^{14} + 9 a^{13} + 6 a^{12} + 4 a^{10} + 18 a^{9} + 14 a^{8} + 10 a^{7} + 12 a^{6} + 7 a^{5} + 8 a^{4} + 4 a^{3} + 15 a^{2} + 13 a + 16\right)\cdot 23^{38} + \left(13 a^{14} + 8 a^{13} + 3 a^{12} + 6 a^{11} + 2 a^{10} + 20 a^{9} + 3 a^{8} + 4 a^{7} + 13 a^{6} + 9 a^{5} + 4 a^{4} + 12 a^{3} + 2 a^{2} + 21 a + 13\right)\cdot 23^{39} + \left(12 a^{14} + 13 a^{13} + 11 a^{12} + 21 a^{11} + 4 a^{10} + 7 a^{9} + 2 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+ a^{2} + a + 6\right)\cdot 23^{57} + \left(18 a^{14} + 16 a^{12} + 5 a^{11} + 14 a^{10} + 6 a^{9} + 13 a^{8} + 22 a^{7} + a^{6} + 12 a^{5} + 17 a^{4} + 7 a^{3} + 5 a + 6\right)\cdot 23^{58} + \left(2 a^{14} + 21 a^{13} + a^{12} + 21 a^{11} + 21 a^{10} + 3 a^{9} + a^{8} + 6 a^{7} + 19 a^{6} + a^{5} + 6 a^{4} + 3 a^{3} + 12 a^{2} + 3 a + 16\right)\cdot 23^{59} + \left(9 a^{14} + 11 a^{13} + 4 a^{12} + 12 a^{11} + 4 a^{10} + 12 a^{9} + 12 a^{8} + 10 a^{7} + 12 a^{6} + 19 a^{5} + 19 a^{4} + 4 a^{3} + 3 a^{2} + 17 a + 3\right)\cdot 23^{60} + \left(18 a^{14} + 22 a^{13} + 15 a^{12} + 2 a^{11} + 13 a^{10} + 16 a^{9} + 4 a^{8} + 11 a^{7} + 9 a^{6} + 10 a^{5} + 15 a^{4} + 11 a^{3} + 17 a^{2} + 7 a + 7\right)\cdot 23^{61} + \left(20 a^{14} + 15 a^{13} + 14 a^{12} + 22 a^{10} + 22 a^{9} + 12 a^{8} + 8 a^{7} + 5 a^{6} + 5 a^{5} + 13 a^{3} + 8 a^{2} + 14 a + 1\right)\cdot 23^{62} + \left(12 a^{13} + 13 a^{12} + 2 a^{11} + 13 a^{10} + 18 a^{9} + 15 a^{8} + 6 a^{7} + 16 a^{6} + 2 a^{5} + 2 a^{4} + 17 a^{3} + 17 a^{2} + 21 a + 3\right)\cdot 23^{63} + \left(10 a^{13} + 22 a^{12} + 9 a^{11} + 8 a^{10} + 22 a^{9} + 4 a^{8} + 21 a^{7} + 3 a^{6} + 17 a^{5} + 5 a^{4} + 19 a^{3} + 5 a^{2} + 10 a + 7\right)\cdot 23^{64} + \left(17 a^{14} + 18 a^{13} + 10 a^{12} + 15 a^{11} + 12 a^{10} + a^{9} + 10 a^{8} + 8 a^{7} + 10 a^{6} + 11 a^{5} + 8 a^{4} + 17 a^{3} + 2 a^{2} + 20 a\right)\cdot 23^{65} + \left(13 a^{14} + 9 a^{13} + 14 a^{12} + 13 a^{11} + 22 a^{10} + 17 a^{9} + 21 a^{8} + 15 a^{7} + 12 a^{6} + 4 a^{5} + 16 a^{4} + 21 a^{3} + 21 a^{2} + 22 a + 15\right)\cdot 23^{66} + \left(15 a^{14} + 22 a^{13} + 2 a^{12} + a^{11} + 22 a^{10} + 16 a^{9} + a^{8} + 19 a^{7} + 9 a^{6} + 19 a^{5} + 14 a^{4} + 22 a^{3} + a^{2} + 12 a + 12\right)\cdot 23^{67} + \left(14 a^{14} + 13 a^{13} + 15 a^{12} + 12 a^{11} + 8 a^{10} + 5 a^{9} + 7 a^{8} + 14 a^{7} + 12 a^{6} + 7 a^{5} + 22 a^{4} + 17 a^{3} + 16 a^{2} + 9 a + 6\right)\cdot 23^{68} + \left(a^{14} + 5 a^{13} + 2 a^{12} + 17 a^{11} + 14 a^{10} + 4 a^{9} + 6 a^{8} + 8 a^{7} + 21 a^{6} + 5 a^{5} + 16 a^{4} + 2 a^{3} + 19 a^{2} + 7 a + 22\right)\cdot 23^{69} + \left(22 a^{14} + 21 a^{13} + 13 a^{12} + 3 a^{11} + 15 a^{9} + 11 a^{8} + 14 a^{7} + 20 a^{6} + 17 a^{5} + 18 a^{4} + 13 a^{3} + a^{2} + a + 13\right)\cdot 23^{70} + \left(15 a^{14} + 2 a^{13} + 18 a^{12} + 3 a^{11} + 6 a^{10} + 11 a^{9} + 22 a^{8} + 18 a^{7} + 7 a^{6} + 8 a^{5} + a^{4} + 20 a^{3} + 3 a^{2} + 19 a + 8\right)\cdot 23^{71} + \left(18 a^{14} + a^{13} + 10 a^{12} + 4 a^{11} + 2 a^{10} + 11 a^{9} + 9 a^{8} + 7 a^{7} + 2 a^{6} + 12 a^{5} + 9 a^{3} + 3 a + 4\right)\cdot 23^{72} + \left(8 a^{14} + 6 a^{13} + 10 a^{12} + 10 a^{11} + 19 a^{10} + 7 a^{9} + 21 a^{8} + 10 a^{7} + 2 a^{6} + 22 a^{5} + 14 a^{4} + a^{3} + 20 a^{2} + 6 a + 18\right)\cdot 23^{73} + \left(8 a^{14} + 18 a^{13} + 20 a^{12} + 3 a^{11} + a^{10} + 2 a^{9} + 17 a^{8} + 8 a^{7} + 15 a^{6} + 21 a^{5} + 22 a^{4} + 2 a^{3} + 14 a^{2} + 17 a + 18\right)\cdot 23^{74} + \left(5 a^{14} + 14 a^{13} + 8 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+ \left(18 a^{14} + 12 a^{13} + 7 a^{12} + 4 a^{11} + 4 a^{10} + a^{9} + 10 a^{8} + 20 a^{7} + 4 a^{5} + 9 a^{4} + 4 a^{3} + 22 a^{2} + 17 a + 20\right)\cdot 23^{81} + \left(12 a^{14} + 21 a^{13} + 11 a^{12} + 20 a^{11} + a^{10} + 7 a^{9} + 9 a^{8} + 9 a^{7} + 15 a^{6} + 3 a^{5} + 18 a^{4} + 3 a^{3} + 13 a^{2} + 10 a + 20\right)\cdot 23^{82} + \left(17 a^{14} + 19 a^{13} + 22 a^{12} + 4 a^{11} + 9 a^{10} + 19 a^{9} + 9 a^{8} + 13 a^{7} + 13 a^{6} + 22 a^{5} + 14 a^{4} + 3 a^{3} + 6 a^{2} + 8 a + 10\right)\cdot 23^{83} + \left(a^{14} + a^{13} + 22 a^{12} + 10 a^{11} + 4 a^{9} + 2 a^{8} + 17 a^{7} + 4 a^{6} + 15 a^{5} + 12 a^{4} + 13 a^{3} + 15 a^{2} + 11 a + 4\right)\cdot 23^{84} + \left(14 a^{14} + 16 a^{13} + 17 a^{12} + 16 a^{11} + 19 a^{10} + 3 a^{9} + 13 a^{8} + 3 a^{7} + 3 a^{6} + 2 a^{5} + 14 a^{4} + a^{3} + 6 a^{2} + 19 a + 1\right)\cdot 23^{85} + \left(21 a^{14} + 6 a^{13} + 17 a^{12} + 18 a^{11} + 7 a^{10} + 9 a^{9} + 9 a^{8} + 12 a^{7} + 19 a^{6} + 16 a^{5} + 10 a^{4} + 13 a^{3} + 20 a + 7\right)\cdot 23^{86} + \left(17 a^{14} + 9 a^{13} + a^{12} + 14 a^{11} + 20 a^{10} + 16 a^{8} + 15 a^{7} + 3 a^{6} + 8 a^{5} + 22 a^{4} + 10 a^{3} + 3 a^{2} + 22 a + 7\right)\cdot 23^{87} + \left(12 a^{14} + 9 a^{13} + 20 a^{12} + 21 a^{11} + 9 a^{9} + 21 a^{8} + 18 a^{7} + 22 a^{6} + 16 a^{5} + 17 a^{4} + 15 a^{3} + 20 a^{2} + 7 a + 14\right)\cdot 23^{88} + \left(8 a^{14} + 21 a^{13} + 12 a^{12} + 20 a^{11} + 22 a^{10} + 2 a^{9} + 5 a^{8} + 19 a^{7} + a^{5} + 7 a^{4} + 16 a^{3} + 16 a^{2} + 15 a + 13\right)\cdot 23^{89} + \left(14 a^{13} + 22 a^{12} + 6 a^{11} + 5 a^{10} + 22 a^{9} + 4 a^{8} + 6 a^{7} + 3 a^{5} + 20 a^{4} + 11 a^{3} + 19 a^{2} + 6\right)\cdot 23^{90} + \left(10 a^{14} + 11 a^{13} + 2 a^{12} + 2 a^{11} + a^{10} + 3 a^{9} + a^{7} + 7 a^{6} + 15 a^{5} + 5 a^{4} + 4 a^{3} + 12 a^{2} + 8 a + 19\right)\cdot 23^{91} + \left(13 a^{14} + 13 a^{13} + 5 a^{12} + 18 a^{11} + 2 a^{10} + 18 a^{9} + 13 a^{8} + 14 a^{6} + 22 a^{5} + 5 a^{3} + 3 a^{2} + 3 a + 21\right)\cdot 23^{92} + \left(9 a^{14} + 22 a^{13} + 10 a^{12} + 13 a^{11} + 13 a^{10} + 5 a^{9} + 14 a^{8} + 9 a^{7} + 10 a^{6} + 12 a^{5} + 2 a^{4} + 16 a^{3} + 15 a^{2} + 6 a + 1\right)\cdot 23^{93} + \left(18 a^{14} + 11 a^{13} + 7 a^{12} + 16 a^{11} + 10 a^{10} + 18 a^{9} + 8 a^{8} + 18 a^{7} + 21 a^{6} + 5 a^{5} + 12 a^{4} + 2 a^{3} + 11 a^{2} + 2 a + 3\right)\cdot 23^{94} + \left(18 a^{13} + 21 a^{12} + 19 a^{11} + 7 a^{10} + 22 a^{8} + 22 a^{6} + 15 a^{5} + 19 a^{4} + 4 a^{3} + 19 a^{2} + 5 a + 20\right)\cdot 23^{95} + \left(14 a^{14} + 7 a^{13} + 16 a^{12} + 13 a^{11} + 12 a^{10} + 16 a^{9} + 14 a^{8} + 9 a^{7} + 4 a^{6} + 12 a^{5} + 13 a^{4} + 19 a^{3} + 21 a^{2} + 3 a\right)\cdot 23^{96} + \left(11 a^{14} + 19 a^{13} + 22 a^{12} + 5 a^{11} + 19 a^{10} + 2 a^{9} + 3 a^{8} + 10 a^{7} + a^{5} + 9 a^{4} + 12 a^{3} + 16 a^{2} + 15 a\right)\cdot 23^{97} + \left(17 a^{14} + 12 a^{13} + 6 a^{12} + 5 a^{9} + 15 a^{8} + 15 a^{7} + 9 a^{6} + 9 a^{5} + 16 a^{4} + 12 a^{3} + 22 a^{2} + 15 a + 9\right)\cdot 23^{98} + \left(19 a^{14} + 12 a^{13} + 7 a^{11} + 20 a^{10} + 20 a^{9} + 12 a^{8} + 21 a^{7} + 11 a^{6} + 7 a^{5} + 21 a^{4} + 18 a^{3} + 8 a^{2} + 9\right)\cdot 23^{99} + \left(16 a^{14} + 14 a^{11} + 22 a^{10} + 4 a^{9} + 7 a^{8} + 9 a^{7} + 5 a^{6} + 12 a^{5} + 13 a^{4} + 18 a^{3} + 9 a^{2} + 8 a + 12\right)\cdot 23^{100} + \left(5 a^{14} + 6 a^{13} + 2 a^{12} + 21 a^{11} + 11 a^{10} + 7 a^{9} + 8 a^{8} + 5 a^{7} + 2 a^{6} + 19 a^{5} + 16 a^{4} + 15 a^{3} + 4 a^{2} + 16 a + 3\right)\cdot 23^{101} + \left(6 a^{14} + 8 a^{13} + 18 a^{12} + 19 a^{11} + 22 a^{10} + 12 a^{9} + 5 a^{8} + 2 a^{7} + 18 a^{6} + 13 a^{5} + 2 a^{4} + 12 a^{2} + 15 a + 3\right)\cdot 23^{102} + \left(13 a^{14} + 6 a^{13} + 8 a^{12} + 10 a^{11} + 9 a^{8} + 18 a^{7} + 11 a^{6} + 11 a^{5} + 17 a^{4} + 2 a^{3} + 11 a^{2} + 18 a + 14\right)\cdot 23^{103} + \left(15 a^{14} + 16 a^{13} + 8 a^{11} + 8 a^{10} + 8 a^{9} + 13 a^{8} + 22 a^{7} + 17 a^{6} + 22 a^{5} + 4 a^{4} + 19 a^{3} + 20 a^{2} + 6 a + 15\right)\cdot 23^{104} + \left(13 a^{14} + a^{13} + 11 a^{12} + 15 a^{11} + 5 a^{10} + 9 a^{9} + 20 a^{8} + 22 a^{7} + 5 a^{6} + 13 a^{5} + 17 a^{4} + 19 a^{3} + 2 a^{2} + 7 a + 11\right)\cdot 23^{105} + \left(8 a^{14} + 20 a^{13} + 6 a^{12} + 9 a^{11} + 17 a^{10} + 4 a^{9} + 18 a^{8} + 11 a^{7} + 18 a^{6} + a^{5} + 6 a^{4} + 17 a^{3} + 11 a^{2} + 4 a + 7\right)\cdot 23^{106} + \left(12 a^{14} + a^{13} + 21 a^{12} + 12 a^{11} + 2 a^{10} + 16 a^{9} + 2 a^{8} + 14 a^{7} + 7 a^{6} + 18 a^{5} + 17 a^{4} + 21 a^{3} + 9 a^{2} + 12 a + 2\right)\cdot 23^{107} + \left(16 a^{14} + 20 a^{13} + 4 a^{12} + 6 a^{11} + 13 a^{10} + 11 a^{9} + 19 a^{8} + 4 a^{7} + 16 a^{6} + 6 a^{5} + 15 a^{4} + 8 a^{3} + 8 a^{2} + 17 a + 12\right)\cdot 23^{108} + \left(9 a^{14} + 9 a^{13} + 8 a^{12} + 19 a^{11} + 2 a^{10} + a^{9} + 20 a^{8} + 3 a^{7} + 11 a^{6} + a^{4} + 11 a^{3} + 11 a^{2} + 22 a + 22\right)\cdot 23^{109} + \left(12 a^{14} + 18 a^{13} + 16 a^{12} + a^{11} + 16 a^{10} + 9 a^{9} + 11 a^{8} + 7 a^{7} + 11 a^{6} + 16 a^{5} + 19 a^{4} + a^{3} + 16 a^{2} + 5 a + 14\right)\cdot 23^{110} + \left(17 a^{14} + 11 a^{13} + 16 a^{12} + 3 a^{11} + 8 a^{10} + 3 a^{9} + 4 a^{8} + 21 a^{7} + 9 a^{6} + 4 a^{5} + 4 a^{4} + 6 a^{3} + 4 a^{2} + 21 a + 2\right)\cdot 23^{111} + \left(13 a^{14} + 14 a^{13} + 12 a^{12} + a^{11} + 7 a^{10} + 14 a^{9} + 14 a^{8} + 11 a^{7} + 19 a^{6} + 9 a^{5} + 8 a^{4} + 4 a^{3} + 21 a + 7\right)\cdot 23^{112} + \left(22 a^{14} + 22 a^{13} + 2 a^{12} + 4 a^{11} + 4 a^{10} + 2 a^{9} + 20 a^{8} + a^{7} + a^{6} + 12 a^{5} + 6 a^{4} + 21 a^{3} + 6 a^{2} + 6 a + 10\right)\cdot 23^{113} + \left(13 a^{14} + 12 a^{13} + 21 a^{12} + 7 a^{10} + 8 a^{9} + 4 a^{8} + 3 a^{7} + 8 a^{6} + 12 a^{5} + 9 a^{4} + 21 a^{3} + 21 a^{2} + 2 a + 6\right)\cdot 23^{114} + \left(12 a^{14} + 7 a^{12} + 13 a^{11} + 3 a^{10} + 8 a^{9} + 9 a^{8} + 21 a^{7} + 14 a^{6} + 13 a^{5} + 14 a^{4} + 18 a^{3} + 10 a^{2} + 5 a + 4\right)\cdot 23^{115} + \left(9 a^{14} + 22 a^{13} + 9 a^{12} + 18 a^{11} + 4 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+ 3\right)\cdot 23^{127} + \left(a^{14} + 13 a^{13} + 18 a^{12} + 9 a^{11} + 11 a^{10} + 11 a^{9} + 12 a^{8} + 3 a^{7} + 2 a^{6} + 10 a^{5} + 2 a^{4} + 6 a^{3} + 7 a^{2} + 13 a + 22\right)\cdot 23^{128} + \left(6 a^{14} + 20 a^{13} + 3 a^{12} + 12 a^{11} + 17 a^{10} + 15 a^{9} + 19 a^{8} + 21 a^{7} + 9 a^{6} + 7 a^{5} + 3 a^{4} + 7 a^{3} + 3 a^{2} + 22 a + 7\right)\cdot 23^{129} + \left(7 a^{14} + 4 a^{13} + 4 a^{12} + 22 a^{11} + 9 a^{10} + 11 a^{9} + 17 a^{8} + 7 a^{7} + 14 a^{6} + 2 a^{5} + 11 a^{4} + 22 a^{3} + 10 a^{2} + 4 a + 3\right)\cdot 23^{130} + \left(8 a^{14} + 14 a^{13} + 13 a^{11} + 16 a^{10} + 9 a^{9} + 2 a^{8} + 2 a^{7} + 14 a^{5} + 16 a^{4} + 7 a^{2} + 18 a + 20\right)\cdot 23^{131} + \left(7 a^{14} + 13 a^{13} + 14 a^{12} + 7 a^{11} + 4 a^{10} + 18 a^{9} + 7 a^{8} + 14 a^{7} + 7 a^{6} + 21 a^{5} + 17 a^{4} + 14 a^{3} + 18 a^{2} + 22 a + 11\right)\cdot 23^{132} + \left(7 a^{14} + 9 a^{13} + 6 a^{12} + 12 a^{11} + 12 a^{10} + 15 a^{9} + 21 a^{8} + 10 a^{7} + 8 a^{6} + 17 a^{5} + 2 a^{4} + 5 a^{3} + 15 a^{2} + 8 a + 10\right)\cdot 23^{133} + \left(a^{14} + 14 a^{13} + 21 a^{12} + 22 a^{11} + 10 a^{10} + 4 a^{9} + 2 a^{8} + 12 a^{7} + 21 a^{6} + 9 a^{5} + 20 a^{4} + 20 a^{3} + 14 a^{2} + 8 a + 12\right)\cdot 23^{134} + \left(15 a^{14} + 12 a^{13} + 2 a^{12} + 3 a^{11} + 9 a^{10} + 17 a^{9} + 16 a^{8} + 8 a^{7} + 11 a^{6} + 15 a^{5} + a^{4} + 16 a^{3} + 8 a^{2} + 2 a + 22\right)\cdot 23^{135} + \left(a^{14} + 6 a^{13} + 20 a^{12} + 2 a^{11} + 9 a^{10} + 19 a^{9} + 12 a^{8} + 9 a^{7} + 13 a^{6} + 6 a^{4} + 9 a^{3} + 5 a^{2} + a + 11\right)\cdot 23^{136} + \left(21 a^{14} + 11 a^{13} + 16 a^{12} + 20 a^{11} + 14 a^{10} + 18 a^{9} + 19 a^{8} + a^{7} + 5 a^{6} + 11 a^{5} + 3 a^{4} + 4 a^{3} + 5 a^{2} + a + 10\right)\cdot 23^{137} + \left(8 a^{14} + 10 a^{13} + 10 a^{12} + 2 a^{11} + 15 a^{10} + 21 a^{9} + 14 a^{8} + 12 a^{7} + 21 a^{6} + 15 a^{5} + 21 a^{4} + 13 a^{3} + 19 a^{2} + 12 a + 14\right)\cdot 23^{138} + \left(13 a^{14} + 2 a^{13} + 2 a^{12} + 10 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a + 8\right)\cdot 23^{144} + \left(11 a^{14} + 19 a^{13} + 19 a^{12} + 3 a^{11} + 9 a^{10} + 12 a^{9} + 22 a^{8} + a^{7} + 7 a^{6} + 5 a^{5} + 9 a^{4} + 4 a^{3} + 16 a^{2} + 8 a + 3\right)\cdot 23^{145} + \left(7 a^{13} + 19 a^{12} + 3 a^{10} + 14 a^{8} + 4 a^{7} + 5 a^{6} + a^{5} + a^{4} + 6 a^{3} + 8 a^{2} + 3 a + 2\right)\cdot 23^{146} + \left(16 a^{14} + 3 a^{13} + 12 a^{12} + 20 a^{11} + a^{10} + 18 a^{9} + 8 a^{8} + a^{7} + 7 a^{6} + 13 a^{5} + 17 a^{4} + 15 a^{3} + 3 a^{2} + 15 a + 4\right)\cdot 23^{147} + \left(22 a^{14} + 9 a^{13} + 14 a^{12} + 11 a^{11} + a^{10} + 7 a^{9} + 8 a^{8} + 9 a^{7} + 18 a^{6} + 4 a^{5} + 20 a^{4} + 20 a^{3} + 16 a^{2} + 18\right)\cdot 23^{148} + \left(7 a^{14} + 11 a^{13} + 10 a^{12} + 22 a^{11} + 13 a^{9} + 19 a^{8} + 11 a^{7} + 20 a^{6} + 18 a^{5} + 13 a^{4} + a^{3} + 7 a^{2} + 22 a + 17\right)\cdot 23^{149} + \left(9 a^{14} + 12 a^{13} + a^{12} + 6 a^{11} + 13 a^{10} + 15 a^{9} + a^{8} + 8 a^{7} + 14 a^{6} + a^{5} + 15 a^{4} + 4 a^{3} + 6 a^{2} + 10 a + 20\right)\cdot 23^{150} + \left(21 a^{14} + a^{13} + 7 a^{12} + a^{11} + 22 a^{10} + 12 a^{9} + 14 a^{8} + 18 a^{7} + 7 a^{6} + 14 a^{5} + 5 a^{4} + a^{3} + 4 a^{2} + 14 a + 14\right)\cdot 23^{151} + \left(20 a^{14} + 14 a^{13} + 7 a^{12} + 20 a^{11} + 6 a^{10} + 20 a^{9} + 18 a^{8} + 5 a^{7} + 20 a^{6} + 8 a^{5} + 16 a^{4} + 20 a^{3} + 4 a^{2} + 13 a + 10\right)\cdot 23^{152} + \left(18 a^{14} + 18 a^{13} + 11 a^{12} + 17 a^{11} + 4 a^{10} + 10 a^{9} + 19 a^{8} + 19 a^{7} + 5 a^{6} + 8 a^{5} + 17 a^{3} + 5 a^{2} + 8 a + 20\right)\cdot 23^{153} + \left(10 a^{14} + 11 a^{13} + 17 a^{12} + 6 a^{11} + 2 a^{10} + 7 a^{9} + 16 a^{8} + 16 a^{7} + 9 a^{6} + 6 a^{5} + 8 a^{3} + 18 a^{2} + 16 a + 1\right)\cdot 23^{154} + \left(22 a^{14} + 12 a^{13} + 17 a^{12} + 14 a^{11} + 4 a^{10} + 16 a^{9} + 16 a^{8} + 19 a^{6} + 11 a^{5} + 12 a^{4} + 20 a^{3} + 21 a^{2} + 9 a + 13\right)\cdot 23^{155} + \left(22 a^{14} + 15 a^{13} + 11 a^{12} + 2 a^{11} + 19 a^{10} + 8 a^{9} + 15 a^{8} + a^{7} + 20 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a^{7} + 8 a^{6} + 8 a^{5} + 21 a^{4} + 3 a^{3} + 5 a^{2} + 12 a + 22\right)\cdot 23^{162} + \left(5 a^{14} + 20 a^{13} + 2 a^{12} + 5 a^{11} + 22 a^{10} + 16 a^{9} + 7 a^{8} + 18 a^{7} + 15 a^{6} + 13 a^{5} + 20 a^{4} + 18 a^{3} + 20 a^{2} + 10 a + 5\right)\cdot 23^{163} + \left(22 a^{14} + 20 a^{13} + 10 a^{12} + a^{11} + 22 a^{10} + 20 a^{9} + 6 a^{8} + a^{7} + 20 a^{6} + 17 a^{5} + 5 a^{4} + 20 a^{3} + 21 a^{2} + 7 a + 21\right)\cdot 23^{164} + \left(18 a^{14} + 22 a^{13} + 22 a^{12} + 19 a^{11} + 7 a^{10} + 17 a^{9} + 6 a^{7} + 14 a^{6} + 22 a^{5} + 21 a^{4} + 21 a^{3} + 13 a^{2} + 8\right)\cdot 23^{165} + \left(11 a^{14} + 7 a^{13} + 3 a^{12} + 14 a^{11} + 20 a^{10} + 10 a^{9} + 14 a^{8} + 21 a^{7} + 9 a^{6} + a^{5} + 15 a^{4} + 10 a^{3} + 12 a^{2} + 15 a + 7\right)\cdot 23^{166} + \left(22 a^{13} + 3 a^{12} + 19 a^{11} + 10 a^{10} + 20 a^{9} + a^{8} + 8 a^{6} + 5 a^{5} + 12 a^{4} + 9 a^{3} + 6 a^{2} + 3 a + 14\right)\cdot 23^{167} + \left(4 a^{14} + 15 a^{13} + 17 a^{12} + 14 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23^{179} + \left(a^{14} + a^{13} + 22 a^{12} + 17 a^{11} + 16 a^{10} + 19 a^{8} + 5 a^{7} + 2 a^{6} + 3 a^{5} + 8 a^{4} + 5 a^{3} + 12 a^{2} + 21 a + 11\right)\cdot 23^{180} + \left(21 a^{14} + 9 a^{13} + 10 a^{12} + 16 a^{11} + 18 a^{10} + 20 a^{9} + 14 a^{8} + 2 a^{7} + 4 a^{6} + 16 a^{5} + 8 a^{4} + 10 a^{3} + 13 a^{2} + 13 a + 2\right)\cdot 23^{181} + \left(6 a^{14} + 2 a^{13} + 20 a^{12} + 4 a^{11} + 17 a^{10} + 11 a^{9} + 13 a^{8} + 9 a^{7} + 6 a^{6} + 10 a^{5} + 9 a^{4} + 8 a^{3} + a^{2} + 17 a + 11\right)\cdot 23^{182} + \left(15 a^{14} + 3 a^{13} + 11 a^{12} + 2 a^{11} + 4 a^{10} + 15 a^{9} + 6 a^{8} + 18 a^{7} + 18 a^{6} + 7 a^{5} + 11 a^{3} + 15 a^{2} + 7 a + 3\right)\cdot 23^{183} + \left(14 a^{14} + 7 a^{13} + 7 a^{12} + 3 a^{11} + 19 a^{10} + 8 a^{8} + 21 a^{7} + 18 a^{6} + 17 a^{5} + 2 a^{4} + 15 a^{3} + 2 a^{2} + 10 a + 17\right)\cdot 23^{184} + \left(14 a^{14} + 12 a^{13} + 8 a^{12} + 13 a^{11} + 3 a^{10} + 7 a^{9} + 2 a^{8} + 20 a^{7} + 18 a^{6} + 13 a^{5} + 20 a^{4} + 3 a + 4\right)\cdot 23^{185} + \left(a^{14} + 17 a^{13} + 6 a^{12} + 12 a^{11} + 17 a^{10} + 2 a^{9} + 10 a^{8} + 6 a^{7} + 17 a^{6} + 12 a^{5} + 10 a^{2} + 8 a + 8\right)\cdot 23^{186} + \left(21 a^{14} + 20 a^{13} + 11 a^{12} + 7 a^{11} + 2 a^{9} + 2 a^{8} + 19 a^{7} + 15 a^{6} + 8 a^{5} + 20 a^{4} + 6 a^{3} + 9 a^{2} + 15 a + 1\right)\cdot 23^{187} + \left(4 a^{14} + 18 a^{13} + 17 a^{12} + 19 a^{11} + 5 a^{10} + 21 a^{9} + 15 a^{8} + 11 a^{7} + 8 a^{6} + 13 a^{5} + 8 a^{4} + 7 a^{3} + 6 a^{2} + 2 a + 3\right)\cdot 23^{188} + \left(20 a^{14} + 21 a^{13} + 17 a^{12} + 4 a^{11} + 5 a^{10} + 3 a^{9} + 20 a^{8} + 19 a^{6} + 9 a^{5} + 8 a^{4} + 22 a^{3} + 11 a^{2} + 21 a + 10\right)\cdot 23^{189} + \left(a^{14} + 20 a^{13} + a^{12} + 5 a^{11} + 7 a^{10} + 18 a^{9} + 14 a^{8} + 2 a^{7} + 17 a^{6} + 21 a^{5} + 4 a^{4} + 16 a^{3} + 15 a^{2} + 21 a + 4\right)\cdot 23^{190} + \left(14 a^{14} + 5 a^{13} + 4 a^{12} + 3 a^{11} + 16 a^{10} + 22 a^{9} + 10 a^{8} + 22 a^{7} + 18 a^{6} + 14 a^{5} + 15 a^{4} + 6 a^{3} + 18 a^{2} + 14 a + 2\right)\cdot 23^{191} + \left(21 a^{14} + 5 a^{13} + 6 a^{12} + 7 a^{11} + 7 a^{10} + 3 a^{9} + 19 a^{8} + 20 a^{7} + 12 a^{6} + 16 a^{5} + 8 a^{4} + 10 a^{3} + 4 a^{2} + 17 a + 6\right)\cdot 23^{192} + \left(16 a^{14} + 9 a^{13} + 21 a^{12} + 4 a^{11} + 22 a^{10} + 21 a^{9} + 12 a^{8} + 15 a^{7} + 19 a^{6} + 22 a^{5} + 14 a^{4} + 20 a^{3} + 16 a^{2} + 11 a + 19\right)\cdot 23^{193} + \left(6 a^{14} + 18 a^{13} + 12 a^{12} + 17 a^{11} + 9 a^{10} + 16 a^{9} + 15 a^{8} + 16 a^{7} + 11 a^{6} + 12 a^{5} + 11 a^{4} + 5 a^{3} + 7 a^{2} + a + 10\right)\cdot 23^{194} + \left(5 a^{14} + 11 a^{13} + 12 a^{12} + 13 a^{11} + 7 a^{10} + 7 a^{9} + 4 a^{8} + 14 a^{7} + 3 a^{6} + 9 a^{5} + 10 a^{4} + 20 a^{3} + 2 a^{2} + 16 a + 7\right)\cdot 23^{195} + \left(9 a^{13} + 15 a^{12} + 9 a^{11} + 15 a^{10} + 3 a^{9} + 14 a^{8} + 3 a^{7} + 11 a^{6} + 12 a^{5} + 10 a^{4} + 19 a^{3} + 15 a^{2} + 12 a + 12\right)\cdot 23^{196} + \left(2 a^{14} + 13 a^{13} + 17 a^{12} + 11 a^{11} + 6 a^{10} + 3 a^{9} + 17 a^{8} + 3 a^{7} + 6 a^{6} + 11 a^{5} + 12 a^{4} + 9 a^{3} + 10 a^{2} + 22 a + 16\right)\cdot 23^{197} + \left(7 a^{14} + 17 a^{13} + 17 a^{11} + 12 a^{10} + 10 a^{9} + 17 a^{8} + 18 a^{7} + 16 a^{6} + 7 a^{5} + 11 a^{4} + 4 a^{3} + 22 a^{2} + a + 8\right)\cdot 23^{198} + \left(22 a^{14} + 20 a^{13} + 22 a^{12} + 16 a^{11} + 16 a^{10} + 22 a^{9} + 6 a^{8} + 22 a^{7} + 19 a^{6} + 6 a^{5} + 7 a^{4} + 20 a^{3} + 5 a^{2} + 2 a + 3\right)\cdot 23^{199} + \left(7 a^{14} + 7 a^{13} + 4 a^{12} + 7 a^{11} + 10 a^{10} + 16 a^{9} + 7 a^{8} + 11 a^{7} + 14 a^{6} + 14 a^{5} + 20 a^{3} + 19 a + 16\right)\cdot 23^{200} + \left(10 a^{14} + 2 a^{12} + a^{11} + 15 a^{10} + a^{9} + 8 a^{8} + 13 a^{7} + 11 a^{6} + 6 a^{5} + 11 a^{4} + 17 a^{3} + 6 a^{2} + 14 a + 11\right)\cdot 23^{201} + \left(19 a^{14} + 7 a^{13} + 9 a^{12} + 7 a^{11} + 4 a^{10} + 15 a^{9} + 14 a^{8} + 4 a^{7} + 6 a^{6} + 2 a^{4} + 5 a^{3} + 21 a^{2} + 14 a + 9\right)\cdot 23^{202} + \left(16 a^{14} + 7 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1\right)\cdot 23^{208} + \left(7 a^{14} + 20 a^{13} + 8 a^{12} + 10 a^{11} + 3 a^{10} + 17 a^{9} + 3 a^{8} + a^{7} + 12 a^{6} + 6 a^{5} + 6 a^{3} + 14 a^{2} + 13\right)\cdot 23^{209} + \left(17 a^{14} + 14 a^{12} + 6 a^{11} + a^{10} + 17 a^{9} + 2 a^{8} + 5 a^{7} + 2 a^{6} + 9 a^{5} + 7 a^{4} + 10 a^{3} + 17 a^{2} + 11 a + 16\right)\cdot 23^{210} + \left(6 a^{14} + 4 a^{13} + 17 a^{12} + 8 a^{11} + a^{10} + 6 a^{9} + 16 a^{8} + 15 a^{7} + 20 a^{6} + 19 a^{5} + 19 a^{4} + 11 a^{3} + 4 a^{2} + 7 a + 22\right)\cdot 23^{211} + \left(11 a^{14} + 10 a^{13} + 17 a^{12} + 20 a^{11} + 20 a^{10} + 3 a^{9} + 18 a^{8} + 22 a^{7} + 16 a^{6} + 22 a^{5} + 7 a^{4} + 12 a^{3} + 17 a + 9\right)\cdot 23^{212} + \left(11 a^{14} + 7 a^{13} + 5 a^{12} + 8 a^{11} + 5 a^{10} + 22 a^{9} + 20 a^{8} + 20 a^{7} + 12 a^{6} + 18 a^{5} + 18 a^{4} + 5 a^{3} + 13 a^{2} + 8 a + 9\right)\cdot 23^{213} + \left(a^{14} + 15 a^{13} + a^{12} + a^{11} + 9 a^{10} + 16 a^{9} + 4 a^{8} + 18 a^{7} + 20 a^{6} + 9 a^{5} + a^{4} + 10 a^{3} + 4 a^{2} + 6 a + 11\right)\cdot 23^{214} + \left(a^{14} + 18 a^{13} + 16 a^{12} + a^{11} + 19 a^{10} + 4 a^{9} + 5 a^{8} + 11 a^{7} + 18 a^{6} + 10 a^{5} + 4 a^{4} + 16 a^{3} + 8 a^{2} + 4 a + 6\right)\cdot 23^{215} + \left(6 a^{14} + 18 a^{13} + 20 a^{12} + 19 a^{11} + 5 a^{10} + 18 a^{8} + 9 a^{7} + 5 a^{6} + 19 a^{5} + 19 a^{4} + 5 a^{3} + 14 a^{2} + 18 a + 4\right)\cdot 23^{216} + \left(8 a^{14} + 12 a^{13} + 3 a^{12} + 16 a^{11} + 11 a^{10} + 18 a^{9} + 19 a^{8} + 5 a^{7} + 15 a^{6} + 17 a^{5} + 7 a^{4} + 4 a^{3} + 18 a^{2} + 2 a + 1\right)\cdot 23^{217} + \left(18 a^{13} + 16 a^{12} + 19 a^{11} + 4 a^{10} + 13 a^{9} + 17 a^{8} + 21 a^{7} + 3 a^{6} + 16 a^{4} + 9 a^{3} + 6 a^{2} + 17 a + 17\right)\cdot 23^{218} + \left(10 a^{14} + 20 a^{13} + 19 a^{11} + 15 a^{10} + 18 a^{9} + 2 a^{7} + 8 a^{6} + 7 a^{5} + 8 a^{4} + 10 a^{3} + 20 a^{2} + 19 a + 22\right)\cdot 23^{219} + \left(13 a^{14} + 7 a^{13} + 16 a^{12} + 21 a^{11} + 7 a^{10} + 5 a^{9} + 10 a^{8} + 12 a^{7} + 3 a^{6} + 5 a^{5} + 12 a^{4} + 11 a^{3} + 14 a^{2} + 12 a + 13\right)\cdot 23^{220} + \left(12 a^{14} + 5 a^{13} + 21 a^{12} + 16 a^{11} + 5 a^{10} + 22 a^{9} + 11 a^{8} + 17 a^{7} + 21 a^{6} + 12 a^{5} + 16 a^{4} + 13 a^{3} + a^{2} + 14 a + 5\right)\cdot 23^{221} + \left(18 a^{14} + 5 a^{13} + 22 a^{12} + 3 a^{11} + 3 a^{10} + 8 a^{9} + 18 a^{8} + 12 a^{7} + 13 a^{6} + a^{5} + 2 a^{4} + 17 a^{3} + 20 a^{2} + 9 a + 20\right)\cdot 23^{222} + \left(12 a^{14} + 18 a^{13} + 2 a^{12} + 11 a^{11} + 19 a^{10} + 15 a^{9} + 4 a^{8} + 2 a^{7} + 20 a^{6} + 21 a^{5} + 16 a^{4} + 15 a^{3} + 10 a^{2} + 22\right)\cdot 23^{223} + \left(12 a^{14} + 16 a^{13} + 7 a^{12} + 13 a^{11} + 4 a^{10} + 2 a^{9} + 13 a^{8} + 9 a^{7} + 10 a^{6} + 16 a^{5} + 21 a^{4} + 2 a^{3} + 6 a^{2} + 17 a + 3\right)\cdot 23^{224} + \left(4 a^{14} + 21 a^{13} + 3 a^{12} + 21 a^{11} + 10 a^{10} + 4 a^{9} + 22 a^{8} + 16 a^{7} + 14 a^{6} + 10 a^{5} + 5 a^{4} + 13 a^{3} + 7 a^{2} + 12 a + 3\right)\cdot 23^{225} + \left(11 a^{14} + 10 a^{13} + 11 a^{12} + 3 a^{11} + 7 a^{10} + 9 a^{9} + 7 a^{8} + 8 a^{7} + 20 a^{6} + 5 a^{5} + 12 a^{4} + 7 a^{3} + 11 a^{2} + 7 a + 2\right)\cdot 23^{226} + \left(17 a^{14} + 4 a^{13} + 15 a^{11} + 4 a^{10} + 18 a^{8} + 2 a^{7} + 18 a^{6} + 4 a^{5} + 5 a^{4} + 15 a^{3} + 6 a^{2} + 16 a + 19\right)\cdot 23^{227} + \left(2 a^{14} + 22 a^{13} + 12 a^{12} + 7 a^{11} + 5 a^{10} + 21 a^{9} + 18 a^{8} + 19 a^{7} + 3 a^{6} + 10 a^{5} + 7 a^{4} + 20 a^{3} + 8 a^{2} + 5 a + 12\right)\cdot 23^{228} + \left(20 a^{14} + 21 a^{13} + 15 a^{12} + 3 a^{11} + 4 a^{10} + 17 a^{9} + 17 a^{8} + 9 a^{7} + 14 a^{6} + 8 a^{4} + 10 a^{3} + 17 a^{2} + 22 a + 2\right)\cdot 23^{229} + \left(3 a^{14} + 18 a^{13} + 11 a^{12} + 9 a^{11} + 7 a^{9} + 22 a^{8} + 16 a^{7} + 15 a^{6} + 4 a^{5} + 13 a^{4} + 9 a^{3} + 8 a^{2} + 9 a + 2\right)\cdot 23^{230} + \left(7 a^{14} + 11 a^{13} + 17 a^{12} + 20 a^{11} + 15 a^{10} + 18 a^{9} + 8 a^{8} + 9 a^{7} + a^{6} + 14 a^{5} + 10 a^{3} + 16 a^{2} + 7 a + 11\right)\cdot 23^{231} + \left(19 a^{14} + 6 a^{13} + 16 a^{11} + 14 a^{10} + 21 a^{9} + 9 a^{8} + 18 a^{7} + 13 a^{5} + 5 a^{4} + 5 a^{3} + 2 a^{2} + 19 a + 17\right)\cdot 23^{232} + \left(18 a^{14} + 7 a^{13} + 2 a^{12} + a^{11} + 11 a^{10} + 7 a^{9} + 10 a^{8} + 8 a^{7} + 5 a^{6} + 21 a^{5} + 5 a^{4} + 14 a^{3} + 11 a^{2} + 19 a + 19\right)\cdot 23^{233} + \left(3 a^{13} + 16 a^{12} + 19 a^{11} + 17 a^{10} + 6 a^{9} + 5 a^{8} + 3 a^{7} + 17 a^{6} + 12 a^{5} + 14 a^{4} + 12 a^{3} + 13 a^{2} + 22 a + 14\right)\cdot 23^{234} + \left(18 a^{14} + 12 a^{13} + a^{12} + 19 a^{11} + 4 a^{10} + 7 a^{9} + 3 a^{8} + 4 a^{7} + 22 a^{6} + 9 a^{5} + 10 a^{4} + 14 a^{3} + 11 a^{2} + 2 a + 1\right)\cdot 23^{235} + \left(14 a^{14} + 15 a^{13} + 18 a^{12} + 13 a^{11} + 5 a^{10} + 5 a^{9} + 8 a^{8} + 19 a^{7} + 14 a^{6} + 18 a^{4} + 6 a^{3} + a^{2} + 2 a\right)\cdot 23^{236} + \left(15 a^{14} + a^{13} + 10 a^{12} + 6 a^{11} + a^{10} + 9 a^{9} + 9 a^{8} + 10 a^{7} + 8 a^{6} + 4 a^{5} + 2 a^{4} + 18 a^{3} + 4 a + 20\right)\cdot 23^{237} + \left(10 a^{14} + 13 a^{13} + 22 a^{12} + 19 a^{11} + 3 a^{10} + 9 a^{8} + 17 a^{7} + 13 a^{6} + 17 a^{5} + 4 a^{4} + 10 a^{3} + 17 a^{2} + 7 a + 22\right)\cdot 23^{238} + \left(7 a^{14} + 17 a^{13} + 12 a^{12} + 22 a^{11} + 8 a^{10} + 10 a^{9} + 7 a^{8} + 2 a^{7} + 18 a^{6} + 12 a^{5} + 17 a^{4} + 18 a^{3} + 11 a^{2} + 3 a + 8\right)\cdot 23^{239} + \left(9 a^{14} + 5 a^{13} + 12 a^{12} + 7 a^{11} + 10 a^{10} + 7 a^{8} + 3 a^{7} + 19 a^{6} + 6 a^{5} + 18 a^{4} + 8 a^{3} + 6 a^{2} + 21 a + 10\right)\cdot 23^{240} + \left(6 a^{14} + 12 a^{13} + 5 a^{12} + 21 a^{11} + 2 a^{10} + 2 a^{9} + 3 a^{8} + 6 a^{7} + 12 a^{6} + 14 a^{5} + 10 a^{4} + 3 a^{3} + 15 a^{2} + 12 a + 21\right)\cdot 23^{241} + \left(9 a^{14} + 4 a^{13} + 10 a^{12} + 19 a^{11} + a^{10} + 22 a^{9} + 18 a^{8} + 18 a^{6} + 5 a^{5} + a^{4} + 21 a^{3} + 16 a^{2} + 8 a + 16\right)\cdot 23^{242} + \left(a^{14} + 11 a^{13} + 6 a^{12} + 20 a^{11} + 15 a^{10} + 14 a^{9} + 21 a^{8} + 19 a^{7} + 3 a^{6} + 12 a^{5} + 9 a^{4} + 9 a^{3} + 2 a^{2} + 22 a + 6\right)\cdot 23^{243} + \left(21 a^{14} + 22 a^{13} + 4 a^{12} + 19 a^{11} + 19 a^{9} + 4 a^{8} + 14 a^{7} + 8 a^{6} + 7 a^{5} + 6 a^{4} + 18 a^{3} + 18 a + 21\right)\cdot 23^{244} + \left(14 a^{14} + 20 a^{13} + 5 a^{12} + 4 a^{11} + 12 a^{10} + 16 a^{9} + 9 a^{8} + 3 a^{7} + 12 a^{6} + 21 a^{5} + 22 a^{4} + 8 a^{3} + 6 a^{2} + 10 a + 2\right)\cdot 23^{245} + \left(a^{14} + 6 a^{13} + 15 a^{12} + 10 a^{11} + 5 a^{10} + 16 a^{8} + 12 a^{7} + 7 a^{6} + 11 a^{5} + 18 a^{4} + 20 a^{3} + 13 a^{2} + 15 a + 2\right)\cdot 23^{246} + \left(3 a^{14} + 15 a^{13} + 11 a^{12} + 9 a^{11} + 16 a^{10} + 4 a^{9} + 18 a^{8} + 3 a^{7} + 11 a^{6} + 12 a^{5} + 8 a^{3} + 22 a^{2} + 13 a + 22\right)\cdot 23^{247} + \left(19 a^{14} + 15 a^{13} + 14 a^{12} + 15 a^{11} + 16 a^{10} + 10 a^{9} + 2 a^{8} + 4 a^{7} + 4 a^{6} + a^{5} + 13 a^{4} + 15 a^{3} + 20 a^{2} + 22 a + 21\right)\cdot 23^{248} + \left(3 a^{14} + 4 a^{13} + 9 a^{12} + 22 a^{11} + 9 a^{10} + 16 a^{9} + 4 a^{8} + 18 a^{7} + 2 a^{6} + 10 a^{5} + 6 a^{4} + 21 a^{3} + 7 a^{2} + 19 a + 11\right)\cdot 23^{249} + \left(12 a^{14} + 7 a^{13} + 5 a^{12} + 5 a^{11} + 9 a^{10} + 3 a^{9} + 2 a^{8} + 18 a^{7} + 3 a^{6} + 21 a^{5} + 14 a^{4} + 21 a^{3} + 7 a^{2} + 3 a + 6\right)\cdot 23^{250} + \left(7 a^{14} + 3 a^{13} + 6 a^{12} + 5 a^{11} + a^{10} + 11 a^{8} + 16 a^{7} + 22 a^{6} + 13 a^{5} + 14 a^{4} + 5 a^{3} + 16 a^{2} + 9 a + 1\right)\cdot 23^{251} + \left(20 a^{14} + a^{13} + 12 a^{12} + 13 a^{11} + 4 a^{10} + 11 a^{9} + 12 a^{8} + 3 a^{7} + 7 a^{6} + 22 a^{5} + 10 a^{4} + 12 a^{3} + 16 a^{2} + 16 a + 9\right)\cdot 23^{252} + \left(a^{14} + 15 a^{13} + 3 a^{12} + 13 a^{11} + 15 a^{10} + 11 a^{8} + 15 a^{7} + 8 a^{6} + 10 a^{5} + a^{4} + 19 a^{3} + 13 a^{2} + 20 a + 5\right)\cdot 23^{253} + \left(4 a^{14} + 5 a^{13} + 6 a^{12} + 13 a^{10} + 19 a^{9} + 6 a^{8} + 14 a^{7} + 18 a^{6} + 2 a^{5} + 22 a^{3} + 11 a^{2} + 13 a + 5\right)\cdot 23^{254} + \left(4 a^{14} + 8 a^{13} + 13 a^{12} + 9 a^{11} + 22 a^{10} + 5 a^{9} + 7 a^{8} + 4 a^{7} + 2 a^{6} + 6 a^{5} + 17 a^{4} + 20 a^{3} + 17 a^{2} + 13 a + 20\right)\cdot 23^{255} + \left(15 a^{14} + 17 a^{12} + 3 a^{11} + 11 a^{10} + 9 a^{9} + 3 a^{8} + 7 a^{7} + 12 a^{6} + 22 a^{5} + 9 a^{4} + 3 a^{3} + 17 a^{2} + 13 a + 21\right)\cdot 23^{256} + \left(4 a^{14} + 18 a^{13} + 8 a^{12} + 20 a^{11} + 17 a^{10} + 14 a^{9} + 6 a^{8} + 14 a^{7} + 22 a^{6} + 2 a^{5} + 12 a^{4} + a^{3} + 16 a^{2} + 19 a + 6\right)\cdot 23^{257} + \left(11 a^{14} + 11 a^{13} + 6 a^{12} + 15 a^{11} + 18 a^{10} + 3 a^{9} + 11 a^{8} + 20 a^{7} + 17 a^{6} + 10 a^{5} + 22 a^{4} + a^{3} + 7 a^{2} + 7 a + 5\right)\cdot 23^{258} + \left(8 a^{14} + 2 a^{13} + 13 a^{11} + 18 a^{10} + 7 a^{9} + 19 a^{8} + 6 a^{7} + 4 a^{6} + 6 a^{5} + 3 a^{4} + 15 a^{3} + 20 a^{2} + 4 a + 21\right)\cdot 23^{259} + \left(5 a^{14} + 6 a^{13} + 11 a^{12} + 7 a^{11} + 8 a^{10} + 20 a^{9} + 17 a^{8} + 18 a^{7} + 8 a^{6} + 5 a^{5} + 16 a^{4} + 20 a^{3} + 13 a^{2} + 8 a + 7\right)\cdot 23^{260} + \left(22 a^{14} + 8 a^{13} + a^{12} + 9 a^{11} + 15 a^{10} + 12 a^{9} + a^{8} + 19 a^{7} + 20 a^{6} + 6 a^{5} + 11 a^{4} + a^{3} + 3 a^{2} + 4 a + 15\right)\cdot 23^{261} + \left(7 a^{14} + 8 a^{13} + 16 a^{12} + 11 a^{11} + 3 a^{10} + a^{9} + 2 a^{8} + a^{7} + a^{6} + 14 a^{5} + 4 a^{4} + a^{3} + 14 a^{2} + 20 a + 10\right)\cdot 23^{262} + \left(4 a^{14} + 15 a^{13} + 5 a^{11} + 3 a^{10} + 16 a^{9} + 22 a^{8} + a^{7} + 5 a^{6} + 22 a^{5} + 10 a^{4} + 16 a^{3} + 2 a^{2} + 10 a + 19\right)\cdot 23^{263} + \left(14 a^{14} + 21 a^{13} + 9 a^{12} + 19 a^{11} + 21 a^{10} + 2 a^{9} + 12 a^{8} + a^{7} + 16 a^{6} + 21 a^{5} + 8 a^{4} + 16 a^{3} + 7 a^{2} + 6 a\right)\cdot 23^{264} + \left(9 a^{14} + 13 a^{13} + 4 a^{12} + 16 a^{11} + 11 a^{10} + 17 a^{9} + 10 a^{8} + 8 a^{6} + 22 a^{5} + 14 a^{4} + 19 a^{3} + 6 a^{2} + 19 a + 3\right)\cdot 23^{265} + \left(22 a^{14} + 15 a^{13} + 5 a^{12} + 2 a^{11} + 10 a^{10} + 10 a^{9} + 2 a^{8} + 8 a^{7} + 22 a^{6} + 9 a^{5} + 2 a^{4} + 13 a^{3} + 17 a^{2} + 10 a\right)\cdot 23^{266} + \left(17 a^{14} + 11 a^{13} + 10 a^{12} + 22 a^{11} + 22 a^{10} + 21 a^{9} + 16 a^{8} + 15 a^{7} + 11 a^{6} + 15 a^{5} + 15 a^{4} + 10 a^{3} + 21 a^{2} + 20 a + 9\right)\cdot 23^{267} + \left(9 a^{14} + 4 a^{13} + 9 a^{12} + 16 a^{11} + 12 a^{10} + 11 a^{9} + 8 a^{8} + 11 a^{7} + 20 a^{6} + 5 a^{5} + 17 a^{4} + 2 a^{3} + 14 a^{2} + 8 a + 18\right)\cdot 23^{268} +O\left(23^{ 269 }\right)$
## Generators of the action on the roots $r_1, \ldots, r_{ 17 }$
Cycle notation $(1,15)(2,10)(3,7)(4,16)(5,8)(6,17)(11,13)(12,14)$ $(1,3,16)(2,7,11)(4,5,14)(6,10,12)(8,13,9)$
## Character values on conjugacy classes
Size Order Action on $r_1, \ldots, r_{ 17 }$ Character value $1$ $1$ $()$ $15$ $255$ $2$ $(1,15)(2,10)(3,7)(4,16)(5,8)(6,17)(11,13)(12,14)$ $-1$ $272$ $3$ $(1,3,16)(2,7,11)(4,5,14)(6,10,12)(8,13,9)$ $0$ $272$ $5$ $(1,12,3,2,11)(4,5,17,15,14)(6,9,7,8,10)$ $0$ $272$ $5$ $(1,3,11,12,2)(4,17,14,5,15)(6,7,10,9,8)$ $0$ $272$ $15$ $(1,3,15,11,4,17,10,12,13,7,6,8,16,5,9)$ $0$ $272$ $15$ $(1,15,4,10,13,6,16,9,3,11,17,12,7,8,5)$ $0$ $272$ $15$ $(1,4,13,16,3,17,7,5,15,10,6,9,11,12,8)$ $0$ $272$ $15$ $(1,12,9,10,5,17,16,4,8,11,6,15,7,3,13)$ $0$ $240$ $17$ $(1,15,3,11,9,8,14,6,17,10,7,16,5,13,2,12,4)$ $-\zeta_{17}^{14} - \zeta_{17}^{3}$ $240$ $17$ $(1,3,9,14,17,7,5,2,4,15,11,8,6,10,16,13,12)$ $-\zeta_{17}^{11} - \zeta_{17}^{6}$ $240$ $17$ $(1,11,14,10,5,12,15,9,6,7,13,4,3,8,17,16,2)$ $-\zeta_{17}^{9} - \zeta_{17}^{8}$ $240$ $17$ $(1,9,17,5,4,11,6,16,12,3,14,7,2,15,8,10,13)$ $-\zeta_{17}^{12} - \zeta_{17}^{5}$ $240$ $17$ $(1,8,7,12,11,17,13,15,14,16,4,9,10,2,3,6,5)$ $-\zeta_{17}^{15} - \zeta_{17}^{2}$ $240$ $17$ $(1,14,5,15,6,13,3,17,2,11,10,12,9,7,4,8,16)$ $\zeta_{17}^{15} + \zeta_{17}^{14} + \zeta_{17}^{13} + \zeta_{17}^{12} + \zeta_{17}^{11} + \zeta_{17}^{10} + \zeta_{17}^{9} + \zeta_{17}^{8} + \zeta_{17}^{7} + \zeta_{17}^{6} + \zeta_{17}^{5} + \zeta_{17}^{4} + \zeta_{17}^{3} + \zeta_{17}^{2} + 1$ $240$ $17$ $(1,6,2,9,16,15,17,12,8,5,3,10,4,14,13,11,7)$ $-\zeta_{17}^{13} - \zeta_{17}^{4}$ $240$ $17$ $(1,17,4,6,12,14,2,8,13,9,5,11,16,3,7,15,10)$ $-\zeta_{17}^{10} - \zeta_{17}^{7}$
The blue line marks the conjugacy class containing complex conjugation.
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CC-MAIN-2020-45
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https://us.metamath.org/mpeuni/islss.html
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crawl-data/CC-MAIN-2024-22/segments/1715971058721.41/warc/CC-MAIN-20240524152541-20240524182541-00708.warc.gz
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Metamath Proof Explorer < Previousย ย Next > Nearby theorems Mirrorsย > ย Homeย > ย MPE Homeย > ย Th. Listย > ย islss Structured versionย ย Visualization versionย ย GIF version
Theorem islssย 19682
Description: The predicate "is a subspace" (of a left module or left vector space). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
Hypotheses
Ref Expression
lssset.f ๐น = (Scalarโ๐)
lssset.b ๐ต = (Baseโ๐น)
lssset.v ๐ = (Baseโ๐)
lssset.p + = (+g๐)
lssset.t ยท = ( ยท๐ ๐)
lssset.s ๐ = (LSubSpโ๐)
Assertion
Ref Expression
islss (๐๐ โ (๐๐๐ โ โ
โง โ๐ฅ๐ต๐๐๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐))
Distinct variable groups: ย ๐ฅ,๐ต ย ๐,๐,๐ฅ,๐ ย ๐,๐,๐,๐ฅ
Allowed substitution hints: ย ๐ต(๐,๐) ย + (๐ฅ,๐,๐) ย ๐(๐ฅ,๐,๐) ย ยท (๐ฅ,๐,๐) ย ๐น(๐ฅ,๐,๐) ย ๐(๐ฅ,๐,๐)
Proof of Theorem islss
Dummy variable ๐ is distinct from all other variables.
StepHypRef Expression
1ย elfvexย 6676 . . 3 (๐ โ (LSubSpโ๐) โ ๐ โ V)
2ย lssset.s . . 3 ๐ = (LSubSpโ๐)
31, 2eleq2sย 2930 . 2 (๐๐๐ โ V)
4ย lssset.v . . . . . . . . 9 ๐ = (Baseโ๐)
5ย fvprcย 6636 . . . . . . . . 9 ๐ โ V โ (Baseโ๐) = โ
)
64, 5syl5eqย 2868 . . . . . . . 8 ๐ โ V โ ๐ = โ
)
76sseq2dย 3975 . . . . . . 7 ๐ โ V โ (๐๐๐ โ โ
))
87biimpcdย 252 . . . . . 6 (๐๐ โ (ยฌ ๐ โ V โ ๐ โ โ
))
9ย ss0ย 4325 . . . . . 6 (๐ โ โ
โ ๐ = โ
)
108, 9syl6ย 35 . . . . 5 (๐๐ โ (ยฌ ๐ โ V โ ๐ = โ
))
1110necon1adย 3024 . . . 4 (๐๐ โ (๐ โ โ
โ ๐ โ V))
1211impย 410 . . 3 ((๐๐๐ โ โ
) โ ๐ โ V)
13123adant3ย 1129 . 2 ((๐๐๐ โ โ
โง โ๐ฅ๐ต๐๐๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐) โ ๐ โ V)
14ย lssset.f . . . . 5 ๐น = (Scalarโ๐)
15ย lssset.b . . . . 5 ๐ต = (Baseโ๐น)
16ย lssset.p . . . . 5 + = (+g๐)
17ย lssset.t . . . . 5 ยท = ( ยท๐ ๐)
1814, 15, 4, 16, 17, 2lsssetย 19681 . . . 4 (๐ โ V โ ๐ = {๐ โ (๐ซ ๐ โ {โ
}) โฃ โ๐ฅ๐ต๐๐ ๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐ })
1918eleq2dย 2897 . . 3 (๐ โ V โ (๐๐๐ โ {๐ โ (๐ซ ๐ โ {โ
}) โฃ โ๐ฅ๐ต๐๐ ๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐ }))
20ย eldifsnย 4692 . . . . . 6 (๐ โ (๐ซ ๐ โ {โ
}) โ (๐ โ ๐ซ ๐๐ โ โ
))
214fvexiย 6657 . . . . . . . 8 ๐ โ V
2221elpw2ย 5221 . . . . . . 7 (๐ โ ๐ซ ๐๐๐)
2322anbi1iย 626 . . . . . 6 ((๐ โ ๐ซ ๐๐ โ โ
) โ (๐๐๐ โ โ
))
2420, 23bitriย 278 . . . . 5 (๐ โ (๐ซ ๐ โ {โ
}) โ (๐๐๐ โ โ
))
2524anbi1iย 626 . . . 4 ((๐ โ (๐ซ ๐ โ {โ
}) โง โ๐ฅ๐ต๐๐๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐) โ ((๐๐๐ โ โ
) โง โ๐ฅ๐ต๐๐๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐))
26ย eleq2ย 2900 . . . . . . . 8 (๐ = ๐ โ (((๐ฅ ยท ๐) + ๐) โ ๐ โ ((๐ฅ ยท ๐) + ๐) โ ๐))
2726raleqbi1dvย 3388 . . . . . . 7 (๐ = ๐ โ (โ๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐ โ โ๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐))
2827raleqbi1dvย 3388 . . . . . 6 (๐ = ๐ โ (โ๐๐ ๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐ โ โ๐๐๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐))
2928ralbidvย 3185 . . . . 5 (๐ = ๐ โ (โ๐ฅ๐ต๐๐ ๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐ โ โ๐ฅ๐ต๐๐๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐))
3029elrabย 3657 . . . 4 (๐ โ {๐ โ (๐ซ ๐ โ {โ
}) โฃ โ๐ฅ๐ต๐๐ ๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐ } โ (๐ โ (๐ซ ๐ โ {โ
}) โง โ๐ฅ๐ต๐๐๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐))
31ย df-3anย 1086 . . . 4 ((๐๐๐ โ โ
โง โ๐ฅ๐ต๐๐๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐) โ ((๐๐๐ โ โ
) โง โ๐ฅ๐ต๐๐๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐))
3225, 30, 313bitr4iย 306 . . 3 (๐ โ {๐ โ (๐ซ ๐ โ {โ
}) โฃ โ๐ฅ๐ต๐๐ ๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐ } โ (๐๐๐ โ โ
โง โ๐ฅ๐ต๐๐๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐))
3319, 32syl6bbย 290 . 2 (๐ โ V โ (๐๐ โ (๐๐๐ โ โ
โง โ๐ฅ๐ต๐๐๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐)))
343, 13, 33pm5.21niiย 383 1 (๐๐ โ (๐๐๐ โ โ
โง โ๐ฅ๐ต๐๐๐๐ ((๐ฅ ยท ๐) + ๐) โ ๐))
Colors of variables: wff setvar class Syntax hints: ย ยฌ wnย 3 ย โ wbย 209 ย โง waย 399 ย โง w3aย 1084 ย = wceqย 1538 ย โ wcelย 2115 ย โ wneย 3007 ย โwralย 3126 ย {crabย 3130 ย Vcvvย 3471 ย โ cdifย 3907 ย โ wssย 3910 ย โ
c0ย 4266 ย ๐ซ cpwย 4512 ย {csnย 4540 ย โcfvย 6328 ย (class class class)coย 7130 ย Basecbsย 16462 ย +gcplusgย 16544 ย Scalarcscaย 16547 ย ยท๐ cvscaย 16548 ย LSubSpclssย 19679 This theorem was proved from axioms: ย ax-mpย 5 ย ax-1ย 6 ย ax-2ย 7 ย ax-3ย 8 ย ax-genย 1797 ย ax-4ย 1811 ย ax-5ย 1912 ย ax-6ย 1971 ย ax-7ย 2016 ย ax-8ย 2117 ย ax-9ย 2125 ย ax-10ย 2146 ย ax-11ย 2162 ย ax-12ย 2178 ย ax-extย 2793 ย ax-sepย 5176 ย ax-nulย 5183 ย ax-powย 5239 ย ax-prย 5303 This theorem depends on definitions: ย df-biย 210 ย df-anย 400 ย df-orย 845 ย df-3anย 1086 ย df-truย 1541 ย df-exย 1782 ย df-nfย 1786 ย df-sbย 2071 ย df-moย 2623 ย df-euย 2654 ย df-clabย 2800 ย df-cleqย 2814 ย df-clelย 2892 ย df-nfcย 2960 ย df-neย 3008 ย df-ralย 3131 ย df-rexย 3132 ย df-rabย 3135 ย df-vย 3473 ย df-sbcย 3750 ย df-difย 3913 ย df-unย 3915 ย df-inย 3917 ย df-ssย 3927 ย df-nulย 4267 ย df-ifย 4441 ย df-pwย 4514 ย df-snย 4541 ย df-prย 4543 ย df-opย 4547 ย df-uniย 4812 ย df-brย 5040 ย df-opabย 5102 ย df-mptย 5120 ย df-idย 5433 ย df-xpย 5534 ย df-relย 5535 ย df-cnvย 5536 ย df-coย 5537 ย df-dmย 5538 ย df-iotaย 6287 ย df-funย 6330 ย df-fvย 6336 ย df-ovย 7133 ย df-lssย 19680 This theorem is referenced by: ย islssd ย 19683 ย lssss ย 19684 ย lssn0 ย 19688 ย lsscl ย 19690 ย islss4 ย 19710 ย lsspropd ย 19765 ย islidl ย 19960 ย ocvlss ย 20792 ย lkrlss ย 36277 ย lclkr ย 38715 ย lclkrs ย 38721 ย lcfr ย 38767
Copyright terms: Public domain W3C validator
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CC-MAIN-2024-22
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latest
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| 0.094667
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