instruction stringlengths 12 30k |
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It depends on your population and the confidence that you need in your results. Typically, you should know something about what you are trying to achieve. If you know your confidence level, confidence interval, and population size, you can choose a sample size that will give you the properly confidence.
Your confide... |
It depends on your population and the confidence that you need in your results. Typically, you should know something about what you are trying to achieve. If you know your confidence level, confidence interval, and population size, you can choose a sample size that will give you the properly confidence.
Your confide... |
Although it's a little basic, I managed to find a ["geometry reference sheet" (direct PDF download)][1]. It's got your standard two- and three-dimensional shapes, plus your law of sines, law of cosines, distance, and midpoint formulas. Although if it's been years since you had to find the volume of a sphere, this kind ... |
Sorry if this question is too basic. It is from Fröhlich and Taylor's "Algebraic Number Theory".
Let $E/F$ be a finite Galois extension of fields, with $G=Gal(E/F)$, and let $K$ and $L$ be two subfields of $E$, containing $F$, such that $K/F$ and $L/F$ are both Galois. Let $M=Gal(E/K)$ and $N=Gal(E/L)$ be normal sub... |
I haven't quite got this straight yet, but I think one way to go is to think about choosing points at random from the positive reals. This answer is going to rather longer than it really needs to be, because I'm thinking about this in a few (closely related) ways, which probably aren't all necessary, and you can decide... |
I haven't quite got this straight yet, but I think one way to go is to think about choosing points at random from the positive reals. This answer is going to rather longer than it really needs to be, because I'm thinking about this in a few (closely related) ways, which probably aren't all necessary, and you can decide... |
Here's the question I've been thinking for 2 days and couldn't find an answer why.
If Y = √Y, where both X and Y are real numbers then which of the following is true?
a) X>=0; Y>=0
b) X<=0; Y>=0
c) X<=0; Y<=0
d) X>=0; Y<=0
e) Either a) or b)
Now, the answer is a) but I don't know why. I know a valu... |
I haven't quite got this straight yet, but I think one way to go is to think about choosing points at random from the positive reals. This answer is going to rather longer than it really needs to be, because I'm thinking about this in a few (closely related) ways, which probably aren't all necessary, and you can decide... |
Use the definition:
Say $f$ is defined on $[-\pi, \pi]$.
If $f(z) = \sum_{-\infty}^{\infty} {c_{n} e^{inz}}$
then
$c_{n} = \frac{1}{2\pi}\int_{-\pi}^{\pi}{f(z)e^{-inz}} dz$
If you put $f(z) = z$, can you work out what $c_{n}$ turns out to be?
To integrate, you can try integration by parts. |
I haven't quite got this straight yet, but I think one way to go is to think about choosing points at random from the positive reals. This answer is going to rather longer than it really needs to be, because I'm thinking about this in a few (closely related) ways, which probably aren't all necessary, and you can decide... |
You can use the formula
$\int_{0}^{\pi}{\sin^{2k+1}(x) dx} = 2\frac{2 \cdot 4 \cdot 6 \cdots 2k}{3\cdot 5 \cdots (2k+1)}$
This is called Walli's product, I believe.
(Aside: how do I get these formulae to look bigger?) |
You can use the formula
$\int_{0}^{\pi}{\sin^{2k+1}(x) dx} = 2\frac{2 \cdot 4 \cdot 6 \cdots 2k}{3\cdot 5 \cdots (2k+1)}$
This is called Walli's product, I believe.
I believe it comes to $\frac{1}{2}\int_{0}^{\pi}{\frac{a\sin x}{1 + (a \sin x)^2}dx}$.
(Aside: how do I get these formulae to look bigger?) |
Is there a natural way to represent all the partitions of an integer set {1,2,3,...,n} as a matrix in the similar way permutations can be mapped to group of matrices? |
At the beginning of [Week 300 of John Baez's blog][1],
Baez gives a proof that the "number" of finite sets (more specifically, the cardinality of the groupoid of all finite sets, where an object in the groupoid counts as 1/n! if it has n! symmetries) equals e.
He then says that this leads to a purely combinatorial ... |
A matrix determinant (naively) can be computed in O(n!) steps, or with a proper LU decomposition O(n^3) steps. This assumes that all the matrix elements are constant. If, however the matrix elements are polynomials (say univariate of max order p) at each step of the LU decomposition an element is multiplied by another ... |
Could you provide a proof of Euler formula?
thanks. |
How to prove Euler formula? |
Could you provide a proof of Euler's formula?
thanks. |
How to prove Euler's formula? |
A standard example is
$$
f(t)= \sum_{n>1} \frac{\sin(nt)}{\log(n)}$$
The conjugate of $f$ is a Fourier series but $f\not\in L^1(\mathbb{T})$ and hence is no Fourier series.
For further explanation see [Katznelson's book page 85.][1]
(Edit: If $f$ is not in $L^1(\mathbb{T})$ it is hard to define Fourier coeffici... |
A matrix determinant (naively) can be computed in O(n!) steps, or with a proper LU decomposition O(n^3) steps. This assumes that all the matrix elements are constant. If, however the matrix elements are polynomials (say univariate of max order p) at each step of the LU decomposition an element is multiplied by another ... |
what does it mean to "test some HTML elements" in a web page? what sort of [numerical] data do you expect to collect - and what, if anything, do you want to infer from what you get?
a particular example may help to clarify what you are driving at.
|
Liouville's Number is defined as $L = \sum_{n=1}^{\infty}(10^{-n!})$. Does it have other applications than just constructing a transcendental number?
(Personally, I would have defined it (as "Steven's Number" :-)) as binary: $S = \sum_{n=1}^{\infty}(2^{-n!})$, since each digit can only be "0" or "1": the corresp... |
I have the following cost function that I want to minimize:
\left F_min = \sum_{k=0}^n a_k x_k
The problem with the above minimization is that, I don't know what `n` is-- it is also a part of the minimization.
There are a few constraints governing how `x` should behave. Also, there are constraints linking ... |
I have a standard linear programming problems I want to solve:
![alt text][1]
The problem with the above minimization is that, I don't know what `n`, the number of variable `x` is-- it is also a part of the minimization.
There are a few constraints governing how `x` should behave. Also, there are constrain... |
I have a standard linear programming problems I want to solve:
![alt text][1]
The problem with the above minimization is that, I don't know what `n`, the number of variable `x` is-- it is also a part of the minimization.
There are a few constraints governing how the vector `x` should behave. Additionally, ... |
I don't know the answer, but in my opinion a lot of people are misinterpreting the question, so perhaps it is worth an answer to try to set this straight.
Here is an analogy: trigonometric series is to Fourier series as power series is to Taylor series. In other words, a trigonometric series is just any series of t... |
I have a standard linear programming problems I want to solve:
$$ \min_x f^T x \text{ such that } \left\\{ \begin{aligned} A\cdot x &\le b, \\\\ A_{eq}\cdot x &= b_{eq}, \\\\ lb \le x &\le ub. \end{aligned} \right. $$
The problem with the above minimization is that, I don't know what $n$, the number of variable... |
How to show the following identity holds?
$\displaystyle\sum_{n=1}^\infty\dfrac{2z}{z^2-n^2}=\pi\cot \pi z-\dfrac{1}{z}\qquad |z|<1$ |
I have a standard linear programming problems I want to solve:
$$ \min_x f^T x \text{ such that } \left\\{ \begin{aligned} A\cdot x &\le b, \\\\ A_{eq}\cdot x &= b_{eq}, \\\\ lb \le x &\le ub. \end{aligned} \right. $$
$F^T$ is a vector that doesn't involve $x$.
The problem with the above minimization is that... |
How do I derive the Beta function using the definition of the beta function as the normalizing constant of the Beta distribution and only common sense random experiments?
I'm pretty sure this is possible, but can't see how.
I can see that
$\sum_{a=0}^n \binom{n}{a} Beta(a+1, n-a+1) = 1$
because we can imagi... |
How do I derive the Beta function using the definition of the beta function as the normalizing constant of the Beta distribution and only common sense random experiments?
I'm pretty sure this is possible, but can't see how.
I can see that
$\sum_{a=0}^n \binom{n}{a} Beta(a+1, n-a+1) = 1$
because we can imagi... |
How do I derive the Beta function using the definition of the beta function as the normalizing constant of the Beta distribution and only common sense random experiments?
I'm pretty sure this is possible, but can't see how.
I can see that
$\sum_{a=0}^n \binom{n}{a} Beta(a+1, n-a+1) = 1$
because we can imagi... |
I have a standard linear programming problems I want to solve:
$$ \min_x f^T x \text{ such that } \left\\{ \begin{aligned} A\cdot x &\le b, \\\\ A_{eq}\cdot x &= b_{eq}, \\\\ lb \le x &\le ub. \end{aligned} \right. $$
$F^T$ is a vector that doesn't involve $x$.
The problem with the above minimization is tha... |
You problem statement is too vague. I guess you mean all four polynomials and their pairwise sums are non-degenerate, otherwise we can simply pick any four straight lines whose slopes are nonzero and of different magnitudes. There should be other restrictions as well, or else {x^2, 2x^2, 3x^2, 4x^2} will also be a so... |
How do I derive the Beta function using the definition of the beta function as the normalizing constant of the Beta distribution and only common sense random experiments?
I'm pretty sure this is possible, but can't see how.
I can see that
$\sum_{a=0}^n {n \choose a} Beta(a+1, n-a+1) = 1$
because we can imag... |
How do I derive the Beta function using the definition of the beta function as the normalizing constant of the Beta distribution and only common sense random experiments?
I'm pretty sure this is possible, but can't see how.
I can see that
$\sum_{a=0}^n {n \choose a} Beta(a+1, n-a+1) = 1$
because we can imag... |
Mathematicians, and esp. number theorists, are used to working with big numbers. I have noted on several occasions that lots of people don't have a clear understanding of big numbers as far as the real world is concerned. I recall a [request](http://groups.google.com/group/sci.math/browse_thread/thread/5da7f8d21fa57d7d... |
How to explain Real Big Numbers? |
how could i convert a non-uniform random variable distribution to a uniform distribution ? Is there a formula ? thanks .
|
I know that the product of two Gaussian functions is also Gaussian function. This is stated in [Wikipedia][1], but I might need to cite a classical (text)book/paper stating this result. A book containing the proof is welcome. Could you name such a book?
[1]: http://en.wikipedia.org/wiki/Gaussian_function |
Mathematicians, and esp. number theorists, are used to working with big numbers. I have noted on several occasions that lots of people don't have a clear understanding of big numbers as far as the real world is concerned. I recall a [request](http://groups.google.com/group/sci.math/browse_thread/thread/5da7f8d21fa57d7d... |
Though I don't quite know that you actually want to hear - *what kind of numbers* do you want to give people insight in, *whom* and *why*? - I'll give a few thoughts.
I) **Real cases**
That's just understanding of natural sciences - numbers of particles in the universe, number of cells in a body ... Try to first ... |
Mathematicians, and esp. number theorists, are used to working with big numbers. I have noted on several occasions that lots of people don't have a clear understanding of big numbers as far as the real world is concerned. I recall a [request](http://groups.google.com/group/sci.math/browse_thread/thread/5da7f8d21fa57d7d... |
Rotations in 3 dimensions are not commutative; however they are in the plane. In classical mechanics, are we allowed to say that angular momentum is a vector because particles only rotate along a single axis? Or do we make some kind of argument that you can assign angular velocity to an object because you can approxima... |
How can angular velocity or angular momentum be a vector? |
The important component of the question here seems to be: can we define a natural binary operation on the set P of partitions of $\\{1,2,\ldots,n\\}$ to form a group G? [Cayley's Theorem][1] tells us that if G exists, it is isomorphic to a subgroup of the symmetric group on G. So G will be isomorphic to a group of pe... |
Back in the day, I was absolutely enthralled by the study of plane curves and their properties (I have [Lockwood][1] and [Zwikker][2] to thank). I learned early on that for the purposes of generating plots on a computer (and for that matter deducing equations of "derived curves" and determining other special properties... |
There are a couple of things at play here, which coincidentally(?) mirror the difference between mathematicians who say "vector" to mean anything can be added commutatively and multiplied by a scalar, and physicists who say "vector" to mean a 3- (or 4-)dimensional quantity that transforms properly under change of coord... |
In <a href="http://terrytao.wordpress.com/2010/01/02/254a-notes-0a-stirlings-formula/">these notes by Terence Tao</a> is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity
$$n! = \int_{0}^{\infty} t^n e^{-t} dt$$
coming from the Gamma function. I have a... |
Wikipedia has separate pages for symmetric group and permutation group, but I don't understand what the difference between them is. A symmetric group on a set is the set of all bijections from the set to itself with composition of functions as the group action. Permutation group on a set is the set of all permutations ... |
Binomial coefficients: how to prove an inequality on the $p$-adic valuation? |
In my attempts to understand just how computer algebra systems "do things", I tried to dig around a bit on Gröbner bases, which are described almost everywhere as "a generalization of the Euclidean algorithm and Gaussian elimination". I've tried to look for examples of Gröbner bases in action, but have been unable to f... |
Very few (if any) mathematicians have significant insight regarding huge natural numbers (cf. various [ultrafinitism][1] arguments). Perhaps the only exceptions are logicians who work with esoteric ordinal notations. This is one of the few ways one can gain any insight into arbitrarily large numbers - using various co... |
Does this surface minimize maximum distance? |
The example that you propose is a bit too general to serve as an introductory pedagogical example (compare e.g. the famous [Kahan SIGSAM problem 9][1] of determining conditions for an ellipse to lie inside the unit circle). Instead, begin with one of the earliest historical examples: Gauss's proof that every symmetric ... |
**Question**:
Given some natural number, we can of course split it up into various sums of other naturals (e.g. $7 = 6 + 1 = 1 + 4 + 2 = \ldots$)
More precisely, for $n \in \mathbb{N}$, we can a find distribution sequence $s_0,\ldots,s_u \in \mathbb{N}$ with
$$\sum_{i=0}^{u}s_i = n$$
Now how can I find a d... |
Could you provide a proof of Euler's formula: $\exp(i t)=\cos(t)+i\sin(t)$ ?
thanks. |
How to prove Euler's formula: $\exp(i t)=\cos(t)+i\sin(t)$ ? |
**Question**:
Given some natural number, we can of course split it up into various sums of other naturals (e.g. $7 = 6 + 1 = 1 + 4 + 2 = \ldots$)
More precisely, for $n \in \mathbb{N}$, we can a find distribution sequence $s_0,\ldots,s_u \in \mathbb{N}$ with
$$\sum_{i=0}^{u}s_i = n$$
Now how can I find a d... |
I don't know the answer, but in my opinion a lot of people are misinterpreting the question, so perhaps it is worth an answer to try to set this straight.
Here is an analogy: trigonometric series is to Fourier series as power series is to Taylor series. In other words, a trigonometric series is just any series of t... |
**Question**:
Given some natural number, we can of course split it up into various sums of other naturals (e.g. $7 = 6 + 1 = 1 + 4 + 2 = \ldots$)
More precisely, for $n \in \mathbb{N}$, we can a find distribution sequence (or _partition_) $s_0,\ldots,s_u \in \mathbb{N}$ with
$$\sum_{i=0}^{u}s_i = n$$
Now h... |
Let me discuss an analogous situation with regard to convex polyhedra.
Archimedes, generalizing what are often called the Platonic Solids or the convex regular polyhedra (there are five of them), seems to have discovered 13 polyhedra with the property that the pattern of faces around each vertex was the same for eve... |
How do I derive the Beta function using the definition of the beta function as the normalizing constant of the Beta distribution and only common sense random experiments?
I'm pretty sure this is possible, but can't see how.
I can see that
$\sum_{a=0}^n {n \choose a} Beta(a+1, n-a+1) = 1$
because we can imag... |
**HINT** Both $\;\rm e^{ix}\;$ and $\;\rm cos(x) + i \; sin(x) \;$ are solutions of $\;\rm y' = i \; y,\; y(0) = 1, \;$ so they are equal by the **uniqueness theorem**. Alternatively, **bisect** into even and odd parts the power series for $\;\rm e^{ix}\;$ , i.e.
$$\begin{align}
\rm f(x) \; &=\; \rm\frac{f(x)+f(-x... |
I came up with this problem after discussion of taxicab geometry in math class... I thought it was a simple problem, but still pretty neat; however, I am as of yet unsure of whether my answer is correct, or logical.
Let $[X]$ be the area of region $X$, and region $S_n$ be represented by the equation $|x-n|+|y-n|=k-n... |
interesting Taxicab Problem? |
I came up with this problem after discussion of taxicab geometry in math class... I thought it was a simple problem, but still pretty neat; however, I am as of yet unsure of whether my answer is correct, or logical.
Let $[X]$ be the area of region $X$, and region $S_n$ be represented by the equation $|x-n|+|y-n|=k-n... |
> How many **unique** strings can be formed by the letters in "aaamottu"?
As a totally incompetent math student, I simply wrote a program to calculate it. And I ended up with 3360 unique strings:
import itertools
unique = {}
for p in itertools.permutations(list("aaamottu"), 8):
unique[p] = ... |
In making up another problem today I came across something odd. I've been thinking it over and I can't exactly place why it's true, but after running a long Python script to check, I haven't yet found a counter example.
Why is $\sum_{n=1}^{m}{n^m}\equiv 0\mod m$ true for all odd $m \ge 3$? The script showed me that ... |
Why does this sum mod out to 0? |
Related to a [question I asked][1] previously, how many integers $m$ are there such that $\displaystyle\sum_{n=1}^{m}{n^m}\equiv 1 \bmod m$? I have only found three: 2, 6, and 42. Are those the only ones? If so, how can that be proven? It is known (see the previous question I asked) that for odd $m$, the sum is $0 \bmo... |
How many integers $m$ are there such that $\displaystyle\sum_{n=1}^{m}{n^m}\equiv 1 \bmod m$? |
I had a question about an instantaneous z score. The question reads as
P(z=2)
I think that this is 0 as z-score is the area under the curve and could be found by integrating the PDF over the range. As the range is 0 the area also zero. Is that correct? |
I had to look at Abramowitz and Stegun just to check that you were saying what I thought you were saying; anyway, if by $P(z)$ you meant
$\frac1{\sqrt{2\pi}}\int_{-\infty}^z \exp\left(-\frac{u^2}{2}\right)\mathrm{d}u$
then no, $P(2)$ is not zero. The integral
$\frac1{\sqrt{2\pi}}\int_{-\infty}^0 \exp\left(-\fr... |
Since $P(z=2)$ is the same as
$\frac1{\sqrt{2\pi}}\int_2^2 \exp\left(-\frac{u^2}{2}\right)\mathrm{d}u$
then yes, the result is zero; this can be interpreted also as saying
$\frac1{\sqrt{2\pi}}\int_{-\infty}^2 \exp\left(-\frac{u^2}{2}\right)\mathrm{d}u-\frac1{\sqrt{2\pi}}\int_{-\infty}^2 \exp\left(-\frac{u^2}{2... |
A train with infinitely many seats, one for each rational number, stops in countably many villages, one for each positive integer, in increasing order, and then finally arrives at the city. At the first village, two women board the train. At the second village, one woman leaves the train to go visit her cousin, and two... |
Here's the question I've been thinking for 2 days and couldn't find an answer why.
If Y = √X, where both X and Y are real numbers then which of the following is true?
a) X>=0; Y>=0
b) X<=0; Y>=0
c) X<=0; Y<=0
d) X>=0; Y<=0
e) Either a) or b)
Now, the answer is a) but I don't know why. I know a valu... |
A standard example is
$$
f(t)= \sum_{n>1} \frac{\sin(nt)}{\log(n)}$$
The conjugate of $f$ is a Fourier series but $f\not\in L^1(\mathbb{T})$ and hence is no Fourier series.
For further explanation see [Katznelson's book page 85.][1]
(Edit: If $f$ is not in $L^1(\mathbb{T})$ it is hard to define Fourier coeffici... |
Motivated by [this][1] question I would like to know whether there is an exact definition of modern mathematics. In which point in time (century, decade) does one think, when speaking about modern mathematics. Does it refer to the Abstract Algebra?
----------
**Edit:** I got -1. If this is not a correct ques... |
The $\Gamma (x)$ function has just one minimum for $x>0$ . This result uses some
properties of the gamma function:
- $\Gamma ^{\prime \prime }(x)>0$ and $\Gamma (x)>0$ for all $x>0$
- $\Gamma (1)=\Gamma (2)=1$.
Observing the following graph (created in SWP) of $y=\Gamma (x)$ this minimum is near $x=3/2$, but ... |
The *vertex* of a parabola is the point on the axis of symmetry that intersects it ([Wikipedia][1]).
The point $(x,y)=\left( 0,0\right) $ is the vertex of the parabola given by $y=ax^{2}$.
By making the change of coordinates
$X=x+\dfrac{b}{2a},Y=y+\dfrac{b^{2}-4ac}{4a}$,
equation $y=ax^{2}+bx+c$ is transfor... |
In section 4 of the article by Afred van der Poorten's [*A Proof That Euler Missed ...*][1] the following inequality is used:
$\nu_{p}\displaystyle\binom{n}{m}$ $\leq\left\lfloor\dfrac{\ln n}{\ln p}\right\rfloor-\nu_{p}(m)\qquad(\ast)$ (*In the original denoted* $\text{ord}$ $_{p}(\cdot)$ *instead of* $\nu_p(\cdot)... |
(1) The Riemann hypothesis (RH) states that all non-trivial zeros of the zeta function have real part 1/2.
(2) The zeta function is intimately connected with the Gamma function via the functional equation.
The second fact suggests that there is an equivalent form of RH which is expressed solely in terms of the Ga... |
How to express the Riemann hypothesis in terms of the Gamma function? |
**HINT**
$$ \frac{d}{dx} \bigl( f(x) + g(x) \bigr) = ? $$ |
Let's introduce their sum function
$$s(x) := f(x) + g(x)$$
The statement to be proven thus becomes $s(x) = 6$.
We know
$$s(0) = 5 + 1 = 6$$
So now what derivative $s'$ do we need such that $s$ does never change it's constant value? |
Related to a [question I asked][1] previously, how many integers $m$ are there such that $\displaystyle\sum_{n=1}^{m}{n^m}\equiv 1 \bmod m$? I have only found three: 2, 6, and 42. Are those the only ones? If so, how can that be proven? It is known (see the previous question I asked) that for odd $m$, the sum is $0 \bmo... |
So, I know that every abelian (commutative) group $G$ is such that, for any subgroup $H$, the left cosets of $H$ in $G$ are the right cosets. I guess this is true even if $G$ is not abelian, but $H$ is (but I'm not sure). Is this enough to characterise the subgroup as abelian, or are there examples of non-abelian subgr... |
Can a non-abelian subgroup be such that the right cosets equal the left cosets? |
Related to a [question I asked][1] previously, how many integers $m$ are there such that $\displaystyle\sum_{n=1}^{m}{n^m}\equiv 1 \bmod m$? I have only found three: 2, 6, and 42. Are those the only ones? If so, how can that be proven? It is known (see the previous question I asked) that for odd $m$, the sum is $0 \bmo... |
I'm trying to express the following function more simply:
$ \begin{align} f(x) = (\sum_{n=3}^{x} \prod_{k=2}^{\lceil \sqrt n \rceil} \lceil \dfrac {n}{k} \rceil - \lfloor \dfrac{n}{k} \rfloor ) + 1 \end{align} $
I know this is kind of a vague request, but I'm trying to get away from the ceiling and floor function... |
Very few (if any) mathematicians have significant insight regarding huge natural numbers (cf. various [ultrafinitism][1] arguments). Perhaps the only exceptions are logicians who work with esoteric ordinal notations. This is one of the few ways one can gain any insight into arbitrarily large numbers - using various co... |
Related to a [question I asked][1] previously, how many integers $m$ are there such that $\displaystyle\sum_{n=1}^{m}{n^m}\equiv 1 \bmod m$? I have only found three: 2, 6, and 42. Are those the only ones? If so, how can that be proven? It is known (see the previous question I asked) that for odd $m$, the sum is $0 \bmo... |
In a recent (and to me, very beautiful) paper, entitled "Relational EPR", [Smerlak and Rovelli][1] present a way of thinking about EPR which relies upon Rovelli's previously published work on relational quantum mechanics (at see arxiv.org/abs/quant-ph/9609002 ). In relational quantum mechanics, there is no non-local... |
What categorical mathematical structure(s) best describe the space of "localized events" in "relational quantum mechanics"? |
[Goodstein's Theorem](http://en.wikipedia.org/wiki/Goodstein%27s_theorem) is the statement that every Goodstein sequence eventually hits 0. It is known to be independent of Peano Arithemtic (PA), and in fact, was the first such purely number theoretic result. It is provable in ZFC.
One way of phrasing this is that... |
Is there a natural model of Peano Arithmetic where Goodstein's theorem fails? |
It depends in what context you want to define that function (for computer implementations?), but following Qiaochu Yuan's post, you can e.g. use set-based notations to state the primality test more clearly.
$$f(x) = \left| \lbrace x \in \lbrace 3,\ldots, x \rbrace \; |\; \forall k \in \lbrace 2,\ldots,\sqrt n \rbrac... |
It can be seen [here][1] that the only numbers for which $n^{m+1}\equiv n \pmod{m}$ is true are 1, 2, 6, 42, and 1806. Through experimentation, it has been found that $\displaystyle\sum_{n=1}^{m}{n^m}\equiv 1 \bmod m$ is true for those numbers, and (as yet unproven) no others. Why is this true?
If there is a simple ... |
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