id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-13100 | Product of Number of Edges, Edges per Face and Faces of Regular Octahedron | The product of the number of edges, edges per face and faces of a regular octahedron is $288$. | A regular octahedron has:
:$12$ edges
and:
:$8$ faces.
Each face is a triangle, and so has $3$ edges.
Hence:
:$12 \times 8 \times 3 = 288$
{{qed}} | The [[Definition:Integer Multiplication|product]] of the number of [[Definition:Edge of Polyhedron|edges]], [[Definition:Edge of Polyhedron|edges]] per [[Definition:Face of Polyhedron|face]] and [[Definition:Face of Polyhedron|faces]] of a [[Definition:Regular Octahedron|regular octahedron]] is $288$. | A [[Definition:Regular Octahedron|regular octahedron]] has:
:$12$ [[Definition:Edge of Polyhedron|edges]]
and:
:$8$ [[Definition:Face of Polyhedron|faces]].
Each [[Definition:Face of Polyhedron|face]] is a [[Definition:Triangle (Geometry)|triangle]], and so has $3$ [[Definition:Edge of Polyhedron|edges]].
Hence:
:$12... | Product of Number of Edges, Edges per Face and Faces of Regular Octahedron | https://proofwiki.org/wiki/Product_of_Number_of_Edges,_Edges_per_Face_and_Faces_of_Regular_Octahedron | https://proofwiki.org/wiki/Product_of_Number_of_Edges,_Edges_per_Face_and_Faces_of_Regular_Octahedron | [
"Regular Octahedra",
"288"
] | [
"Definition:Multiplication/Integers",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Face",
"Definition:Octahedron/Regular"
] | [
"Definition:Octahedron/Regular",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Face",
"Definition:Triangle (Geometry)",
"Definition:Polyhedron/Edge"
] |
proofwiki-13101 | Ordinals are Well-Ordered/Corollary | Let $A$ be a set of ordinals.
Let $\Epsilon {\restriction_A}$ denote the epsilon restriction on $A$.
Then $A$ is strictly well-ordered by $\Epsilon {\restriction_A}$. | Let $A$ be a set of ordinals.
Let $\Epsilon {\restriction_A}$ denote the epsilon restriction on $A$.
It is to be shown that $\Epsilon {\restriction_A}$ is antireflexive.
{{AimForCont}} there is a $a \in A$ such that $a \in a$.
Then by {{Corollary|Transitive Set is Proper Subset of Ordinal iff Element of Ordinal}}, we h... | Let $A$ be a [[Definition:Set|set]] of [[Definition:Ordinal|ordinals]].
Let $\Epsilon {\restriction_A}$ denote the [[Definition:Epsilon Restriction|epsilon restriction]] on $A$.
Then $A$ is [[Definition:Strict Well-Ordering|strictly well-ordered]] by $\Epsilon {\restriction_A}$. | Let $A$ be a [[Definition:Set|set]] of [[Definition:Ordinal|ordinals]].
Let $\Epsilon {\restriction_A}$ denote the [[Definition:Epsilon Restriction|epsilon restriction]] on $A$.
It is to be shown that $\Epsilon {\restriction_A}$ is [[Definition:Antireflexive Relation|antireflexive]].
{{AimForCont}} there is a $a \i... | Ordinals are Well-Ordered/Corollary | https://proofwiki.org/wiki/Ordinals_are_Well-Ordered/Corollary | https://proofwiki.org/wiki/Ordinals_are_Well-Ordered/Corollary | [
"Ordinals",
"Ordinals are Well-Ordered"
] | [
"Definition:Set",
"Definition:Ordinal",
"Definition:Epsilon Relation/Restriction",
"Definition:Strict Well-Ordering"
] | [
"Definition:Set",
"Definition:Ordinal",
"Definition:Epsilon Relation/Restriction",
"Definition:Antireflexive Relation",
"Definition:Contradiction",
"Definition:Antireflexive Relation",
"Strict Subset Relation is Transitive",
"Definition:Transitive Relation",
"Ordinals are Well-Ordered",
"Definitio... |
proofwiki-13102 | Product of Number of Edges, Edges per Face and Faces of Regular Dodecahedron | The product of the number of edges, edges per face and faces of a regular dodecahedron is $1800$. | A regular dodecahedron has:
:$30$ edges
and:
:$12$ faces.
Each face is a pentagon, and so has $5$ edges.
Hence:
:$30 \times 12 \times 5 = 1800$
{{qed}} | The [[Definition:Integer Multiplication|product]] of the number of [[Definition:Edge of Polyhedron|edges]], [[Definition:Edge of Polyhedron|edges]] per [[Definition:Face of Polyhedron|face]] and [[Definition:Face of Polyhedron|faces]] of a [[Definition:Regular Dodecahedron|regular dodecahedron]] is $1800$. | A [[Definition:Regular Dodecahedron|regular dodecahedron]] has:
:$30$ [[Definition:Edge of Polyhedron|edges]]
and:
:$12$ [[Definition:Face of Polyhedron|faces]].
Each [[Definition:Face of Polyhedron|face]] is a [[Definition:Pentagon|pentagon]], and so has $5$ [[Definition:Edge of Polyhedron|edges]].
Hence:
:$30 \time... | Product of Number of Edges, Edges per Face and Faces of Regular Dodecahedron | https://proofwiki.org/wiki/Product_of_Number_of_Edges,_Edges_per_Face_and_Faces_of_Regular_Dodecahedron | https://proofwiki.org/wiki/Product_of_Number_of_Edges,_Edges_per_Face_and_Faces_of_Regular_Dodecahedron | [
"Regular Dodecahedra",
"1800"
] | [
"Definition:Multiplication/Integers",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Face",
"Definition:Dodecahedron/Regular"
] | [
"Definition:Dodecahedron/Regular",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Face",
"Definition:Pentagon",
"Definition:Polyhedron/Edge"
] |
proofwiki-13103 | Product of Number of Edges, Edges per Face and Faces of Regular Icosahedron | The product of the number of edges, edges per face and faces of a regular icosahedron is $1800$. | A regular icosahedron has:
:$30$ edges
and:
:$20$ faces.
Each face is a triangle, and so has $3$ edges.
Hence:
:$30 \times 20 \times 3 = 1800$
{{qed}} | The [[Definition:Integer Multiplication|product]] of the number of [[Definition:Edge of Polyhedron|edges]], [[Definition:Edge of Polyhedron|edges]] per [[Definition:Face of Polyhedron|face]] and [[Definition:Face of Polyhedron|faces]] of a [[Definition:Regular Icosahedron|regular icosahedron]] is $1800$. | A [[Definition:Regular Icosahedron|regular icosahedron]] has:
:$30$ [[Definition:Edge of Polyhedron|edges]]
and:
:$20$ [[Definition:Face of Polyhedron|faces]].
Each [[Definition:Face of Polyhedron|face]] is a [[Definition:Triangle (Geometry)|triangle]], and so has $3$ [[Definition:Edge of Polyhedron|edges]].
Hence:
:... | Product of Number of Edges, Edges per Face and Faces of Regular Icosahedron | https://proofwiki.org/wiki/Product_of_Number_of_Edges,_Edges_per_Face_and_Faces_of_Regular_Icosahedron | https://proofwiki.org/wiki/Product_of_Number_of_Edges,_Edges_per_Face_and_Faces_of_Regular_Icosahedron | [
"Regular Icosahedra",
"1800"
] | [
"Definition:Multiplication/Integers",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Face",
"Definition:Icosahedron/Regular"
] | [
"Definition:Icosahedron/Regular",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Face",
"Definition:Triangle (Geometry)",
"Definition:Polyhedron/Edge"
] |
proofwiki-13104 | Smallest 5th Power equal to Sum of 5 other 5th Powers | The smallest positive integer whose fifth power can be expressed as the sum of $5$ other distinct positive fifth powers is $72$:
:$72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5$ | {{begin-eqn}}
{{eqn | l = 19^5 + 43^5 + 46^5 + 47^5 + 67^5
| r = 2 \, 476 \, 099
| c =
}}
{{eqn | o =
| ro= +
| r = 147 \, 008 \, 443
| c =
}}
{{eqn | o =
| ro= +
| r = 205 \, 962 \, 976
| c =
}}
{{eqn | o =
| ro= +
| r = 229 \, 345 \, 007
| c =
}... | The smallest [[Definition:Positive Integer|positive integer]] whose [[Definition:Fifth Power|fifth power]] can be expressed as the [[Definition:Integer Addition|sum]] of $5$ other [[Definition:Distinct Elements|distinct]] [[Definition:Positive Integer|positive]] [[Definition:Fifth Power|fifth powers]] is $72$:
:$72^5 ... | {{begin-eqn}}
{{eqn | l = 19^5 + 43^5 + 46^5 + 47^5 + 67^5
| r = 2 \, 476 \, 099
| c =
}}
{{eqn | o =
| ro= +
| r = 147 \, 008 \, 443
| c =
}}
{{eqn | o =
| ro= +
| r = 205 \, 962 \, 976
| c =
}}
{{eqn | o =
| ro= +
| r = 229 \, 345 \, 007
| c =
}... | Smallest 5th Power equal to Sum of 5 other 5th Powers | https://proofwiki.org/wiki/Smallest_5th_Power_equal_to_Sum_of_5_other_5th_Powers | https://proofwiki.org/wiki/Smallest_5th_Power_equal_to_Sum_of_5_other_5th_Powers | [
"Fifth Powers",
"72"
] | [
"Definition:Positive/Integer",
"Definition:Fifth Power",
"Definition:Addition/Integers",
"Definition:Distinct/Plural",
"Definition:Positive/Integer",
"Definition:Fifth Power"
] | [
"Definition:Distinct/Plural",
"Definition:Positive/Integer",
"Definition:Fifth Power"
] |
proofwiki-13105 | Smallest Consecutive Even Nontotients | The smallest pair of consecutive even nontotients is $74$ and $76$. | From the sequence of nontotients:
{{:Definition:Nontotient/Sequence}}
Hence, by inspection, it can be seen that $74$ and $76$ are the smallest such pair.
{{qed}} | The smallest [[Definition:Doubleton|pair]] of consecutive [[Definition:Even Integer|even]] [[Definition:Nontotient|nontotients]] is $74$ and $76$. | From the [[Definition:Nontotient/Sequence|sequence of nontotients]]:
{{:Definition:Nontotient/Sequence}}
Hence, by inspection, it can be seen that $74$ and $76$ are the smallest such [[Definition:Doubleton|pair]].
{{qed}} | Smallest Consecutive Even Nontotients | https://proofwiki.org/wiki/Smallest_Consecutive_Even_Nontotients | https://proofwiki.org/wiki/Smallest_Consecutive_Even_Nontotients | [
"Nontotients"
] | [
"Definition:Doubleton",
"Definition:Even Integer",
"Definition:Nontotient"
] | [
"Definition:Nontotient/Sequence",
"Definition:Doubleton"
] |
proofwiki-13106 | Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers | :$79 = 15 \times 1^4 + 4 \times 2^4$ | We have $1^4 = 1, 2^4 = 16, 3^4 = 81 > 79$.
Hence for each $n < 79$, we can only use $1^4$ and $2^4$ in our sum.
Write $n = 2^4 a + 1^4 b$.
We can use the greedy algorithm to generate these expressions, since replacing $2^4$ with $16 \times 1^4$ increases the number of fourth powers required.
Suppose $n < 64$.
By Divis... | :$79 = 15 \times 1^4 + 4 \times 2^4$ | We have $1^4 = 1, 2^4 = 16, 3^4 = 81 > 79$.
Hence for each $n < 79$, we can only use $1^4$ and $2^4$ in our [[Definition:Integer Addition|sum]].
Write $n = 2^4 a + 1^4 b$.
We can use the [[Definition:Greedy Algorithm|greedy algorithm]] to generate these expressions, since replacing $2^4$ with $16 \times 1^4$ increas... | Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers | https://proofwiki.org/wiki/Smallest_Number_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | https://proofwiki.org/wiki/Smallest_Number_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | [
"Fourth Powers",
"Hilbert-Waring Theorem",
"79"
] | [] | [
"Definition:Addition/Integers",
"Definition:Greedy Algorithm",
"Definition:Fourth Power",
"Division Theorem",
"Definition:Positive/Integer",
"Definition:Fourth Power",
"Definition:Fourth Power",
"Definition:Fourth Power",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition... |
proofwiki-13107 | Reciprocal of 81 | :$\dfrac 1 {81} = 0 \cdotp \dot 01234 \, 567 \dot 9$ | Performing the calculation using long division:
<pre>
0.0123456790...
----------------
81)1.0000000000000
81
--
190
162
---
280
243
---
370
324
---
460
405
---
550
486
---
640
567
... | :$\dfrac 1 {81} = 0 \cdotp \dot 01234 \, 567 \dot 9$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.0123456790...
----------------
81)1.0000000000000
81
--
190
162
---
280
243
---
370
324
---
460
405
---
550
486
--... | Reciprocal of 81 | https://proofwiki.org/wiki/Reciprocal_of_81 | https://proofwiki.org/wiki/Reciprocal_of_81 | [
"81",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division"
] |
proofwiki-13108 | Reciprocal of Square of 1 Less than Number Base | Let $b \in \Z$ be an integer such that $b > 2$.
Let $n = \paren {b - 1}^2$.
The reciprocal of $n$, expressed in base $b$, recurs with period $b - 1$:
:$\dfrac 1 n = \sqbrk {0 \cdotp \dot 012 \ldots c \dot d}_b = \sqbrk {0 \cdotp 012 \ldots cd012 \ldots}_b$
where:
:$c = b - 3$
:$d = b - 1$ | By Basis Representation Theorem, the number $\sqbrk {12 \ldots cd}_b$ can be written as:
{{begin-eqn}}
{{eqn | l = \sqbrk {12 \ldots cd}_b
| r = 1 \times b^{b - 3} + 2 \times b^{b - 4} + \dots + \paren {b - 3} \times b + \paren {b - 1}
}}
{{eqn | r = 1 + \sum_{k \mathop = 0}^{b - 3} \paren {b - 2 - k} b^k
}}
{{eq... | Let $b \in \Z$ be an [[Definition:Integer|integer]] such that $b > 2$.
Let $n = \paren {b - 1}^2$.
The [[Definition:Reciprocal|reciprocal]] of $n$, expressed in [[Definition:Number Base|base $b$]], [[Definition:Recurring Basis Expansion|recurs]] with [[Definition:Period of Recurrence|period]] $b - 1$:
:$\dfrac 1 n ... | By [[Basis Representation Theorem]], the number $\sqbrk {12 \ldots cd}_b$ can be written as:
{{begin-eqn}}
{{eqn | l = \sqbrk {12 \ldots cd}_b
| r = 1 \times b^{b - 3} + 2 \times b^{b - 4} + \dots + \paren {b - 3} \times b + \paren {b - 1}
}}
{{eqn | r = 1 + \sum_{k \mathop = 0}^{b - 3} \paren {b - 2 - k} b^k
}}... | Reciprocal of Square of 1 Less than Number Base | https://proofwiki.org/wiki/Reciprocal_of_Square_of_1_Less_than_Number_Base | https://proofwiki.org/wiki/Reciprocal_of_Square_of_1_Less_than_Number_Base | [
"Reciprocals",
"Reciprocal of Square of 1 Less than Number Base"
] | [
"Definition:Integer",
"Definition:Reciprocal",
"Definition:Number Base",
"Definition:Basis Expansion/Recurrence",
"Definition:Basis Expansion/Recurrence/Period"
] | [
"Basis Representation Theorem",
"Sum of Arithmetic-Geometric Sequence",
"Definition:Basis Expansion/Recurrence",
"Sum of Infinite Geometric Sequence",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-13109 | Square of 1 Less than Number Base | Let $b \in \Z$ be an integer such that $b > 2$.
Let $n = b - 1$.
The square of $n$ is expressed in base $b$ as:
:$n^2 = \left[{c1}\right]_b$
where $c = b - 2$. | {{begin-eqn}}
{{eqn | l = n^2
| r = \left({b - 1}\right)^2
| c =
}}
{{eqn | r = b^2 - 2 b + 1
| c =
}}
{{eqn | r = b \left({b - 2}\right) + 1
| c =
}}
{{end-eqn}}
The result follows by definition of number base.
{{qed}} | Let $b \in \Z$ be an [[Definition:Integer|integer]] such that $b > 2$.
Let $n = b - 1$.
The [[Definition:Square (Algebra)|square]] of $n$ is expressed in [[Definition:Number Base|base $b$]] as:
:$n^2 = \left[{c1}\right]_b$
where $c = b - 2$. | {{begin-eqn}}
{{eqn | l = n^2
| r = \left({b - 1}\right)^2
| c =
}}
{{eqn | r = b^2 - 2 b + 1
| c =
}}
{{eqn | r = b \left({b - 2}\right) + 1
| c =
}}
{{end-eqn}}
The result follows by definition of [[Definition:Number Base|number base]].
{{qed}} | Square of 1 Less than Number Base | https://proofwiki.org/wiki/Square_of_1_Less_than_Number_Base | https://proofwiki.org/wiki/Square_of_1_Less_than_Number_Base | [
"Square Numbers",
"Square of 1 Less than Number Base"
] | [
"Definition:Integer",
"Definition:Square/Function",
"Definition:Number Base"
] | [
"Definition:Number Base"
] |
proofwiki-13110 | Positive Integers whose Square Root equals Sum of Digits | The following positive integers have a square root that equals the sum of their digits:
:$0, 1, 81$
and there are no more. | {{begin-eqn}}
{{eqn | l = \sqrt 0
| r = 0
| c =
}}
{{eqn | l = \sqrt 1
| r = 1
| c =
}}
{{eqn | l = \sqrt {81}
| r = 9
| c =
}}
{{eqn | r = 8 + 1
| c =
}}
{{end-eqn}}
By considering the square roots, we are looking for positive integers with a square for which its sum of di... | The following [[Definition:Positive Integer|positive integers]] have a [[Definition:Square Root|square root]] that equals the [[Definition:Integer Addition|sum]] of their [[Definition:Digit|digits]]:
:$0, 1, 81$
and there are no more. | {{begin-eqn}}
{{eqn | l = \sqrt 0
| r = 0
| c =
}}
{{eqn | l = \sqrt 1
| r = 1
| c =
}}
{{eqn | l = \sqrt {81}
| r = 9
| c =
}}
{{eqn | r = 8 + 1
| c =
}}
{{end-eqn}}
By considering the [[Definition:Square Root|square roots]], we are looking for [[Definition:Positive Inte... | Positive Integers whose Square Root equals Sum of Digits | https://proofwiki.org/wiki/Positive_Integers_whose_Square_Root_equals_Sum_of_Digits | https://proofwiki.org/wiki/Positive_Integers_whose_Square_Root_equals_Sum_of_Digits | [
"Square Roots"
] | [
"Definition:Positive/Integer",
"Definition:Square Root",
"Definition:Addition/Integers",
"Definition:Digit"
] | [
"Definition:Square Root",
"Definition:Positive/Integer",
"Definition:Square Number",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:D... |
proofwiki-13111 | Closed Form for Heptagonal Numbers | The closed-form expression for the $n$th heptagonal number is:
:$H_n = \dfrac {n \paren {5 n - 3} } 2$ | Heptagonal numbers are $k$-gonal numbers where $k = 7$.
From Closed Form for Polygonal Numbers we have that:
:$\map P {k, n} = \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$
Hence:
{{begin-eqn}}
{{eqn | l = H_n
| r = \frac n 2 \paren {\paren {7 - 2} n - 7 + 4}
| c = Closed Form for Polygonal Numbers
}}
{{eqn... | The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Heptagonal Number|heptagonal number]] is:
:$H_n = \dfrac {n \paren {5 n - 3} } 2$ | [[Definition:Heptagonal Number|Heptagonal numbers]] are [[Definition:Polygonal Number|$k$-gonal numbers]] where $k = 7$.
From [[Closed Form for Polygonal Numbers]] we have that:
:$\map P {k, n} = \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$
Hence:
{{begin-eqn}}
{{eqn | l = H_n
| r = \frac n 2 \paren {\paren {... | Closed Form for Heptagonal Numbers | https://proofwiki.org/wiki/Closed_Form_for_Heptagonal_Numbers | https://proofwiki.org/wiki/Closed_Form_for_Heptagonal_Numbers | [
"Heptagonal Numbers",
"Closed Forms"
] | [
"Definition:Closed Form Expression",
"Definition:Heptagonal Number"
] | [
"Definition:Heptagonal Number",
"Definition:Polygonal Number",
"Closed Form for Polygonal Numbers",
"Closed Form for Polygonal Numbers"
] |
proofwiki-13112 | Integers for which Divisor Sum of Phi equals Divisor Sum | The following positive integers have the property that the divisor sum of their Euler $\phi$ value equals their divisor sum:
:$\map {\sigma_1} {\map \phi n} = \map {\sigma_1} n$
:$1, 87, 362, 1257, 1798, 5002, 9374, \ldots$
{{OEIS|A033631}} | {{begin-eqn}}
{{eqn | l = \map {\sigma_1} {\map \phi 1}
| r = \map {\sigma_1} 1
| c = {{EulerPhiLink|1}}
}}
{{eqn | r = 1
| c = {{DSFLink|1}}
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {\map \phi {87} }
| r = \map {\sigma_1} {56}
| c = {{EulerPhiLink|87}}
}}
{{eqn | r = 120
... | The following [[Definition:Positive Integer|positive integers]] have the property that the [[Definition:Divisor Sum Function|divisor sum]] of their [[Definition:Euler Phi Function|Euler $\phi$ value]] equals their [[Definition:Divisor Sum Function|divisor sum]]:
:$\map {\sigma_1} {\map \phi n} = \map {\sigma_1} n$
:$1... | {{begin-eqn}}
{{eqn | l = \map {\sigma_1} {\map \phi 1}
| r = \map {\sigma_1} 1
| c = {{EulerPhiLink|1}}
}}
{{eqn | r = 1
| c = {{DSFLink|1}}
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {\map \phi {87} }
| r = \map {\sigma_1} {56}
| c = {{EulerPhiLink|87}}
}}
{{eqn | r = 120... | Integers for which Divisor Sum of Phi equals Divisor Sum | https://proofwiki.org/wiki/Integers_for_which_Divisor_Sum_of_Phi_equals_Divisor_Sum | https://proofwiki.org/wiki/Integers_for_which_Divisor_Sum_of_Phi_equals_Divisor_Sum | [
"Divisor Sum Function",
"Euler Phi Function",
"Integers for which Divisor Sum of Phi equals Divisor Sum"
] | [
"Definition:Positive/Integer",
"Definition:Divisor Sum Function",
"Definition:Euler Phi Function",
"Definition:Divisor Sum Function"
] | [] |
proofwiki-13113 | Prime Gaps of 8 | The following pairs of consecutive prime numbers are those whose difference is $8$:
:$\tuple {89, 97}, \tuple {359, 367}, \tuple {389, 397}, \tuple {401, 409}, \ldots$
{{OEIS|A031926}} | Demonstrated by listing the prime gaps. | The following [[Definition:Ordered Pair|pairs]] of consecutive [[Definition:Prime Number|prime numbers]] are those whose [[Definition:Integer Multiplication|difference]] is $8$:
:$\tuple {89, 97}, \tuple {359, 367}, \tuple {389, 397}, \tuple {401, 409}, \ldots$
{{OEIS|A031926}} | Demonstrated by listing the [[Definition:Prime Gap|prime gaps]]. | Prime Gaps of 8 | https://proofwiki.org/wiki/Prime_Gaps_of_8 | https://proofwiki.org/wiki/Prime_Gaps_of_8 | [
"Prime Gaps"
] | [
"Definition:Ordered Pair",
"Definition:Prime Number",
"Definition:Multiplication/Integers"
] | [
"Definition:Prime Gap"
] |
proofwiki-13114 | Smallest Cunningham Chain of the First Kind of Length 6 | The smallest Cunningham chain of the first kind of length $6$ is:
:$\tuple {89, 179, 359, 719, 1439, 2879}$ | By definition, a Cunningham chain of the first kind is a sequence of prime numbers $\tuple {p_1, p_2, \ldots, p_n}$ such that:
: $p_{k + 1} = 2 p_k + 1$
: $\dfrac {p_1 - 1} 2$ is not prime
: $2 p_n + 1$ is not prime.
Thus each term except the last is a Sophie Germain prime.
{{:Definition:Sophie Germain Prime/Sequence}}... | The smallest [[Definition:Cunningham Chain of the First Kind|Cunningham chain of the first kind]] of [[Definition:Length of Sequence|length]] $6$ is:
:$\tuple {89, 179, 359, 719, 1439, 2879}$ | By definition, a [[Definition:Cunningham Chain of the First Kind|Cunningham chain of the first kind]] is a [[Definition:Integer Sequence|sequence]] of [[Definition:Prime Number|prime numbers]] $\tuple {p_1, p_2, \ldots, p_n}$ such that:
: $p_{k + 1} = 2 p_k + 1$
: $\dfrac {p_1 - 1} 2$ is not [[Definition:Prime Number|p... | Smallest Cunningham Chain of the First Kind of Length 6 | https://proofwiki.org/wiki/Smallest_Cunningham_Chain_of_the_First_Kind_of_Length_6 | https://proofwiki.org/wiki/Smallest_Cunningham_Chain_of_the_First_Kind_of_Length_6 | [
"Cunningham Chains"
] | [
"Definition:Cunningham Chain/First Kind",
"Definition:Length of Sequence"
] | [
"Definition:Cunningham Chain/First Kind",
"Definition:Integer Sequence",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Term of Sequence",
"Definition:Sophie Germain Prime",
"Definition:Mapping",
"Definition:Sophie Germain Prime",
"Definition:Prime Num... |
proofwiki-13115 | Sequence of Sum of Squares of Digits | For a positive integer $n$, let $\map f n$ be the integer created by adding the squares of digits of $n$.
Let $m \in \Z_{>0}$ be expressed in decimal notation.
Let $\sequence {S_m}_{n \mathop \in \Z_{>0} }$ be the sequence defined as follows:
:$n_k = \begin{cases} m & : n = 1 \\
\map f {n_{k - 1} } & : n > 1 \end{cases... | First note that:
:$1^2 + 9^2 + 9^2 = 163$
:$9^2 + 9^2 + 9^2 = 243$
and it can be seen that for a positive integer $m$ larger than $199$, $\map f m < m$.
Thus it is necessary to investigate numbers only up as far as that.
Starting from the bottom, we have that:
:$\map f 1 = 1^2 = 1$
and so $\sequence {S_1} = 1, 1, 1, \l... | For a [[Definition:Positive Integer|positive integer]] $n$, let $\map f n$ be the [[Definition:Integer|integer]] created by [[Definition:Integer Addition|adding]] the [[Definition:Square (Algebra)|squares]] of [[Definition:Digit|digits]] of $n$.
Let $m \in \Z_{>0}$ be expressed in [[Definition:Decimal Notation|decimal... | First note that:
:$1^2 + 9^2 + 9^2 = 163$
:$9^2 + 9^2 + 9^2 = 243$
and it can be seen that for a [[Definition:Positive Integer|positive integer]] $m$ larger than $199$, $\map f m < m$.
Thus it is necessary to investigate numbers only up as far as that.
Starting from the bottom, we have that:
:$\map f 1 = 1^2 = 1$
... | Sequence of Sum of Squares of Digits | https://proofwiki.org/wiki/Sequence_of_Sum_of_Squares_of_Digits | https://proofwiki.org/wiki/Sequence_of_Sum_of_Squares_of_Digits | [
"Sums of Squares",
"Recreational Mathematics"
] | [
"Definition:Positive/Integer",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Square/Function",
"Definition:Digit",
"Definition:Decimal Notation",
"Definition:Integer Sequence"
] | [
"Definition:Positive/Integer",
"Definition:Integer Sequence",
"Definition:Set",
"Definition:Integer Sequence",
"Definition:Set",
"Definition:Positive/Integer",
"Definition:Element",
"Definition:Positive/Integer",
"Definition:Zero Digit",
"Definition:Digit"
] |
proofwiki-13116 | Necessary and Sufficient Condition for First Order System to be Mutually Consistent | Let $\mathbf y$, $\boldsymbol \psi$ be $n$-dimensional vectors.
Let $g$ be a twice differentiable mapping.
Let
:$(1): \quad \map {\boldsymbol \psi} {x, \mathbf y} = \map {\mathbf y'} {x, \mathbf y}$
:$(2): \quad \mathbf p \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } = \map {g_{\mathbf y} } {x, \mathbf... | === Necessary condition ===
Take the partial derivative of $(3)$ {{WRT|Differentiation}} $x$:
:$(4): \quad \frac {\partial^2 \map g {x, \mathbf y} } {\partial \mathbf y \partial x} = -\dfrac \partial {\partial \mathbf y} \map H {x, \mathbf y, \dfrac {\partial g} {\partial \mathbf y} }$
By the Schwarz-Clairaut Theorem, ... | Let $\mathbf y$, $\boldsymbol \psi$ be $n$-[[Definition:Dimension|dimensional]] [[Definition:Vector|vectors]].
Let $g$ be a [[Definition:Second Derivative|twice]] [[Definition:Differentiable Mapping|differentiable mapping]].
Let
:$(1): \quad \map {\boldsymbol \psi} {x, \mathbf y} = \map {\mathbf y'} {x, \mathbf y}$
... | === Necessary condition ===
Take the [[Definition:Partial Derivative|partial derivative]] of $(3)$ {{WRT|Differentiation}} $x$:
:$(4): \quad \frac {\partial^2 \map g {x, \mathbf y} } {\partial \mathbf y \partial x} = -\dfrac \partial {\partial \mathbf y} \map H {x, \mathbf y, \dfrac {\partial g} {\partial \mathbf y} ... | Necessary and Sufficient Condition for First Order System to be Mutually Consistent | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_First_Order_System_to_be_Mutually_Consistent | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_First_Order_System_to_be_Mutually_Consistent | [
"Calculus of Variations"
] | [
"Definition:Dimension",
"Definition:Vector",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Differentiable Mapping",
"Definition:Canonical Variable",
"Definition:Boundary Condition",
"Definition:Mutually Consistent Boundary Conditions",
"Definition:Mapping",
"Definition:Ha... | [
"Definition:Partial Derivative",
"Schwarz-Clairaut Theorem",
"Definition:Partial Derivative",
"Euler's Equation for Vanishing Variation in Canonical Variables"
] |
proofwiki-13117 | 2-Digit Numbers forming Longest Reverse-and-Add Sequence | Let $m \in \Z_{>0}$ be a positive integer expressed in decimal notation.
Let $\map r m$ be the reverse-and-add process on $m$.
Let $r$ be applied iteratively to $m$.
The $2$-digit integers $m$ which need the largest number of iterations before reaching a palindromic number are $89$ and $98$, both needing $24$ iteration... | The sequence obtained by iterating $r$ on $89$ is:
:$89, 187, 968, 1837, 9218, 17347, 91718, 173437, 907808, 1716517, 8872688,$
:$17735476, 85189247, 159487405, 664272356, 1317544822, 3602001953, 7193004016, 13297007933,$
:$47267087164, 93445163438, 176881317877, 955594506548, 170120002107, 8713200023178$
{{OEIS|A03367... | Let $m \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]] expressed in [[Definition:Decimal Notation|decimal notation]].
Let $\map r m$ be the [[Definition:Reverse-and-Add|reverse-and-add process]] on $m$.
Let $r$ be applied iteratively to $m$.
The $2$-[[Definition:Digit|digit]] [[Definition:Positiv... | The [[Definition:Integer Sequence|sequence]] obtained by iterating $r$ on $89$ is:
:$89, 187, 968, 1837, 9218, 17347, 91718, 173437, 907808, 1716517, 8872688,$
:$17735476, 85189247, 159487405, 664272356, 1317544822, 3602001953, 7193004016, 13297007933,$
:$47267087164, 93445163438, 176881317877, 955594506548, 1701200021... | 2-Digit Numbers forming Longest Reverse-and-Add Sequence | https://proofwiki.org/wiki/2-Digit_Numbers_forming_Longest_Reverse-and-Add_Sequence | https://proofwiki.org/wiki/2-Digit_Numbers_forming_Longest_Reverse-and-Add_Sequence | [
"Reverse-and-Add",
"89",
"98"
] | [
"Definition:Positive/Integer",
"Definition:Decimal Notation",
"Definition:Reverse-and-Add",
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Palindromic Number"
] | [
"Definition:Integer Sequence",
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Integer Sequence",
"Definition:Equivalence Relation",
"Definition:Positive/Integer",
"Definition:Positive/Integer",
"Definition:Equivalence Class",
"Definition:Positive/Integer",
"Definition:Po... |
proofwiki-13118 | Reciprocal of 89 | :$\dfrac 1 {89} = 0 \cdotp \dot 01123 \, 59550 \, 56179 \, 77528 \, 08988 \, 76404 \, 49438 \, 20224 \, 719 \dot 1$ | Performing the calculation using long division:
<pre>
0.0112359550561797752808988764044943820224719101...
------------------------------------------------------
89)1.0000000000000000000000000000000000000000000000000
89 534 712 801 89
-- --- --- --- ---
... | :$\dfrac 1 {89} = 0 \cdotp \dot 01123 \, 59550 \, 56179 \, 77528 \, 08988 \, 76404 \, 49438 \, 20224 \, 719 \dot 1$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.0112359550561797752808988764044943820224719101...
------------------------------------------------------
89)1.0000000000000000000000000000000000000000000000000
89 534 712 801 89
-- --- ... | Reciprocal of 89 | https://proofwiki.org/wiki/Reciprocal_of_89 | https://proofwiki.org/wiki/Reciprocal_of_89 | [
"89",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division"
] |
proofwiki-13119 | Cauchy Sequence is Bounded/Real Numbers | Every Cauchy sequence in $\R$ is bounded. | Let $\sequence {a_n}$ be a Cauchy sequence in $\R$.
Then there exists $N \in \N$ such that:
:$\size {a_m - a_n} < 1$
for all $m, n \ge N$.
So for all $m \ge N$, we have:
{{begin-eqn}}
{{eqn | l = \size {a_m}
| r = \size {a_N + a_m - a_N}
}}
{{eqn | o = \le
| r = \size {a_N} + \size {a_m - a_N}
| c = ... | Every [[Definition:Real Cauchy Sequence|Cauchy sequence in $\R$]] is [[Definition:Bounded Real Sequence|bounded]]. | Let $\sequence {a_n}$ be a [[Definition:Real Cauchy Sequence|Cauchy sequence in $\R$]].
Then there exists $N \in \N$ such that:
:$\size {a_m - a_n} < 1$
for all $m, n \ge N$.
So for all $m \ge N$, we have:
{{begin-eqn}}
{{eqn | l = \size {a_m}
| r = \size {a_N + a_m - a_N}
}}
{{eqn | o = \le
| r = \siz... | Cauchy Sequence is Bounded/Real Numbers/Proof 1 | https://proofwiki.org/wiki/Cauchy_Sequence_is_Bounded/Real_Numbers | https://proofwiki.org/wiki/Cauchy_Sequence_is_Bounded/Real_Numbers/Proof_1 | [
"Real Analysis",
"Cauchy Sequences",
"Cauchy Sequence is Bounded"
] | [
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Bounded Sequence/Real"
] | [
"Definition:Cauchy Sequence/Real Numbers",
"Triangle Inequality",
"Definition:Bounded Sequence/Real"
] |
proofwiki-13120 | Cauchy Sequence is Bounded/Real Numbers | Every Cauchy sequence in $\R$ is bounded. | Let $\sequence {a_n}$ be a Cauchy sequence in $\R$.
Then there exists $N \in \N$ such that:
:$\size {a_m - a_n} < 1$
for all $m, n \ge N$.
Note that for $m \le N$:
{{begin-eqn}}
{{eqn | l = \size {a_m}
| o = \le
| r = \max \set {\size {a_1}, \size {a_2}, \dotsc, \size {a_N} }
}}
{{eqn |o = <
| r = \m... | Every [[Definition:Real Cauchy Sequence|Cauchy sequence in $\R$]] is [[Definition:Bounded Real Sequence|bounded]]. | Let $\sequence {a_n}$ be a [[Definition:Real Cauchy Sequence|Cauchy sequence in $\R$]].
Then there exists $N \in \N$ such that:
:$\size {a_m - a_n} < 1$
for all $m, n \ge N$.
Note that for $m \le N$:
{{begin-eqn}}
{{eqn | l = \size {a_m}
| o = \le
| r = \max \set {\size {a_1}, \size {a_2}, \dotsc, \size ... | Cauchy Sequence is Bounded/Real Numbers/Proof 2 | https://proofwiki.org/wiki/Cauchy_Sequence_is_Bounded/Real_Numbers | https://proofwiki.org/wiki/Cauchy_Sequence_is_Bounded/Real_Numbers/Proof_2 | [
"Real Analysis",
"Cauchy Sequences",
"Cauchy Sequence is Bounded"
] | [
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Bounded Sequence/Real"
] | [
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Bounded Sequence/Real"
] |
proofwiki-13121 | Reciprocal of 89 as Sum of Fibonacci Numbers by Negative Powers of 10 | :$\ds \sum_{k \mathop \ge 0} \dfrac {F_k} {10^{k + 1} } = \dfrac 1 {89}$
where $F_k$ is the $k$th Fibonacci number:
:$F_0 = 0, F_1 = 1, F_k = F_{k - 1} + F_{k - 2}$
That is:
<pre>
1 / 89 = 0.0
+ 0.01
+ 0.001
+ 0.0002
+ 0.00003
+ 0.000005
+ 0.0000008
+ 0.00000013
+... | First we note that from Reciprocal of $89$:
:$\dfrac 1 {89} = 0 \cdotp \dot 01123 \, 59550 \, 56179 \, 77528 \, 08988 \, 76404 \, 49438 \, 20224 \, 719 \dot 1$
We have that:
:$89 = 10^2 - 10 - 1$
So:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 0} \dfrac {F_k} {10^{k + 1} }
| r = \dfrac 1 {10} \sum_{k \mathop \g... | :$\ds \sum_{k \mathop \ge 0} \dfrac {F_k} {10^{k + 1} } = \dfrac 1 {89}$
where $F_k$ is the $k$th [[Definition:Fibonacci Number|Fibonacci number]]:
:$F_0 = 0, F_1 = 1, F_k = F_{k - 1} + F_{k - 2}$
That is:
<pre>
1 / 89 = 0.0
+ 0.01
+ 0.001
+ 0.0002
+ 0.00003
+ 0.000005
+ 0.0... | First we note that from [[Reciprocal of 89|Reciprocal of $89$]]:
:$\dfrac 1 {89} = 0 \cdotp \dot 01123 \, 59550 \, 56179 \, 77528 \, 08988 \, 76404 \, 49438 \, 20224 \, 719 \dot 1$
We have that:
:$89 = 10^2 - 10 - 1$
So:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 0} \dfrac {F_k} {10^{k + 1} }
| r = \dfrac ... | Reciprocal of 89 as Sum of Fibonacci Numbers by Negative Powers of 10 | https://proofwiki.org/wiki/Reciprocal_of_89_as_Sum_of_Fibonacci_Numbers_by_Negative_Powers_of_10 | https://proofwiki.org/wiki/Reciprocal_of_89_as_Sum_of_Fibonacci_Numbers_by_Negative_Powers_of_10 | [
"89",
"Reciprocals",
"Fibonacci Numbers"
] | [
"Definition:Fibonacci Number"
] | [
"Reciprocal of 89",
"Generating Function for Fibonacci Numbers"
] |
proofwiki-13122 | Fermat Pseudoprime/Base 3/Examples/91 | The smallest Fermat pseudoprime to base $3$ is $91$:
:$3^{91} \equiv 3 \pmod {91}$
despite the fact that $91$ is not prime:
:$91 = 7 \times 13$ | We have that:
{{begin-eqn}}
{{eqn | l = 3^{91}
| r = 26 \, 183 \, 890 \, 704 \, 263 \, 137 \, 277 \, 674 \, 192 \, 438 \, 430 \, 182 \, 020 \, 124 \, 347
| c =
}}
{{eqn | r = 26 \, 183 \, 890 \, 704 \, 263 \, 137 \, 277 \, 674 \, 192 \, 438 \, 430 \, 182 \, 020 \, 124 \, 344 + 3
| c =
}}
{{eqn | r =... | The smallest [[Definition:Fermat Pseudoprime to Base 3|Fermat pseudoprime to base $3$]] is $91$:
:$3^{91} \equiv 3 \pmod {91}$
despite the fact that $91$ is not [[Definition:Prime Number|prime]]:
:$91 = 7 \times 13$ | We have that:
{{begin-eqn}}
{{eqn | l = 3^{91}
| r = 26 \, 183 \, 890 \, 704 \, 263 \, 137 \, 277 \, 674 \, 192 \, 438 \, 430 \, 182 \, 020 \, 124 \, 347
| c =
}}
{{eqn | r = 26 \, 183 \, 890 \, 704 \, 263 \, 137 \, 277 \, 674 \, 192 \, 438 \, 430 \, 182 \, 020 \, 124 \, 344 + 3
| c =
}}
{{eqn | r ... | Fermat Pseudoprime/Base 3/Examples/91 | https://proofwiki.org/wiki/Fermat_Pseudoprime/Base_3/Examples/91 | https://proofwiki.org/wiki/Fermat_Pseudoprime/Base_3/Examples/91 | [
"Fermat Pseudoprimes",
"91"
] | [
"Definition:Fermat Pseudoprime/Base 3",
"Definition:Prime Number"
] | [
"Definition:Fermat Pseudoprime/Base 3",
"Definition:Composite Number",
"Definition:Fermat Pseudoprime/Base 3",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Square Modulo 4",
"Square Modulo 5",
"Definition:Prime Number",
"Fermat's Little Theorem",
"Definition:Pri... |
proofwiki-13123 | 91 is Pseudoprime to 35 Bases less than 91 | $91$ is a Fermat pseudoprime in $35$ bases less than itself:
:$3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 27, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 64, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88, 90$ | By definition of a Fermat pseudoprime, we need to check for $a < 91$:
:$a^{90} \equiv 1 \pmod {91}$
is satisfied or not.
By Chinese Remainder Theorem, this is equivalent to checking whether:
:$a^{90} \equiv 1 \pmod 7$
and:
:$a^{90} \equiv 1 \pmod {13}$
are both satisfied.
If $a$ is a multiple of $7$ or $13$, $a^{90} \n... | $91$ is a [[Definition:Fermat Pseudoprime|Fermat pseudoprime]] in $35$ bases less than itself:
:$3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 27, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 64, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88, 90$ | By definition of a [[Definition:Fermat Pseudoprime|Fermat pseudoprime]], we need to check for $a < 91$:
:$a^{90} \equiv 1 \pmod {91}$
is satisfied or not.
By [[Chinese Remainder Theorem]], this is equivalent to checking whether:
:$a^{90} \equiv 1 \pmod 7$
and:
:$a^{90} \equiv 1 \pmod {13}$
are both satisfied.
If ... | 91 is Pseudoprime to 35 Bases less than 91 | https://proofwiki.org/wiki/91_is_Pseudoprime_to_35_Bases_less_than_91 | https://proofwiki.org/wiki/91_is_Pseudoprime_to_35_Bases_less_than_91 | [
"Fermat Pseudoprimes",
"91"
] | [
"Definition:Fermat Pseudoprime"
] | [
"Definition:Fermat Pseudoprime",
"Chinese Remainder Theorem",
"Fermat's Little Theorem",
"Fermat's Little Theorem"
] |
proofwiki-13124 | Even Integers not Sum of 2 Twin Primes | The following even integers cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes:
:$2, 4, 94, 96, 98, 400, 402, 404, \ldots$
{{OEIS|A007534}}
{{expand|add a page to the effect that it is conjectured that this list is complete.}} | {{ProofWanted|Can be demonstrated by brute force. A pseudocode program and a printout may be adequate, but first we need to develop a standard language-agnostic {{ProofWiki}} pseudocode.}} | The following [[Definition:Even Integer|even integers]] cannot be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Prime Number|prime numbers]] which are each one of a [[Definition:Twin Primes|pair of twin primes]]:
:$2, 4, 94, 96, 98, 400, 402, 404, \ldots$
{{OEIS|A007534}}
{{expand|add a page... | {{ProofWanted|Can be demonstrated by brute force. A pseudocode program and a printout may be adequate, but first we need to develop a standard language-agnostic {{ProofWiki}} pseudocode.}} | Even Integers not Sum of 2 Twin Primes | https://proofwiki.org/wiki/Even_Integers_not_Sum_of_2_Twin_Primes | https://proofwiki.org/wiki/Even_Integers_not_Sum_of_2_Twin_Primes | [
"Twin Primes",
"94"
] | [
"Definition:Even Integer",
"Definition:Addition/Integers",
"Definition:Prime Number",
"Definition:Twin Primes"
] | [] |
proofwiki-13125 | Reciprocal of 97 | :$\dfrac 1 {97} = 0 \cdotp \dot 01030 \, 92783 \, 50515 \, 46391 \, 75257 \, 73195 \, 87628 \, 86597 \, 93814 \, 43298 \, 96907 \, 21649 \, 48453 \, 60824 \, 74226 \, 80412 \, 37113 \, 40206 \, 18556 \, \dot 7$ | Performing the calculation using long division:
<pre>
0.01030927835051546391752577319587628865979381443298969072164948453608247422680412371134020618556701...
------------------------------------------------------------------------------------------------------
97)1.0000000000000000000000000000000000000000000000000... | :$\dfrac 1 {97} = 0 \cdotp \dot 01030 \, 92783 \, 50515 \, 46391 \, 75257 \, 73195 \, 87628 \, 86597 \, 93814 \, 43298 \, 96907 \, 21649 \, 48453 \, 60824 \, 74226 \, 80412 \, 37113 \, 40206 \, 18556 \, \dot 7$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.01030927835051546391752577319587628865979381443298969072164948453608247422680412371134020618556701...
------------------------------------------------------------------------------------------------------
97)1.0000000000000000000... | Reciprocal of 97 | https://proofwiki.org/wiki/Reciprocal_of_97 | https://proofwiki.org/wiki/Reciprocal_of_97 | [
"97",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division"
] |
proofwiki-13126 | Central Field is Field of Functional | Let $\mathbf y$ be an $N$-dimensional vector.
Let $J$ be a functional, such that:
:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
Let the following be a central field:
:$\map {\mathbf y'} x = \map {\boldsymbol \psi} {x, \mathbf y}$
Then this central field is a field of functional $J$. | Suppose:
:$\ds \map g {x, \mathbf y} = \int_c^{\paren {x, \mathbf y} } \map F {x, \hat {\mathbf y}, \hat {\mathbf y}'} \rd x$
where $\hat {\mathbf y}$ is an extremal of $J$ connecting points $c$ and $\tuple {x, \mathbf y}$.
{{explain|Extremum or extremal? Can this be made consistent?}}
Define a field of directions in $... | Let $\mathbf y$ be an [[Definition:Dimension|$N$-dimensional]] [[Definition:Vector|vector]].
Let $J$ be a [[Definition:Real Functional|functional]], such that:
:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
Let the following be a [[Definition:Central Field|central field]]:
:$\map {\m... | Suppose:
:$\ds \map g {x, \mathbf y} = \int_c^{\paren {x, \mathbf y} } \map F {x, \hat {\mathbf y}, \hat {\mathbf y}'} \rd x$
where $\hat {\mathbf y}$ is an [[Definition:Extremum of Functional|extremal]] of $J$ connecting [[Definition:Point|points]] $c$ and $\tuple {x, \mathbf y}$.
{{explain|Extremum or extremal? Ca... | Central Field is Field of Functional | https://proofwiki.org/wiki/Central_Field_is_Field_of_Functional | https://proofwiki.org/wiki/Central_Field_is_Field_of_Functional | [
"Calculus of Variations"
] | [
"Definition:Dimension",
"Definition:Vector",
"Definition:Functional/Real",
"Definition:Central Field",
"Definition:Central Field",
"Definition:Field of Directions/Functional"
] | [
"Definition:Extremum/Functional",
"Definition:Point",
"Definition:Field of Directions/Functional",
"Definition:Canonical Variable",
"Definition:Geodetic Distance",
"Definition:Slope",
"Definition:Line/Curve",
"Definition:Point",
"Definition:Geodetic Distance",
"Definition:Hamilton-Jacobi Equation"... |
proofwiki-13127 | Reciprocal of 98 | :$\dfrac 1 {98} = 0 \cdotp 0 \dot 1020 \, 40816 \, 32653 \, 06122 \, 44897 \, 95918 \, 36734 \, 69387 \, 75 \dot 5$ | Performing the calculation using long division:
<pre>
0.01020408163265306122448979591836734693877551...
--------------------------------------------------
98)1.00000000000000000000000000000000000000000000000
98 196 196 490 294 686
-- --- --- --- --- ---
200 ... | :$\dfrac 1 {98} = 0 \cdotp 0 \dot 1020 \, 40816 \, 32653 \, 06122 \, 44897 \, 95918 \, 36734 \, 69387 \, 75 \dot 5$ | Performing the calculation using long division:
<pre>
0.01020408163265306122448979591836734693877551...
--------------------------------------------------
98)1.00000000000000000000000000000000000000000000000
98 196 196 490 294 686
-- --- --- --- --- ---
200 ... | Reciprocal of 98 | https://proofwiki.org/wiki/Reciprocal_of_98 | https://proofwiki.org/wiki/Reciprocal_of_98 | [
"98",
"Examples of Reciprocals"
] | [] | [
"Sum of Infinite Geometric Sequence/Corollary 1"
] |
proofwiki-13128 | Even Integers not Expressible as Sum of 3, 5 or 7 with Prime | The even integers that cannot be expressed as the sum of $2$ prime numbers where one of those primes is $3$, $5$ or $7$ begins:
:$98, 122, 124, 126, 128, 148, 150, \ldots$
{{OEIS|A283555}} | These are the primes which coincide with the upper end of a prime gap greater than $6$.
These can be found at:
:$89$ to $97$: prime gap of $8$
:$113$ to $127$: prime gap of $14$
:$139$ to $149$: prime gap of $10$
and so on.
We have that:
{{begin-eqn}}
{{eqn | l = 98
| r = 19 + 79
}}
{{eqn | l = 122
| r = 13... | The [[Definition:Even Integer|even integers]] that cannot be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Prime Number|prime numbers]] where one of those [[Definition:Prime Number|primes]] is $3$, $5$ or $7$ begins:
:$98, 122, 124, 126, 128, 148, 150, \ldots$
{{OEIS|A283555}} | These are the [[Definition:Prime Number|primes]] which coincide with the upper end of a [[Definition:Prime Gap|prime gap]] greater than $6$.
These can be found at:
:$89$ to $97$: [[Definition:Prime Gap|prime gap]] of $8$
:$113$ to $127$: [[Definition:Prime Gap|prime gap]] of $14$
:$139$ to $149$: [[Definition:Prime Ga... | Even Integers not Expressible as Sum of 3, 5 or 7 with Prime | https://proofwiki.org/wiki/Even_Integers_not_Expressible_as_Sum_of_3,_5_or_7_with_Prime | https://proofwiki.org/wiki/Even_Integers_not_Expressible_as_Sum_of_3,_5_or_7_with_Prime | [
"98",
"Prime Numbers"
] | [
"Definition:Even Integer",
"Definition:Addition/Integers",
"Definition:Prime Number",
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime Gap",
"Definition:Prime Gap",
"Definition:Prime Gap",
"Definition:Prime Gap"
] |
proofwiki-13129 | Reciprocal of 99 | :$\dfrac 1 {99} = 0 \cdotp \dot 0 \dot 1$ | Performing the calculation using long division:
<pre>
0.0101...
----------
99)1.0000000
99
--
100
99
---
...
</pre>
{{qed}} | :$\dfrac 1 {99} = 0 \cdotp \dot 0 \dot 1$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.0101...
----------
99)1.0000000
99
--
100
99
---
...
</pre>
{{qed}} | Reciprocal of 99 | https://proofwiki.org/wiki/Reciprocal_of_99 | https://proofwiki.org/wiki/Reciprocal_of_99 | [
"99",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division"
] |
proofwiki-13130 | Repdigit Number consisting of Instances of 9 is Kaprekar | A repdigit number that consists entirely of the digit $9$ is a Kaprekar number. | We note as examples:
{{begin-eqn}}
{{eqn | l = 9^2
| r = 81
}}
{{eqn | l = 8 + 1
| r = 9
}}
{{eqn | l = 99^2
| r = 9801
}}
{{eqn | l = 98 + 01
| r = 99
}}
{{eqn | l = 999^2
| r = 998001
}}
{{eqn | l = 998 + 001
| r = 999
}}
{{end-eqn}}
Now we consider:
{{begin-eqn}}
{{eqn | l = \pare... | A [[Definition:Repdigit Number|repdigit number]] that consists entirely of the [[Definition:Digit|digit]] $9$ is a [[Definition:Kaprekar Number|Kaprekar number]]. | We note as examples:
{{begin-eqn}}
{{eqn | l = 9^2
| r = 81
}}
{{eqn | l = 8 + 1
| r = 9
}}
{{eqn | l = 99^2
| r = 9801
}}
{{eqn | l = 98 + 01
| r = 99
}}
{{eqn | l = 999^2
| r = 998001
}}
{{eqn | l = 998 + 001
| r = 999
}}
{{end-eqn}}
Now we consider:
{{begin-eqn}}
{{eqn | l = \... | Repdigit Number consisting of Instances of 9 is Kaprekar | https://proofwiki.org/wiki/Repdigit_Number_consisting_of_Instances_of_9_is_Kaprekar | https://proofwiki.org/wiki/Repdigit_Number_consisting_of_Instances_of_9_is_Kaprekar | [
"Kaprekar Numbers",
"Repdigit Numbers"
] | [
"Definition:Repdigit Number",
"Definition:Digit",
"Definition:Kaprekar Number"
] | [
"Sum of Geometric Sequence",
"Definition:Kaprekar Number"
] |
proofwiki-13131 | Numbers equal to Sum of Primes not Greater than its Prime Counting Function Value | Let $\map \pi n: \Z_{\ge 0} \to \Z_{\ge 0}$ denote the prime-counting function:
:$\map \pi n =$ the count of the primes less than $n$
Consider the equation:
:$\ds n = \sum_{p \mathop \le \map \pi n} p$
where $p \le \pi \left({n}\right)$ denotes the primes not greater than $\pi \left({n}\right)$.
Then $n$ is one of:
:$5... | We have that:
{{begin-eqn}}
{{eqn | l = \map \pi 5
| r = 3
| c =
}}
{{eqn | ll= \leadsto
| l = 2 + 3
| r = 5
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map \pi {17}
| r = 7
| c =
}}
{{eqn | ll= \leadsto
| l = 2 + 3 + 5 + 7
| r = 17
| c =
}}
{{end-eqn}... | Let $\map \pi n: \Z_{\ge 0} \to \Z_{\ge 0}$ denote the [[Definition:Prime-Counting Function|prime-counting function]]:
:$\map \pi n =$ the count of the [[Definition:Prime Number|primes]] less than $n$
Consider the equation:
:$\ds n = \sum_{p \mathop \le \map \pi n} p$
where $p \le \pi \left({n}\right)$ denotes the [[... | We have that:
{{begin-eqn}}
{{eqn | l = \map \pi 5
| r = 3
| c =
}}
{{eqn | ll= \leadsto
| l = 2 + 3
| r = 5
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map \pi {17}
| r = 7
| c =
}}
{{eqn | ll= \leadsto
| l = 2 + 3 + 5 + 7
| r = 17
| c =
}}
{{end-e... | Numbers equal to Sum of Primes not Greater than its Prime Counting Function Value | https://proofwiki.org/wiki/Numbers_equal_to_Sum_of_Primes_not_Greater_than_its_Prime_Counting_Function_Value | https://proofwiki.org/wiki/Numbers_equal_to_Sum_of_Primes_not_Greater_than_its_Prime_Counting_Function_Value | [
"Prime Numbers",
"Prime-Counting Function"
] | [
"Definition:Prime-Counting Function",
"Definition:Prime Number",
"Definition:Prime Number"
] | [] |
proofwiki-13132 | Smallest Seventh Power which is Sum of 8 other Seventh Powers | The smallest seventh power that can be expressed as the sum of $8$ other seventh powers is $102^7$:
:$102^7 = 12^7 + 35^7 + 53^7 + 58^7 + 64^7 + 83^7 + 85^7 + 90^7$ | {{ProofWanted|Brute force}} | The smallest [[Definition:Seventh Power|seventh power]] that can be expressed as the [[Definition:Integer Addition|sum]] of $8$ other [[Definition:Seventh Power|seventh powers]] is $102^7$:
:$102^7 = 12^7 + 35^7 + 53^7 + 58^7 + 64^7 + 83^7 + 85^7 + 90^7$ | {{ProofWanted|Brute force}} | Smallest Seventh Power which is Sum of 8 other Seventh Powers | https://proofwiki.org/wiki/Smallest_Seventh_Power_which_is_Sum_of_8_other_Seventh_Powers | https://proofwiki.org/wiki/Smallest_Seventh_Power_which_is_Sum_of_8_other_Seventh_Powers | [
"Seventh Powers",
"102"
] | [
"Definition:Seventh Power",
"Definition:Addition/Integers",
"Definition:Seventh Power"
] | [] |
proofwiki-13133 | 103 is Smallest Prime whose Period of Reciprocal is One Third of Maximal | :$\dfrac 1 {103} = 0 \cdotp \dot 00970 \, 87378 \, 64077 \, 66990 \, 29126 \, 21359 \, 223 \dot 3$ | From Reciprocal of $103$:
{{:Reciprocal of 103}}
and so by counting it can be seen that its period of recurrence is $34$.
By Maximum Period of Reciprocal of Prime, the maximum period of recurrence of the reciprocal of $p$ when expressed in decimal notation is $p - 1$.
Therefore in order for a prime number to have its p... | :$\dfrac 1 {103} = 0 \cdotp \dot 00970 \, 87378 \, 64077 \, 66990 \, 29126 \, 21359 \, 223 \dot 3$ | From [[Reciprocal of 103|Reciprocal of $103$]]:
{{:Reciprocal of 103}}
and so by counting it can be seen that its [[Definition:Period of Recurrence|period of recurrence]] is $34$.
By [[Maximum Period of Reciprocal of Prime]], the maximum [[Definition:Period of Recurrence|period of recurrence]] of the [[Definition:Rec... | 103 is Smallest Prime whose Period of Reciprocal is One Third of Maximal | https://proofwiki.org/wiki/103_is_Smallest_Prime_whose_Period_of_Reciprocal_is_One_Third_of_Maximal | https://proofwiki.org/wiki/103_is_Smallest_Prime_whose_Period_of_Reciprocal_is_One_Third_of_Maximal | [
"103",
"Examples of Reciprocals"
] | [] | [
"Reciprocal of 103",
"Definition:Basis Expansion/Recurrence/Period",
"Maximum Period of Reciprocal of Prime",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Reciprocal",
"Definition:Decimal Notation",
"Definition:Prime Number",
"Definition:Basis Expansion/Recurrence/Period",
"Definition... |
proofwiki-13134 | Odd Integers whose Smaller Odd Coprimes are Prime | Let $n \in \Z_{>0}$ be an odd positive integer such that all smaller odd integers greater than $1$ which are coprime to it are prime.
The complete list of such $n$ is as follows:
:$1, 3, 5, 7, 9, 15, 21, 45, 105$
{{OEIS|A327823}} | First it is demonstrated that $105$ itself satisfies this property.
Let $d \in \Z_{> 1}$ be odd and coprime to $105$.
Then $d$ does not have $3$, $5$ or $7$ as a prime factor.
Thus $d$ must have at least one odd prime as a divisor which is $11$ or greater.
The smallest such composite number is $11^2$.
But $11^2 = 121 >... | Let $n \in \Z_{>0}$ be an [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive integer]] such that all smaller [[Definition:Odd Integer|odd integers]] greater than $1$ which are [[Definition:Coprime Integers|coprime]] to it are [[Definition:Prime Number|prime]].
The complete list of such $n$ is as fol... | First it is demonstrated that $105$ itself satisfies this property.
Let $d \in \Z_{> 1}$ be [[Definition:Odd Integer|odd]] and [[Definition:Coprime Integers|coprime]] to $105$.
Then $d$ does not have $3$, $5$ or $7$ as a [[Definition:Prime Factor|prime factor]].
Thus $d$ must have at least one [[Definition:Odd Prime... | Odd Integers whose Smaller Odd Coprimes are Prime | https://proofwiki.org/wiki/Odd_Integers_whose_Smaller_Odd_Coprimes_are_Prime | https://proofwiki.org/wiki/Odd_Integers_whose_Smaller_Odd_Coprimes_are_Prime | [
"Coprime Integers",
"105"
] | [
"Definition:Odd Integer",
"Definition:Positive/Integer",
"Definition:Odd Integer",
"Definition:Coprime/Integers",
"Definition:Prime Number"
] | [
"Definition:Odd Integer",
"Definition:Coprime/Integers",
"Definition:Prime Factor",
"Definition:Odd Prime",
"Definition:Divisor (Algebra)/Integer",
"Definition:Composite Number",
"Definition:Odd Prime",
"Definition:Odd Integer",
"Definition:Coprime/Integers",
"Definition:Prime Number",
"Definiti... |
proofwiki-13135 | Sequences of 3 Consecutive Integers with Rising Phi | The following ordered triples of consecutive integers have $\phi$ values which are strictly increasing:
:$105, 106, 107$
:$165, 166, 167$
:$315, 316, 317$ | {{begin-eqn}}
{{eqn | l = \map \phi {105}
| r = 48
| c = {{EulerPhiLink|105}}
}}
{{eqn | l = \map \phi {106}
| r = 52
| c = {{EulerPhiLink|106}}
}}
{{eqn | l = \map \phi {107}
| r = 106
| c = Euler Phi Function of Prime: $107$ is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map \p... | The following [[Definition:Ordered Triple|ordered triples]] of consecutive [[Definition:Integer|integers]] have [[Definition:Euler Phi Function|$\phi$]] values which are [[Definition:Strictly Increasing Mapping|strictly increasing]]:
:$105, 106, 107$
:$165, 166, 167$
:$315, 316, 317$ | {{begin-eqn}}
{{eqn | l = \map \phi {105}
| r = 48
| c = {{EulerPhiLink|105}}
}}
{{eqn | l = \map \phi {106}
| r = 52
| c = {{EulerPhiLink|106}}
}}
{{eqn | l = \map \phi {107}
| r = 106
| c = [[Euler Phi Function of Prime]]: $107$ is [[Definition:Prime Number|prime]]
}}
{{end-eqn}}
... | Sequences of 3 Consecutive Integers with Rising Phi | https://proofwiki.org/wiki/Sequences_of_3_Consecutive_Integers_with_Rising_Phi | https://proofwiki.org/wiki/Sequences_of_3_Consecutive_Integers_with_Rising_Phi | [
"Euler Phi Function"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Integer",
"Definition:Euler Phi Function",
"Definition:Strictly Increasing/Mapping"
] | [
"Euler Phi Function of Prime",
"Definition:Prime Number",
"Euler Phi Function of Prime",
"Definition:Prime Number",
"Euler Phi Function of Prime",
"Definition:Prime Number"
] |
proofwiki-13136 | Integers whose Divisor Sum equals Half Phi times Divisor Count | The following positive integers $n$ have the property where:
:$\map {\sigma_1} n = \dfrac {\map \phi n \times \map {\sigma_0} n} 2$
where:
:$\map {\sigma_1} n$ denotes the divisor sum function: the sum of the divisors of $n$
:$\map \phi n$ denotes the Euler $\phi$ function: the count of positive integers smaller than o... | We have:
{{begin-eqn}}
{{eqn | l = \map \phi {35}
| r = 24
| c = {{EulerPhiLink|35}}
}}
{{eqn | l = \map {\sigma_0} {35}
| r = 4
| c = {{DCFLink|35}}
}}
{{eqn | ll= \leadsto
| l = \map \phi {35} \times \map {\sigma_0} {35}
| r = \dfrac {24 \times 4} 2
| c =
}}
{{eqn | r = 48
... | The following [[Definition:Positive Integer|positive integers]] $n$ have the property where:
:$\map {\sigma_1} n = \dfrac {\map \phi n \times \map {\sigma_0} n} 2$
where:
:$\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]: the [[Definition:Integer Addition|sum]] of the [[Definitio... | We have:
{{begin-eqn}}
{{eqn | l = \map \phi {35}
| r = 24
| c = {{EulerPhiLink|35}}
}}
{{eqn | l = \map {\sigma_0} {35}
| r = 4
| c = {{DCFLink|35}}
}}
{{eqn | ll= \leadsto
| l = \map \phi {35} \times \map {\sigma_0} {35}
| r = \dfrac {24 \times 4} 2
| c =
}}
{{eqn | r = 48
... | Integers whose Divisor Sum equals Half Phi times Divisor Count | https://proofwiki.org/wiki/Integers_whose_Divisor_Sum_equals_Half_Phi_times_Divisor_Count | https://proofwiki.org/wiki/Integers_whose_Divisor_Sum_equals_Half_Phi_times_Divisor_Count | [
"Divisor Sum Function",
"Euler Phi Function",
"Divisor Count Function"
] | [
"Definition:Positive/Integer",
"Definition:Divisor Sum Function",
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Euler Phi Function",
"Definition:Positive/Integer",
"Definition:Coprime/Integers",
"Definition:Divisor Count Function",
"Definition:Divisor (Algebra)/... | [] |
proofwiki-13137 | Smallest Polyomino with Hole | The smallest polyomino with a hole is the heptomino in the form of a $3 \times 3$
square with the center $1 \times 1$ square and a corner $1 \times 1$ square missing:
:200px | From 35 Hexominoes and by inspection, none of the $35$ hexominoes has a hole.
From Number of Heptominoes and by inspection, exactly one of the $108$ heptominoes has a hole.
This is the smallest polyomino with a hole.
{{qed}} | The smallest [[Definition:Polyomino|polyomino]] with a hole is the [[Definition:Heptomino|heptomino]] in the form of a $3 \times 3$
[[Definition:Square (Geometry)|square]] with the center $1 \times 1$ [[Definition:Square (Geometry)|square]] and a corner $1 \times 1$ [[Definition:Square (Geometry)|square]] missing:
:... | From [[35 Hexominoes]] and by inspection, none of the $35$ [[Definition:Hexomino|hexominoes]] has a hole.
From [[Number of Heptominoes]] and by inspection, exactly one of the $108$ [[Definition:Heptomino|heptominoes]] has a hole.
This is the smallest [[Definition:Polyomino|polyomino]] with a hole.
{{qed}} | Smallest Polyomino with Hole | https://proofwiki.org/wiki/Smallest_Polyomino_with_Hole | https://proofwiki.org/wiki/Smallest_Polyomino_with_Hole | [
"Polyominoes",
"Heptominoes"
] | [
"Definition:Polyomino",
"Definition:Heptomino",
"Definition:Quadrilateral/Square",
"Definition:Quadrilateral/Square",
"Definition:Quadrilateral/Square",
"File:HeptominoWithHole.png"
] | [
"35 Hexominoes",
"Definition:Hexomino",
"Number of Heptominoes",
"Definition:Heptomino",
"Definition:Polyomino"
] |
proofwiki-13138 | Magic Constant of Smallest Prime Magic Square | The magic constant of the smallest prime magic square is $111$. | The smallest prime magic square (including $1$) is:
{{:Prime Magic Square/Examples/Order 3/Smallest}}
As can be seen by inspection, the sums of the elements in the rows, columns and diagonals are $111$:
{{begin-eqn}}
{{eqn | l = 67 + 1 + 43
| r = 111
}}
{{eqn | l = 13 + 37 + 61
| r = 111
}}
{{eqn | l = 31 +... | The [[Definition:Magic Constant|magic constant]] of the smallest [[Definition:Prime Magic Square|prime magic square]] is $111$. | The [[Prime Magic Square/Examples/Order 3/Smallest|smallest prime magic square (including $1$)]] is:
{{:Prime Magic Square/Examples/Order 3/Smallest}}
As can be seen by inspection, the [[Definition:Integer Addition|sums]] of the [[Definition:Element of Array|elements]] in the [[Definition:Row of Array|rows]], [[Defin... | Magic Constant of Smallest Prime Magic Square | https://proofwiki.org/wiki/Magic_Constant_of_Smallest_Prime_Magic_Square | https://proofwiki.org/wiki/Magic_Constant_of_Smallest_Prime_Magic_Square | [
"Prime Magic Squares",
"111"
] | [
"Definition:Magic Square/Magic Constant",
"Definition:Prime Magic Square"
] | [
"Prime Magic Square/Examples/Order 3/Smallest",
"Definition:Addition/Integers",
"Definition:Array/Element",
"Definition:Array/Row",
"Definition:Array/Column",
"Definition:Array/Diagonal"
] |
proofwiki-13139 | Sequence of Palindromic Lucky Numbers | The The sequence of lucky numbers which are also palindromic begins:
:$1, 3, 7, 9, 33, 99, 111, 141, 151, 171, \ldots$
{{OEIS|A031161}} | {{:Definition:Lucky Number/Sequence}}
Of these, the palindromic ones can be picked out by inspection.
{{qed}} | The The [[Definition:Integer Sequence|sequence]] of [[Definition:Lucky Number|lucky numbers]] which are also [[Definition:Palindromic Number|palindromic]] begins:
:$1, 3, 7, 9, 33, 99, 111, 141, 151, 171, \ldots$
{{OEIS|A031161}} | {{:Definition:Lucky Number/Sequence}}
Of these, the [[Definition:Palindromic Number|palindromic]] ones can be picked out by inspection.
{{qed}} | Sequence of Palindromic Lucky Numbers | https://proofwiki.org/wiki/Sequence_of_Palindromic_Lucky_Numbers | https://proofwiki.org/wiki/Sequence_of_Palindromic_Lucky_Numbers | [
"Palindromic Numbers",
"Lucky Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Lucky Number",
"Definition:Palindromic Number"
] | [
"Definition:Palindromic Number"
] |
proofwiki-13140 | Sequence of Square Lucky Numbers | The sequence of lucky numbers which are also square begins:
:$1, 9, 25, 49, 169, 289, 361, 529, \ldots$
{{OEIS|A031162}} | {{:Definition:Lucky Number/Sequence}}
Of these, the square ones can be picked out by inspection.
{{qed}}
Category:Lucky Numbers
Category:Square Numbers
926u5u5kr0ql3e1xd83oaoyatuxjcbw | The [[Definition:Integer Sequence|sequence]] of [[Definition:Lucky Number|lucky numbers]] which are also [[Definition:Square Number|square]] begins:
:$1, 9, 25, 49, 169, 289, 361, 529, \ldots$
{{OEIS|A031162}} | {{:Definition:Lucky Number/Sequence}}
Of these, the [[Definition:Square Number|square]] ones can be picked out by inspection.
{{qed}}
[[Category:Lucky Numbers]]
[[Category:Square Numbers]]
926u5u5kr0ql3e1xd83oaoyatuxjcbw | Sequence of Square Lucky Numbers | https://proofwiki.org/wiki/Sequence_of_Square_Lucky_Numbers | https://proofwiki.org/wiki/Sequence_of_Square_Lucky_Numbers | [
"Lucky Numbers",
"Square Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Lucky Number",
"Definition:Square Number"
] | [
"Definition:Square Number",
"Category:Lucky Numbers",
"Category:Square Numbers"
] |
proofwiki-13141 | Conditions for Extremal Embedding in Field of Functional | Let $J$ be a functional such that:
:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
Let $\gamma$ be an extremal of $J$, defined by $\mathbf y = \map {\mathbf y} x$ for $x \in \closedint a b$.
Suppose:
:$\forall x \in \closedint a b: \map \det {F_{\mathbf y' \mathbf y'} } \ne 0$
Suppose no ... | Let $c \in \R$ be conjugate to $a$, such that $c < a$.
By assumption:
:$c \notin \closedint a b$
Hence, there exists a set $\closedint c b$ such that:
:$\closedint c b = \closedint c a \cup \closedint a b$
where $\size {c - a} > 0$.
By there exists a real point between two real points:
:$\exists \epsilon: \size {c - a}... | Let $J$ be a [[Definition:Real Functional|functional]] such that:
:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
Let $\gamma$ be an [[Definition:Extremum of Functional|extremal]] of $J$, defined by $\mathbf y = \map {\mathbf y} x$ for $x \in \closedint a b$.
Suppose:
:$\forall x \in ... | Let $c \in \R$ be [[Definition:Conjugate Point (Calculus of Variations)|conjugate]] to $a$, such that $c < a$.
By assumption:
:$c \notin \closedint a b$
Hence, there exists a [[Definition:Set|set]] $\closedint c b$ such that:
:$\closedint c b = \closedint c a \cup \closedint a b$
where $\size {c - a} > 0$.
By [[t... | Conditions for Extremal Embedding in Field of Functional | https://proofwiki.org/wiki/Conditions_for_Extremal_Embedding_in_Field_of_Functional | https://proofwiki.org/wiki/Conditions_for_Extremal_Embedding_in_Field_of_Functional | [
"Calculus of Variations"
] | [
"Definition:Functional/Real",
"Definition:Extremum/Functional",
"Definition:Conjugate Point (Calculus of Variations)",
"Definition:Extremal Embedding in Field of Functional"
] | [
"Definition:Conjugate Point (Calculus of Variations)",
"Definition:Set",
"there exists a real point between two real points",
"Definition:Interval/Ordered Set/Closed",
"Definition:Definition",
"Definition:Mapping",
"Definition:Interval/Ordered Set/Closed",
"Definition:Conjugate Point (Calculus of Vari... |
proofwiki-13142 | Sequence of Smallest Consecutive Composite Numbers longer than 100 | The $1$st prime gap greater than $100$ is between $370 \, 261$ and $370 \, 373$, of length $112$.
That is, the sequence of the smallest consecutive composite positive integers longer than $100$ is that of $111$ such, from $370 \, 262$ to $370 \, 372$. | {{ProofWanted|Brute force?}} | The $1$st [[Definition:Prime Gap|prime gap]] greater than $100$ is between $370 \, 261$ and $370 \, 373$, of length $112$.
That is, the [[Definition:Integer Sequence|sequence]] of the smallest consecutive [[Definition:Composite Number|composite]] [[Definition:Positive Integer|positive integers]] longer than $100$ is t... | {{ProofWanted|Brute force?}} | Sequence of Smallest Consecutive Composite Numbers longer than 100 | https://proofwiki.org/wiki/Sequence_of_Smallest_Consecutive_Composite_Numbers_longer_than_100 | https://proofwiki.org/wiki/Sequence_of_Smallest_Consecutive_Composite_Numbers_longer_than_100 | [
"Prime Gaps"
] | [
"Definition:Prime Gap",
"Definition:Integer Sequence",
"Definition:Composite Number",
"Definition:Positive/Integer"
] | [] |
proofwiki-13143 | Difference between Two Squares equal to Repunit | The sequence of differences of two squares that each make a repunit begins:
{{begin-eqn}}
{{eqn | l = 1^2 - 0^2
| r = 1
| c =
}}
{{eqn | l = 6^2 - 5^2
| r = 11
| c =
}}
{{eqn | l = 20^2 - 17^2
| r = 111
| c =
}}
{{eqn | l = 56^2 - 45^2
| r = 1111
| c =
}}
{{eqn | l = ... | Let $x^2 - y^2 = R_n$ for some $n$, where $R_n$ denotes the $n$-digit repunit.
From Integer as Difference between Two Squares:
:$R_n$ has at least two distinct divisors of the same parity that multiply to $R_n$.
Then from Difference of Two Squares:
:$x = \dfrac {a + b} 2$
:$y = \dfrac {a - b} 2$
where:
:$R_n = a b$
for... | The [[Definition:Integer Sequence|sequence]] of [[Difference of Two Squares|differences of two squares]] that each make a [[Definition:Repunit|repunit]] begins:
{{begin-eqn}}
{{eqn | l = 1^2 - 0^2
| r = 1
| c =
}}
{{eqn | l = 6^2 - 5^2
| r = 11
| c =
}}
{{eqn | l = 20^2 - 17^2
| r = 111... | Let $x^2 - y^2 = R_n$ for some $n$, where $R_n$ denotes the [[Definition:Digit|$n$-digit]] [[Definition:Repunit|repunit]].
From [[Integer as Difference between Two Squares]]:
:$R_n$ has at least two [[Definition:Distinct|distinct]] [[Definition:Divisor of Integer|divisors]] of the same [[Definition:Parity|parity]] th... | Difference between Two Squares equal to Repunit | https://proofwiki.org/wiki/Difference_between_Two_Squares_equal_to_Repunit | https://proofwiki.org/wiki/Difference_between_Two_Squares_equal_to_Repunit | [
"Square Numbers",
"Repunits",
"Difference between Two Squares equal to Repunit"
] | [
"Definition:Integer Sequence",
"Difference of Two Squares",
"Definition:Repunit"
] | [
"Definition:Digit",
"Definition:Repunit",
"Integer as Difference between Two Squares",
"Definition:Distinct",
"Definition:Divisor (Algebra)/Integer",
"Definition:Parity",
"Definition:Multiplication/Integers",
"Difference of Two Squares",
"Definition:Parity",
"Definition:Odd Integer",
"Definition... |
proofwiki-13144 | Smallest 3-Digit Permutable Prime | The smallest $3$-digit permutable prime is $113$. | :$113$ is prime.
:$131$ is prime.
:$311$ is prime.
Consider the $3$-digit primes smaller than $113$:
:$101, 103, 107, 109$
They all contain a zero.
Thus, for each of these, at least one permutation ends in a zero.
Hence it is divisible by $10$ and so is not prime.
{{qed}} | The smallest [[Definition:Digit|$3$-digit]] [[Definition:Permutable Prime|permutable prime]] is $113$. | :$113$ is [[Definition:Prime Number|prime]].
:$131$ is [[Definition:Prime Number|prime]].
:$311$ is [[Definition:Prime Number|prime]].
Consider the $3$-digit [[Definition:Prime Number|primes]] smaller than $113$:
:$101, 103, 107, 109$
They all contain a [[Definition:Zero Digit|zero]].
Thus, for each of these, at ... | Smallest 3-Digit Permutable Prime | https://proofwiki.org/wiki/Smallest_3-Digit_Permutable_Prime | https://proofwiki.org/wiki/Smallest_3-Digit_Permutable_Prime | [
"Permutable Primes",
"113"
] | [
"Definition:Digit",
"Definition:Permutable Prime"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Zero Digit",
"Definition:Permutation",
"Definition:Zero Digit",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number"
] |
proofwiki-13145 | Infinite Group has Infinite Number of Subgroups | Let $\struct {G, \circ}$ be an infinite group.
Then $\struct {G, \circ}$ has an infinite number of distinct subgroups. | There are two cases to consider: either $\struct {G, \circ}$ has an infinite cyclic subgroup, or it does not. | Let $\struct {G, \circ}$ be an [[Definition:Infinite Group|infinite group]].
Then $\struct {G, \circ}$ has an [[Definition:Infinite Set|infinite]] number of [[Definition:Distinct|distinct]] [[Definition:Subgroup|subgroups]]. | There are two cases to consider: either $\struct {G, \circ}$ has an [[Definition:Infinite Cyclic Group|infinite cyclic]] [[Definition:Subgroup|subgroup]], or it does not. | Infinite Group has Infinite Number of Subgroups | https://proofwiki.org/wiki/Infinite_Group_has_Infinite_Number_of_Subgroups | https://proofwiki.org/wiki/Infinite_Group_has_Infinite_Number_of_Subgroups | [
"Infinite Groups"
] | [
"Definition:Infinite Group",
"Definition:Infinite Set",
"Definition:Distinct",
"Definition:Subgroup"
] | [
"Definition:Infinite Cyclic Group",
"Definition:Subgroup",
"Definition:Infinite Cyclic Group",
"Definition:Subgroup",
"Definition:Subgroup",
"Definition:Infinite Cyclic Group",
"Definition:Subgroup",
"Definition:Subgroup",
"Definition:Subgroup"
] |
proofwiki-13146 | Group is Finite iff Finite Number of Subgroups | Let $\struct {G, \circ}$ be a group.
Then $G$ is finite {{iff}} $\struct {G, \circ}$ has a finite number of subgroups. | === Necessary Condition ===
Suppose that $\struct {G, \circ}$ is a finite group.
Let $\struct {H, \circ}$ be a subgroup of $\struct {G, \circ}$.
$H \subseteq G$ by definition.
Therefore:
:$H \in \powerset G$
where $\powerset G$ denotes the power set of $G$.
By Power Set of Finite Set is Finite, $\powerset G$ is finite.... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Then $G$ is [[Definition:Finite Set|finite]] {{iff}} $\struct {G, \circ}$ has a [[Definition:Finite Set|finite number]] of [[Definition:Subgroup|subgroups]]. | === Necessary Condition ===
Suppose that $\struct {G, \circ}$ is a [[Definition:Finite Group|finite group]].
Let $\struct {H, \circ}$ be a [[Definition:Subgroup|subgroup]] of $\struct {G, \circ}$.
$H \subseteq G$ by definition.
Therefore:
:$H \in \powerset G$
where $\powerset G$ denotes the [[Definition:Power Set|p... | Group is Finite iff Finite Number of Subgroups | https://proofwiki.org/wiki/Group_is_Finite_iff_Finite_Number_of_Subgroups | https://proofwiki.org/wiki/Group_is_Finite_iff_Finite_Number_of_Subgroups | [
"Finite Groups"
] | [
"Definition:Group",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Subgroup"
] | [
"Definition:Finite Group",
"Definition:Subgroup",
"Definition:Power Set",
"Power Set of Finite Set is Finite",
"Definition:Finite Set",
"Definition:Subset",
"Subset of Finite Set is Finite",
"Definition:Finite Set",
"Definition:Subgroup"
] |
proofwiki-13147 | 3-Digit Permutable Primes | The $3$-digit permutable primes are:
:$311, 199, 337$
and their anagrams, and no other. | It is confirmed that:
:$113, 131, 311$ are all prime
:$199, 919, 991$ are all prime
:$337, 373, 733$ are all prime.
From Digits of Permutable Prime, all permutable primes contain digits in the set:
:$\left\{ {1, 3, 7, 9}\right\}$
The sum of a $3$-digit repdigit number is divisible by $3$.
By Divisibility by 3 it follow... | The [[Definition:Digit|$3$-digit]] [[Definition:Permutable Prime|permutable primes]] are:
:$311, 199, 337$
and their [[Definition:Anagram|anagrams]], and no other. | It is confirmed that:
:$113, 131, 311$ are all [[Definition:Prime Number|prime]]
:$199, 919, 991$ are all [[Definition:Prime Number|prime]]
:$337, 373, 733$ are all [[Definition:Prime Number|prime]].
From [[Digits of Permutable Prime]], all [[Definition:Permutable Prime|permutable primes]] contain [[Definition:Digi... | 3-Digit Permutable Primes | https://proofwiki.org/wiki/3-Digit_Permutable_Primes | https://proofwiki.org/wiki/3-Digit_Permutable_Primes | [
"Permutable Primes"
] | [
"Definition:Digit",
"Definition:Permutable Prime",
"Definition:Anagram"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Digits of Permutable Prime",
"Definition:Permutable Prime",
"Definition:Digit",
"Definition:Set",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Repdigit Number",
"Definition:Divisor (Algebra)/Inte... |
proofwiki-13148 | Digits of Permutable Prime | Let $p$ be a permutable prime with more than $1$ digit.
Then $p$ contains only digits from the set:
:$\left\{ {1, 3, 7, 9}\right\}$ | First note that from 3-Digit Permutable Primes, both $337$ and $199$ are permutable primes.
Hence it follows that all the elements of $\left\{ {1, 3, 7, 9}\right\}$ appear in at least one permutable prime.
Let $p$ contain an even digit $d$.
Then at least one anagram $p'$ of $p$ has $d$ at the end.
From Divisibility by ... | Let $p$ be a [[Definition:Permutable Prime|permutable prime]] with more than $1$ [[Definition:Digit|digit]].
Then $p$ contains only [[Definition:Digit|digits]] from the [[Definition:Set|set]]:
:$\left\{ {1, 3, 7, 9}\right\}$ | First note that from [[3-Digit Permutable Primes]], both $337$ and $199$ are [[Definition:Permutable Prime|permutable primes]].
Hence it follows that all the [[Definition:Element|elements]] of $\left\{ {1, 3, 7, 9}\right\}$ appear in at least one [[Definition:Permutable Prime|permutable prime]].
Let $p$ contain an [... | Digits of Permutable Prime | https://proofwiki.org/wiki/Digits_of_Permutable_Prime | https://proofwiki.org/wiki/Digits_of_Permutable_Prime | [
"Permutable Primes"
] | [
"Definition:Permutable Prime",
"Definition:Digit",
"Definition:Digit",
"Definition:Set"
] | [
"3-Digit Permutable Primes",
"Definition:Permutable Prime",
"Definition:Element",
"Definition:Permutable Prime",
"Definition:Even Integer",
"Definition:Digit",
"Definition:Anagram",
"Divisibility by 2",
"Definition:Even Integer",
"Definition:Digit",
"Definition:Composite",
"Definition:Digit",
... |
proofwiki-13149 | Divisibility by 5 | An integer $N$ expressed in decimal notation is divisible by $5$ {{iff}} the units digit of $N$ is divisible by $5$.
That is:
:$N = \sqbrk {a_n \ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $5$
{{iff}}:
:$a_0$ is divisible by $5$. | Let $N$ be divisible by $5$.
Then:
{{begin-eqn}}
{{eqn | l = N
| o = \equiv
| r = 0 \pmod 5
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{k \mathop = 0}^n a_k 10^k
| o = \equiv
| r = 0 \pmod 5
}}
{{eqn | ll= \leadstoandfrom
| l = a_0 + 10 \sum_{k \mathop = 1}^n a_k 10^{k - 1}
| o ... | An [[Definition:Integer|integer]] $N$ expressed in [[Definition:Decimal Notation|decimal notation]] is [[Definition:Divisor of Integer|divisible]] by $5$ {{iff}} the [[Definition:Units Digit|units digit]] of $N$ is [[Definition:Divisor of Integer|divisible]] by $5$.
That is:
:$N = \sqbrk {a_n \ldots a_2 a_1 a_0}_{10}... | Let $N$ be [[Definition:Divisor of Integer|divisible]] by $5$.
Then:
{{begin-eqn}}
{{eqn | l = N
| o = \equiv
| r = 0 \pmod 5
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{k \mathop = 0}^n a_k 10^k
| o = \equiv
| r = 0 \pmod 5
}}
{{eqn | ll= \leadstoandfrom
| l = a_0 + 10 \sum_{k \math... | Divisibility by 5 | https://proofwiki.org/wiki/Divisibility_by_5 | https://proofwiki.org/wiki/Divisibility_by_5 | [
"Divisibility Tests",
"5"
] | [
"Definition:Integer",
"Definition:Decimal Notation",
"Definition:Divisor (Algebra)/Integer",
"Definition:Units Digit",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-13150 | Prime Repdigit Number is Repunit | Let $b \in \Z_{>0}$ be an integer greater than $1$.
Let $n \in \Z$ expressed in base $b$ be a repdigit number with more than $1$ digit.
Let $n$ be prime.
Then $n$ is a repunit (in base $b$). | Let $n$ be a repdigit number with $k$ digits.
Then by the Basis Representation Theorem:
:$\ds n = \sum_{j \mathop = 0}^k m b^j$
for some $m$ such that $1 \le m < b$.
Let $m \ge 2$.
Then:
:$\ds n = m \sum_{j \mathop = 0}^k b^j$
and so has $m$ as a divisor.
Hence $n$ is not prime.
The result follows by the Rule of Transp... | Let $b \in \Z_{>0}$ be an [[Definition:Integer|integer]] greater than $1$.
Let $n \in \Z$ expressed in [[Definition:Number Base|base $b$]] be a [[Definition:Repdigit Number|repdigit number]] with more than $1$ [[Definition:Digit|digit]].
Let $n$ be [[Definition:Prime Number|prime]].
Then $n$ is a [[Definition:Repun... | Let $n$ be a [[Definition:Repdigit Number|repdigit number]] with $k$ [[Definition:Digit|digits]].
Then by the [[Basis Representation Theorem]]:
:$\ds n = \sum_{j \mathop = 0}^k m b^j$
for some $m$ such that $1 \le m < b$.
Let $m \ge 2$.
Then:
:$\ds n = m \sum_{j \mathop = 0}^k b^j$
and so has $m$ as a [[Definition:... | Prime Repdigit Number is Repunit | https://proofwiki.org/wiki/Prime_Repdigit_Number_is_Repunit | https://proofwiki.org/wiki/Prime_Repdigit_Number_is_Repunit | [
"Repdigit Numbers",
"Repunits",
"Repunit Primes"
] | [
"Definition:Integer",
"Definition:Number Base",
"Definition:Repdigit Number",
"Definition:Digit",
"Definition:Prime Number",
"Definition:Repunit",
"Definition:Number Base"
] | [
"Definition:Repdigit Number",
"Definition:Digit",
"Basis Representation Theorem",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number",
"Rule of Transposition",
"Category:Repdigit Numbers",
"Category:Repunits",
"Category:Repunit Primes"
] |
proofwiki-13151 | 2-Digit Permutable Primes | The $2$-digit permutable primes are:
:$11, 13, 17, 37, 79$
and their anagrams, and no other. | It is confirmed that:
:$13$ and $31$ are both prime
:$17$ and $71$ are both prime
:$37$ and $73$ are both prime.
:$79$ and $97$ are both prime.
From Digits of Permutable Prime, all permutable primes contain digits in the set:
:$\left\{ {1, 3, 7, 9}\right\}$
By Prime Repdigit Number is Repunit, all $2$-digit repdigit nu... | The [[Definition:Digit|$2$-digit]] [[Definition:Permutable Prime|permutable primes]] are:
:$11, 13, 17, 37, 79$
and their [[Definition:Anagram|anagrams]], and no other. | It is confirmed that:
:$13$ and $31$ are both [[Definition:Prime Number|prime]]
:$17$ and $71$ are both [[Definition:Prime Number|prime]]
:$37$ and $73$ are both [[Definition:Prime Number|prime]].
:$79$ and $97$ are both [[Definition:Prime Number|prime]].
From [[Digits of Permutable Prime]], all [[Definition:Permu... | 2-Digit Permutable Primes | https://proofwiki.org/wiki/2-Digit_Permutable_Primes | https://proofwiki.org/wiki/2-Digit_Permutable_Primes | [
"Permutable Primes"
] | [
"Definition:Digit",
"Definition:Permutable Prime",
"Definition:Anagram"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Digits of Permutable Prime",
"Definition:Permutable Prime",
"Definition:Digit",
"Definition:Set",
"Prime Repdigit Number is Repunit",
"Definition:Digit",
"Definition:Repdigit Number",
"... |
proofwiki-13152 | Prime Gaps of 14 | The following pairs of consecutive prime numbers are those whose difference is $14$:
:$\tuple {113, 127}, \tuple {293, 307}, \tuple {317, 331}, \ldots$
{{OEIS|A031932}} | Demonstrated by listing the prime gaps. | The following [[Definition:Ordered Pair|pairs]] of consecutive [[Definition:Prime Number|prime numbers]] are those whose [[Definition:Integer Multiplication|difference]] is $14$:
:$\tuple {113, 127}, \tuple {293, 307}, \tuple {317, 331}, \ldots$
{{OEIS|A031932}} | Demonstrated by listing the [[Definition:Prime Gap|prime gaps]]. | Prime Gaps of 14 | https://proofwiki.org/wiki/Prime_Gaps_of_14 | https://proofwiki.org/wiki/Prime_Gaps_of_14 | [
"Prime Gaps"
] | [
"Definition:Ordered Pair",
"Definition:Prime Number",
"Definition:Multiplication/Integers"
] | [
"Definition:Prime Gap"
] |
proofwiki-13153 | Prime Gaps of 18 | After the prime gap of $14$ between the pairs of consecutive prime numbers:
:$\left({113, 127}\right), \left({293, 307}\right), \left({317, 331}\right), \ldots$
the next prime gap which is greater than $14$ is between the pair of consecutive prime numbers:
:$\left({523, 541}\right)$
for a prime gap of $18$. | Demonstrated by listing the prime gaps. | After the [[Definition:Prime Gap|prime gap]] of $14$ between the [[Definition:Ordered Pair|pairs]] of consecutive [[Definition:Prime Number|prime numbers]]:
:$\left({113, 127}\right), \left({293, 307}\right), \left({317, 331}\right), \ldots$
the next [[Definition:Prime Gap|prime gap]] which is greater than $14$ is bet... | Demonstrated by listing the [[Definition:Prime Gap|prime gaps]]. | Prime Gaps of 18 | https://proofwiki.org/wiki/Prime_Gaps_of_18 | https://proofwiki.org/wiki/Prime_Gaps_of_18 | [
"Prime Gaps"
] | [
"Definition:Prime Gap",
"Definition:Ordered Pair",
"Definition:Prime Number",
"Definition:Prime Gap",
"Definition:Ordered Pair",
"Definition:Prime Number",
"Definition:Prime Gap"
] | [
"Definition:Prime Gap"
] |
proofwiki-13154 | Smallest Number which is Sum of 4 Triples with Equal Products | The smallest positive integer which is the sum of $4$
distinct ordered triples, each of which has the same product, is $118$:
{{begin-eqn}}
{{eqn | l = 118
| r = 14 + 50 + 54
}}
{{eqn | r = 15 + 40 + 63
| c =
}}
{{eqn | r = 18 + 30 + 70
| c =
}}
{{eqn | r = 21 + 25 + 72
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = 14 \times 50 \times 54
| r = \paren {2 \times 7} \times \paren {2 \times 5^2} \times \paren {2 \times 3^3}
}}
{{eqn | r = 2^3 \times 3^3 \times 5^2 \times 7
| c =
}}
{{eqn | r = 37 \, 800
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 15 \times 40 \times 63
| r = \par... | The smallest [[Definition:Positive Integer|positive integer]] which is the [[Definition:Integer Addition|sum]] of $4$
[[Definition:Distinct|distinct]] [[Definition:Ordered Triple|ordered triples]], each of which has the same [[Definition:Integer Multiplication|product]], is $118$:
{{begin-eqn}}
{{eqn | l = 118
... | {{begin-eqn}}
{{eqn | l = 14 \times 50 \times 54
| r = \paren {2 \times 7} \times \paren {2 \times 5^2} \times \paren {2 \times 3^3}
}}
{{eqn | r = 2^3 \times 3^3 \times 5^2 \times 7
| c =
}}
{{eqn | r = 37 \, 800
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 15 \times 40 \times 63
| r = \p... | Smallest Number which is Sum of 4 Triples with Equal Products | https://proofwiki.org/wiki/Smallest_Number_which_is_Sum_of_4_Triples_with_Equal_Products | https://proofwiki.org/wiki/Smallest_Number_which_is_Sum_of_4_Triples_with_Equal_Products | [
"118"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Multiplication/Integers"
] | [] |
proofwiki-13155 | Interval containing Prime Number of forms 4n - 1, 4n + 1, 6n - 1, 6n + 1 | Let $n \in \Z$ be an integer such that $n \ge 118$.
Then between $n$ and $\dfrac {4 n} 3$ there exists at least one prime number of each of the forms:
:$4 m - 1, 4 m + 1, 6 m - 1, 6 m + 1$ | {{questionable|See below.}}
It is demonstrated that the result is true for $n = 118$:
:$\dfrac {4 \times 118} 3 = 157 \cdotp \dot 3$
The primes between $118$ and $157$ are:
{{begin-eqn}}
{{eqn | l = 127
| r = 4 \times 32 - 1
}}
{{eqn | l =
| r = 6 \times 21 + 1
}}
{{eqn | l = 131
| r = 4 \times 33 - ... | Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n \ge 118$.
Then between $n$ and $\dfrac {4 n} 3$ there exists at least one [[Definition:Prime Number|prime number]] of each of the forms:
:$4 m - 1, 4 m + 1, 6 m - 1, 6 m + 1$ | {{questionable|See below.}}
It is demonstrated that the result is true for $n = 118$:
:$\dfrac {4 \times 118} 3 = 157 \cdotp \dot 3$
The [[Definition:Prime Number|primes]] between $118$ and $157$ are:
{{begin-eqn}}
{{eqn | l = 127
| r = 4 \times 32 - 1
}}
{{eqn | l =
| r = 6 \times 21 + 1
}}
{{eqn | l ... | Interval containing Prime Number of forms 4n - 1, 4n + 1, 6n - 1, 6n + 1 | https://proofwiki.org/wiki/Interval_containing_Prime_Number_of_forms_4n_-_1,_4n_+_1,_6n_-_1,_6n_+_1 | https://proofwiki.org/wiki/Interval_containing_Prime_Number_of_forms_4n_-_1,_4n_+_1,_6n_-_1,_6n_+_1 | [
"Prime Numbers"
] | [
"Definition:Integer",
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number"
] |
proofwiki-13156 | Sum of Cubes of 5 Consecutive Integers which is Square | The following sequences of $5$ consecutive (strictly) positive integers have cubes that sum to squares:
:$1, 2, 3, 4, 5$
:$25, 26, 27, 28, 29$
:$96, 97, 98, 99, 100$
:$118, 119, 120, 121, 122$
No other such sequence of $5$ consecutive positive integers has the same property.
However, if we allow sequences containing ze... | {{begin-eqn}}
{{eqn | l = 1^3 + 2^3 + 3^3 + 4^3 + 5^3
| r = 1 + 8 + 27 + 64 + 125
| c =
}}
{{eqn | r = 225
| c =
}}
{{eqn | r = 15^2
| c = also see Sum of Sequence of Cubes
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 25^3 + 26^3 + 27^3 + 28^3 + 29^3
| r = 15 \, 625 + 17 \, 576 + 19 \, 683 +... | The following [[Definition:Integer Sequence|sequences]] of $5$ consecutive [[Definition:Strictly Positive Integer|(strictly) positive integers]] have [[Definition:Cube Number|cubes]] that [[Definition:Integer Addition|sum]] to [[Definition:Square Number|squares]]:
:$1, 2, 3, 4, 5$
:$25, 26, 27, 28, 29$
:$96, 97, 98,... | {{begin-eqn}}
{{eqn | l = 1^3 + 2^3 + 3^3 + 4^3 + 5^3
| r = 1 + 8 + 27 + 64 + 125
| c =
}}
{{eqn | r = 225
| c =
}}
{{eqn | r = 15^2
| c = also see [[Sum of Sequence of Cubes]]
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 25^3 + 26^3 + 27^3 + 28^3 + 29^3
| r = 15 \, 625 + 17 \, 576 + 19 \,... | Sum of Cubes of 5 Consecutive Integers which is Square | https://proofwiki.org/wiki/Sum_of_Cubes_of_5_Consecutive_Integers_which_is_Square | https://proofwiki.org/wiki/Sum_of_Cubes_of_5_Consecutive_Integers_which_is_Square | [
"Sums of Cubes"
] | [
"Definition:Integer Sequence",
"Definition:Strictly Positive/Integer",
"Definition:Cube Number",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Integer Sequence",
"Definition:Zero (Number)",
"Definition:Negative... | [
"Sum of Sequence of Cubes",
"Sum of Sequence of Cubes",
"Definition:Degenerate Case",
"Definition:Integer Sequence",
"Definition:Integer",
"Definition:Cube Number",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Integer Sequence",
"Binomial Theorem/Examples/Cube of Sum",
... |
proofwiki-13157 | Smallest Number to appear 6 Times in Pascal's Triangle | The smallest positive integer greater than $1$ to appear $6$ times in Pascal's Triangle is $120$. | We have:
:$\dbinom {120} 1 = \dbinom {16} 2 = \dbinom {10} 3 = \dbinom {10} 7 = \dbinom {16} {14} = \dbinom {120} {119} = 120$
To verify that this is the smallest, we look at binomial coefficients that are no more than $120$.
Observe that for $n > 120$, $1 \le k \le n - 1$:
:$\dbinom n k \ge \dbinom n 1 = n > 120$
For ... | The smallest [[Definition:Positive Integer|positive integer]] greater than $1$ to appear $6$ times in [[Definition:Pascal's Triangle|Pascal's Triangle]] is $120$. | We have:
:$\dbinom {120} 1 = \dbinom {16} 2 = \dbinom {10} 3 = \dbinom {10} 7 = \dbinom {16} {14} = \dbinom {120} {119} = 120$
To verify that this is the smallest, we look at [[Definition:binomial Coefficient|binomial coefficients]] that are no more than $120$.
Observe that for $n > 120$, $1 \le k \le n - 1$:
:$\dbin... | Smallest Number to appear 6 Times in Pascal's Triangle | https://proofwiki.org/wiki/Smallest_Number_to_appear_6_Times_in_Pascal's_Triangle | https://proofwiki.org/wiki/Smallest_Number_to_appear_6_Times_in_Pascal's_Triangle | [
"Pascal's Triangle",
"120"
] | [
"Definition:Positive/Integer",
"Definition:Pascal's Triangle"
] | [
"Definition:binomial Coefficient"
] |
proofwiki-13158 | Smallest n such that 6 n + 1 and 6 n - 1 are both Composite | The smallest positive integer $n$ such that $6 n + 1$ and $6 n - 1$ are both composite is $20$. | Running through the positive integers in turn:
{{begin-eqn}}
{{eqn | l = 6 \times 1 - 1
| r = 5
| c = which is prime
}}
{{eqn | l = 6 \times 1 + 1
| r = 7
| c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 2 - 1
| r = 11
| c = which is prime
}}
{{eqn | l = 6 \time... | The smallest [[Definition:Positive Integer|positive integer]] $n$ such that $6 n + 1$ and $6 n - 1$ are both [[Definition:Composite Number|composite]] is $20$. | Running through the [[Definition:Positive Integer|positive integers]] in turn:
{{begin-eqn}}
{{eqn | l = 6 \times 1 - 1
| r = 5
| c = which is [[Definition:Prime Number|prime]]
}}
{{eqn | l = 6 \times 1 + 1
| r = 7
| c = which is [[Definition:Prime Number|prime]]
}}
{{end-eqn}}
{{begin-eqn}}
... | Smallest n such that 6 n + 1 and 6 n - 1 are both Composite | https://proofwiki.org/wiki/Smallest_n_such_that_6_n_+_1_and_6_n_-_1_are_both_Composite | https://proofwiki.org/wiki/Smallest_n_such_that_6_n_+_1_and_6_n_-_1_are_both_Composite | [
"Prime Numbers",
"20"
] | [
"Definition:Positive/Integer",
"Definition:Composite Number"
] | [
"Definition:Positive/Integer",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Composite Number",
"Definition:Prime Number",
"Definition:Prime ... |
proofwiki-13159 | Smallest Number with 16 Divisors | The smallest positive integer with $16$ divisors is $120$. | From {{DCFLink|120}}:
:$\map {\sigma_0} {120} = 16$
The result is a specific instance of Smallest Number with $2^n$ Divisors:
:$120 = 2 \times 3 \times 4 \times 5$
{{qed}} | The smallest [[Definition:Positive Integer|positive integer]] with $16$ [[Definition:Divisor of Integer|divisors]] is $120$. | From {{DCFLink|120}}:
:$\map {\sigma_0} {120} = 16$
The result is a specific instance of [[Smallest Number with 2^n Divisors|Smallest Number with $2^n$ Divisors]]:
:$120 = 2 \times 3 \times 4 \times 5$
{{qed}} | Smallest Number with 16 Divisors | https://proofwiki.org/wiki/Smallest_Number_with_16_Divisors | https://proofwiki.org/wiki/Smallest_Number_with_16_Divisors | [
"Divisor Count Function",
"120"
] | [
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Smallest Number with 2^n Divisors"
] |
proofwiki-13160 | Fermat Set is Diophantine Quadruple | The Fermat set $F = \left\{{1, 3, 8, 120}\right\}$ is a Diophantine quadruple:
:$\forall a, b \in F: a \ne b: a b + 1 = n^2$
for some $n \in \Z$. | {{begin-eqn}}
{{eqn | l = 1 \times 3 + 1
| r = 4
| c =
}}
{{eqn | r = 2^2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 1 \times 8 + 1
| r = 9
| c =
}}
{{eqn | r = 3^2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 1 \times 120 + 1
| r = 121
| c =
}}
{{eqn | r = 11^... | The [[Definition:Fermat Set|Fermat set]] $F = \left\{{1, 3, 8, 120}\right\}$ is a [[Definition:Diophantine Quadruple|Diophantine quadruple]]:
:$\forall a, b \in F: a \ne b: a b + 1 = n^2$
for some $n \in \Z$. | {{begin-eqn}}
{{eqn | l = 1 \times 3 + 1
| r = 4
| c =
}}
{{eqn | r = 2^2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 1 \times 8 + 1
| r = 9
| c =
}}
{{eqn | r = 3^2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 1 \times 120 + 1
| r = 121
| c =
}}
{{eqn | r =... | Fermat Set is Diophantine Quadruple | https://proofwiki.org/wiki/Fermat_Set_is_Diophantine_Quadruple | https://proofwiki.org/wiki/Fermat_Set_is_Diophantine_Quadruple | [
"Diophantine m-Tuples"
] | [
"Definition:Fermat Set",
"Definition:Diophantine m-Tuple/Quadruple"
] | [] |
proofwiki-13161 | Fermat Set cannot be Extended to Diophantine Quintuple | The Fermat set $F = \set {1, 3, 8, 120}$ cannot be extended to a Diophantine quintuple. | {{ProofWanted|Can be found in the link below.}} | The [[Definition:Fermat Set|Fermat set]] $F = \set {1, 3, 8, 120}$ cannot be extended to a [[Definition:Diophantine Quintuple|Diophantine quintuple]]. | {{ProofWanted|Can be found in the link below.}} | Fermat Set cannot be Extended to Diophantine Quintuple | https://proofwiki.org/wiki/Fermat_Set_cannot_be_Extended_to_Diophantine_Quintuple | https://proofwiki.org/wiki/Fermat_Set_cannot_be_Extended_to_Diophantine_Quintuple | [
"Diophantine m-Tuples"
] | [
"Definition:Fermat Set",
"Definition:Diophantine m-Tuple/Quintuple"
] | [] |
proofwiki-13162 | Ratio of 360 to Aliquot Sum | $360$ has the property that its ratio to its aliquot sum is $4 : 9$. | The aliquot sum of an integer $n$ is the integer sum of the aliquot parts of $n$.
That is, the aliquot sum of $360$ is the divisor sum of $360$ minus $360$.
Thus:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {360} - 360
| r = 1170 - 360
| c = {{DSFLink|360}}
}}
{{eqn | r = 810
| c =
}}
{{eqn | ll= \lea... | $360$ has the property that its [[Definition:Ratio|ratio]] to its [[Definition:Aliquot Sum|aliquot sum]] is $4 : 9$. | The [[Definition:Aliquot Sum|aliquot sum]] of an [[Definition:Integer|integer]] $n$ is the [[Definition:Integer Addition|integer sum]] of the [[Definition:Aliquot Part|aliquot parts]] of $n$.
That is, the [[Definition:Aliquot Sum|aliquot sum]] of $360$ is the [[Definition:Divisor Sum Function|divisor sum]] of $360$ m... | Ratio of 360 to Aliquot Sum | https://proofwiki.org/wiki/Ratio_of_360_to_Aliquot_Sum | https://proofwiki.org/wiki/Ratio_of_360_to_Aliquot_Sum | [
"360",
"Aliquot Sums"
] | [
"Definition:Ratio",
"Definition:Aliquot Sum"
] | [
"Definition:Aliquot Sum",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)/Integer/Aliquot Part",
"Definition:Aliquot Sum",
"Definition:Divisor Sum Function"
] |
proofwiki-13163 | Are All Triperfect Numbers Even?/Progress/Minimum Size | It has been established that an odd triperfect number, if one were to exist, would be greater than $10^{70}$.
If it does not have $3$ as a prime factor, then it is greater than $10^{108}$. | {{ProofWanted|Details}} | It has been established that an [[Definition:Odd Integer|odd]] [[Definition:Triperfect Number|triperfect number]], if one were to exist, would be greater than $10^{70}$.
If it does not have $3$ as a [[Definition:Prime Factor|prime factor]], then it is greater than $10^{108}$. | {{ProofWanted|Details}} | Are All Triperfect Numbers Even?/Progress/Minimum Size | https://proofwiki.org/wiki/Are_All_Triperfect_Numbers_Even?/Progress/Minimum_Size | https://proofwiki.org/wiki/Are_All_Triperfect_Numbers_Even?/Progress/Minimum_Size | [
"Are All Triperfect Numbers Even?"
] | [
"Definition:Odd Integer",
"Definition:Triperfect Number",
"Definition:Prime Factor"
] | [] |
proofwiki-13164 | Multiply Perfect Number of Order 8 | {{tidy|use {{TL|begin-itemize}} or {{TL|begin-eqn}}, whatever ingenuity you like}}
The number defined as:
:$n = 2^{65} \times 3^{23} \times 5^9 \times 7^{12} \times 11^3 \times 13^3 \times 17^2 \times 19^2 \times 23 \times 29^2 \times 31^2$
::${} \times 37 \times 41 \times 53 \times 61 \times 67^2 \times 71^2 \times 73... | From Divisor Sum Function is Multiplicative, we may take each prime factor separately and form $\map {\sigma_1} n$ as the product of the divisor sum of each.
Each of the prime factors which occur with multiplicity $1$ will be treated first.
A prime factor $p$ contributes towards the combined $\sigma_1$ a factor $p + 1$... | {{tidy|use {{TL|begin-itemize}} or {{TL|begin-eqn}}, whatever ingenuity you like}}
The number defined as:
:$n = 2^{65} \times 3^{23} \times 5^9 \times 7^{12} \times 11^3 \times 13^3 \times 17^2 \times 19^2 \times 23 \times 29^2 \times 31^2$
::${} \times 37 \times 41 \times 53 \times 61 \times 67^2 \times 71^2 \times 73... | From [[Divisor Sum Function is Multiplicative]], we may take each [[Definition:Prime Factor|prime factor]] separately and form $\map {\sigma_1} n$ as the [[Definition:Integer Multiplication|product]] of the [[Definition:Divisor Sum Function|divisor sum]] of each.
Each of the [[Definition:Prime Factor|prime factors]] ... | Multiply Perfect Number of Order 8 | https://proofwiki.org/wiki/Multiply_Perfect_Number_of_Order_8 | https://proofwiki.org/wiki/Multiply_Perfect_Number_of_Order_8 | [
"Multiply Perfect Numbers"
] | [
"Definition:Multiply Perfect Number",
"Definition:Multiply Perfect Number/Order"
] | [
"Divisor Sum Function is Multiplicative",
"Definition:Prime Factor",
"Definition:Multiplication/Integers",
"Definition:Divisor Sum Function",
"Definition:Prime Factor",
"Definition:Prime Decomposition/Multiplicity",
"Definition:Prime Factor",
"Definition:Divisor (Algebra)/Integer",
"Divisor Sum of P... |
proofwiki-13165 | 121 is Square Number in All Bases greater than 2 | Let $b \in \Z$ be an integer such that $b \ge 3$.
Let $n$ be a positive integer which can be expressed in base $b$ as $\sqbrk {121}_b$.
Then $n$ is a square number. | Consider $\sqbrk {11}_b$.
By the Basis Representation Theorem:
:$\sqbrk {11}_b = b + 1$
Thus:
{{begin-eqn}}
{{eqn | l = {\sqbrk {11}_b}^2
| r = \paren {b + 1}^2
| c =
}}
{{eqn | r = b^2 + 2 b + 1
| c = Square of Sum
}}
{{eqn | r = \sqbrk {121}_b
| c =
}}
{{end-eqn}}
Thus:
:$\sqbrk {121}_b = {\... | Let $b \in \Z$ be an [[Definition:Integer|integer]] such that $b \ge 3$.
Let $n$ be a [[Definition:Positive Integer|positive integer]] which can be expressed in [[Definition:Number Base|base $b$]] as $\sqbrk {121}_b$.
Then $n$ is a [[Definition:Square Number|square number]]. | Consider $\sqbrk {11}_b$.
By the [[Basis Representation Theorem]]:
:$\sqbrk {11}_b = b + 1$
Thus:
{{begin-eqn}}
{{eqn | l = {\sqbrk {11}_b}^2
| r = \paren {b + 1}^2
| c =
}}
{{eqn | r = b^2 + 2 b + 1
| c = [[Square of Sum]]
}}
{{eqn | r = \sqbrk {121}_b
| c =
}}
{{end-eqn}}
Thus:
:$\sqbrk {... | 121 is Square Number in All Bases greater than 2 | https://proofwiki.org/wiki/121_is_Square_Number_in_All_Bases_greater_than_2 | https://proofwiki.org/wiki/121_is_Square_Number_in_All_Bases_greater_than_2 | [
"Square Numbers",
"Number Bases"
] | [
"Definition:Integer",
"Definition:Positive/Integer",
"Definition:Number Base",
"Definition:Square Number"
] | [
"Basis Representation Theorem",
"Square of Sum",
"Definition:Square Number"
] |
proofwiki-13166 | Cube of 11 is Palindromic | ::$11^3 = 1331$ | From Square of 11 is Palindromic:
:$11^2 = 121$
Thus:
<pre>
121
x 11
-----
121
1210
-----
1331
</pre>{{qed}} | ::$11^3 = 1331$ | From [[Square of Small-Digit Palindromic Number is Palindromic/Examples/11|Square of 11 is Palindromic]]:
:$11^2 = 121$
Thus:
<pre>
121
x 11
-----
121
1210
-----
1331
</pre>{{qed}} | Cube of 11 is Palindromic | https://proofwiki.org/wiki/Cube_of_11_is_Palindromic | https://proofwiki.org/wiki/Cube_of_11_is_Palindromic | [
"Cube Numbers",
"11",
"1331",
"Palindromic Numbers"
] | [] | [
"Square of Small-Digit Palindromic Number is Palindromic/Examples/11"
] |
proofwiki-13167 | Fourth Power of 11 is Palindromic | ::$11^4 = 14641$ | From Square of 11 is Palindromic:
:$11^2 = 121$
Then from Square of 121 is Palindromic:
:$121^2 = 14641 = \paren {11^2}^2 = 11^4$
{{qed}} | ::$11^4 = 14641$ | From [[Square of Small-Digit Palindromic Number is Palindromic/Examples/11|Square of 11 is Palindromic]]:
:$11^2 = 121$
Then from [[Square of Small-Digit Palindromic Number is Palindromic/Examples/121|Square of 121 is Palindromic]]:
:$121^2 = 14641 = \paren {11^2}^2 = 11^4$
{{qed}} | Fourth Power of 11 is Palindromic | https://proofwiki.org/wiki/Fourth_Power_of_11_is_Palindromic | https://proofwiki.org/wiki/Fourth_Power_of_11_is_Palindromic | [
"Fourth Powers",
"11",
"14,641",
"Palindromic Numbers"
] | [] | [
"Square of Small-Digit Palindromic Number is Palindromic/Examples/11",
"Square of Small-Digit Palindromic Number is Palindromic/Examples/121"
] |
proofwiki-13168 | Palindromes in Base 10 and Base 3 | The following $n \in \Z$ are palindromic in both decimal and ternary:
:$0, 1, 2, 4, 8, 121, 151, 212, 242, 484, 656, 757, \ldots$
{{OEIS|A007633}} | :{| border="1"
|-
! align="center" style = "padding: 2px 10px" | $n_{10}$
! align="center" style = "padding: 2px 10px" | $n_3$
|-
| align="right" style = "padding: 2px 10px" | $0$
| align="right" style = "padding: 2px 10px" | $0$
|-
| align="right" style = "padding: 2px 10px" | $1$
| align="right" style = "padding: 2... | The following $n \in \Z$ are [[Definition:Palindromic Number|palindromic]] in both [[Definition:Decimal Notation|decimal]] and [[Definition:Ternary Notation|ternary]]:
:$0, 1, 2, 4, 8, 121, 151, 212, 242, 484, 656, 757, \ldots$
{{OEIS|A007633}} | :{| border="1"
|-
! align="center" style = "padding: 2px 10px" | $n_{10}$
! align="center" style = "padding: 2px 10px" | $n_3$
|-
| align="right" style = "padding: 2px 10px" | $0$
| align="right" style = "padding: 2px 10px" | $0$
|-
| align="right" style = "padding: 2px 10px" | $1$
| align="right" style = "padding: 2... | Palindromes in Base 10 and Base 3 | https://proofwiki.org/wiki/Palindromes_in_Base_10_and_Base_3 | https://proofwiki.org/wiki/Palindromes_in_Base_10_and_Base_3 | [
"Palindromic Numbers",
"3",
"10"
] | [
"Definition:Palindromic Number",
"Definition:Decimal Notation",
"Definition:Ternary Notation"
] | [] |
proofwiki-13169 | Square Numbers which are Sum of Consecutive Powers | The only two square numbers which are the sum of consecutive powers of a positive integer are $121$ and $400$:
:$121 = 3^0 + 3^1 + 3^2 + 3^3 + 3^4 = 11^2$
:$400 = 7^0 + 7^1 + 7^2 + 7^3 = 20^2$ | :$121 = 1 + 3 + 9 + 27 + 81$
:$400 = 1 + 7 + 49 + 343$
{{ProofWanted|It remains to be shown that these are the only such square numbers.}} | The only two [[Definition:Square Number|square numbers]] which are the [[Definition:Integer Addition|sum]] of consecutive [[Definition:Integer Power|powers]] of a [[Definition:Positive Integer|positive integer]] are $121$ and $400$:
:$121 = 3^0 + 3^1 + 3^2 + 3^3 + 3^4 = 11^2$
:$400 = 7^0 + 7^1 + 7^2 + 7^3 = 20^2$ | :$121 = 1 + 3 + 9 + 27 + 81$
:$400 = 1 + 7 + 49 + 343$
{{ProofWanted|It remains to be shown that these are the only such [[Definition:Square Number|square numbers]].}} | Square Numbers which are Sum of Consecutive Powers | https://proofwiki.org/wiki/Square_Numbers_which_are_Sum_of_Consecutive_Powers | https://proofwiki.org/wiki/Square_Numbers_which_are_Sum_of_Consecutive_Powers | [
"Square Numbers",
"121",
"400"
] | [
"Definition:Square Number",
"Definition:Addition/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Positive/Integer"
] | [
"Definition:Square Number"
] |
proofwiki-13170 | Numbers whose Difference equals Difference between Cube and Seventh Power | The following $2$ pairs of integers are the only ones known which exhibit this pattern:
{{begin-eqn}}
{{eqn | l = \size {5^3 - 2^7}
| r = 5 - 2
}}
{{eqn | l = \size {13^3 - 3^7}
| r = 13 - 3
}}
{{end-eqn}} | We have:
{{begin-eqn}}
{{eqn | l = \size {5^3 - 2^7}
| r = \size {125 - 128}
| c =
}}
{{eqn | r = 3
| c =
}}
{{eqn | r = 5 - 2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \size {13^3 - 3^7}
| r = \size {2197 - 2187}
| c =
}}
{{eqn | r = 10
| c =
}}
{{eqn | r = 13 - 3
... | The following $2$ pairs of [[Definition:Integer|integers]] are the only ones known which exhibit this pattern:
{{begin-eqn}}
{{eqn | l = \size {5^3 - 2^7}
| r = 5 - 2
}}
{{eqn | l = \size {13^3 - 3^7}
| r = 13 - 3
}}
{{end-eqn}} | We have:
{{begin-eqn}}
{{eqn | l = \size {5^3 - 2^7}
| r = \size {125 - 128}
| c =
}}
{{eqn | r = 3
| c =
}}
{{eqn | r = 5 - 2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \size {13^3 - 3^7}
| r = \size {2197 - 2187}
| c =
}}
{{eqn | r = 10
| c =
}}
{{eqn | r = 13 - 3... | Numbers whose Difference equals Difference between Cube and Seventh Power | https://proofwiki.org/wiki/Numbers_whose_Difference_equals_Difference_between_Cube_and_Seventh_Power | https://proofwiki.org/wiki/Numbers_whose_Difference_equals_Difference_between_Cube_and_Seventh_Power | [
"Powers"
] | [
"Definition:Integer"
] | [] |
proofwiki-13171 | Reciprocal of 8 | The reciprocal of $8$ can be expressed as the summation of the powers of $2$ multiplied by the reciprocal powers of $10$:
:$\dfrac 1 8 = \ds \sum_{k \mathop \ge 0} \dfrac {2^k} {10^{k + 1} }$
<pre>
.1
2
4
8
16
32
64
128
256
512
1024
....
-------------
.1249... | We have that:
:$\dfrac {2^k} {10^{k + 1} } = \dfrac 1 {10} \paren {\dfrac 1 5}^k$
and so by Sum of Infinite Geometric Sequence:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 0} \dfrac {2^k} {10^{k + 1} }
| r = \dfrac {1/10} {1 - 1/5}
| c =
}}
{{eqn | r = \dfrac 1 {10 \paren {4 / 5} }
| c =
}}
{{eqn ... | The [[Definition:Reciprocal|reciprocal]] of $8$ can be expressed as the [[Definition:Summation|summation]] of the [[Definition:Integer Power|powers]] of $2$ [[Definition:Integer Multiplication|multiplied]] by the [[Definition:Reciprocal|reciprocal]] [[Definition:Integer Power|powers]] of $10$:
:$\dfrac 1 8 = \ds \sum_... | We have that:
:$\dfrac {2^k} {10^{k + 1} } = \dfrac 1 {10} \paren {\dfrac 1 5}^k$
and so by [[Sum of Infinite Geometric Sequence]]:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 0} \dfrac {2^k} {10^{k + 1} }
| r = \dfrac {1/10} {1 - 1/5}
| c =
}}
{{eqn | r = \dfrac 1 {10 \paren {4 / 5} }
| c =
}}
... | Reciprocal of 8 | https://proofwiki.org/wiki/Reciprocal_of_8 | https://proofwiki.org/wiki/Reciprocal_of_8 | [
"8"
] | [
"Definition:Reciprocal",
"Definition:Summation",
"Definition:Power (Algebra)/Integer",
"Definition:Multiplication/Integers",
"Definition:Reciprocal",
"Definition:Power (Algebra)/Integer"
] | [
"Sum of Infinite Geometric Sequence"
] |
proofwiki-13172 | Triangles with Integer Area and Integer Sides in Arithmetical Sequence | The triangles with the following sides in arithmetic sequence have integer areas:
:$3, 4, 5$
:$13, 14, 15$
:$15, 28, 41$
:$15, 26, 37$
Their areas are:
:$6, 84, 126, 156$ | From Heron's Formula, the area $A$ of $\triangle ABC$ is given by:
:$A = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $s = \dfrac{a + b + c} 2$ is the semiperimeter of $\triangle ABC$.
For $3, 4, 5$:
{{begin-eqn}}
{{eqn | l = s
| r = \frac {3 + 4 + 5} 2
| c =
}}
{{eqn | r = 6
| c = ... | The [[Definition:Triangle (Geometry)|triangles]] with the following [[Definition:Side of Polygon|sides]] in [[Definition:Arithmetic Sequence|arithmetic sequence]] have [[Definition:Integer|integer]] [[Definition:Area|areas]]:
:$3, 4, 5$
:$13, 14, 15$
:$15, 28, 41$
:$15, 26, 37$
Their [[Definition:Area|areas]] are:
:$6... | From [[Heron's Formula]], the [[Definition:Area|area]] $A$ of $\triangle ABC$ is given by:
:$A = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $s = \dfrac{a + b + c} 2$ is the [[Definition:Semiperimeter|semiperimeter]] of $\triangle ABC$.
For $3, 4, 5$:
{{begin-eqn}}
{{eqn | l = s
| r = \frac {... | Triangles with Integer Area and Integer Sides in Arithmetical Sequence | https://proofwiki.org/wiki/Triangles_with_Integer_Area_and_Integer_Sides_in_Arithmetical_Sequence | https://proofwiki.org/wiki/Triangles_with_Integer_Area_and_Integer_Sides_in_Arithmetical_Sequence | [
"Areas of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Arithmetic Sequence",
"Definition:Integer",
"Definition:Area",
"Definition:Area"
] | [
"Heron's Formula",
"Definition:Area",
"Definition:Semiperimeter"
] |
proofwiki-13173 | Equivalence of Definitions of Weierstrass E-Function | Let $\mathbf y, \mathbf z, \mathbf w$ be $n$-dimensional vectors.
Let $\mathbf y$ be such that $\map{\mathbf y} a=A$ and $\map{\mathbf y} b=B$.
Let $J$ be a functional such that:
:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
{{TFAE|def = Weierstrass E-Function}} | === Definition 1 implies Definition 2 ===
By Definition 1:
:$\map E {x, \mathbf y, \mathbf z, \mathbf w} = \map F {x, \mathbf y, \mathbf w} - \map F {x, \mathbf y, \mathbf z} + \paren {\mathbf w - \mathbf z} \map {F_{\mathbf y'} } {x, \mathbf y, \mathbf z}$
By Taylor's Theorem, where expansion is done around $\mathbf w... | Let $\mathbf y, \mathbf z, \mathbf w$ be [[Definition:Dimension|$n$-dimensional]] [[Definition:Vector|vectors]].
Let $\mathbf y$ be such that $\map{\mathbf y} a=A$ and $\map{\mathbf y} b=B$.
Let $J$ be a [[Definition:Real Functional|functional]] such that:
:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y,... | === Definition 1 implies Definition 2 ===
By Definition 1:
:$\map E {x, \mathbf y, \mathbf z, \mathbf w} = \map F {x, \mathbf y, \mathbf w} - \map F {x, \mathbf y, \mathbf z} + \paren {\mathbf w - \mathbf z} \map {F_{\mathbf y'} } {x, \mathbf y, \mathbf z}$
By [[Taylor's Theorem]], where expansion is done around $\m... | Equivalence of Definitions of Weierstrass E-Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Weierstrass_E-Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Weierstrass_E-Function | [
"Calculus of Variations"
] | [
"Definition:Dimension",
"Definition:Vector",
"Definition:Functional/Real"
] | [
"Taylor's Theorem",
"Definition:Taylor Series/Remainder",
"Definition:Weierstrass E-Function"
] |
proofwiki-13174 | Conditions for Strong Minimum of Functional | Let $\mathbf y$ be an $n$-dimensional vector such that:
:$\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$
Let $J$ be a functional such that:
:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
Let $\gamma$ be an extremal curve of $J$.
Let the following be the field of the functional $J$:... | By definition, the increment of $J$ is:
:$\ds \Delta J = \int_{\gamma^*} \map F {x, \mathbf y, \mathbf y'} \rd x - \int_\gamma \map F {x, \mathbf y, \mathbf y'} \rd x$
where $\gamma$ and $\gamma^*$ are curves described by $\paren {x, \map {\mathbf y} x}$ and $\paren {x, \paren {\mathbf y^*} x}$ respectively, such that:... | Let $\mathbf y$ be an [[Definition:Dimension|$n$-dimensional]] [[Definition:Vector|vector]] such that:
:$\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$
Let $J$ be a [[Definition:Real Functional|functional]] such that:
:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
Let $\gamma$ b... | By definition, the [[Definition:Increment of Functional|increment]] of $J$ is:
:$\ds \Delta J = \int_{\gamma^*} \map F {x, \mathbf y, \mathbf y'} \rd x - \int_\gamma \map F {x, \mathbf y, \mathbf y'} \rd x$
where $\gamma$ and $\gamma^*$ are [[Definition:Curve|curves]] described by $\paren {x, \map {\mathbf y} x}$ and... | Conditions for Strong Minimum of Functional | https://proofwiki.org/wiki/Conditions_for_Strong_Minimum_of_Functional | https://proofwiki.org/wiki/Conditions_for_Strong_Minimum_of_Functional | [
"Calculus of Variations"
] | [
"Definition:Dimension",
"Definition:Vector",
"Definition:Functional/Real",
"Definition:Extremum/Functional",
"Definition:Line/Curve",
"Definition:Field of Directions/Functional",
"Definition:Open Region",
"Definition:Field of Directions/Functional",
"Definition:Vector",
"Definition:Weierstrass E-F... | [
"Definition:Increment/Functional",
"Definition:Line/Curve",
"Definition:Hilbert's Invariant Integral",
"Definition:Canonical Variable",
"Definition:Integration/Integrand",
"Definition:Definite Integral",
"Definition:Path (Topology)/Endpoint",
"Definition:Definite Integral",
"Definition:Boundary Cond... |
proofwiki-13175 | Factor of Mersenne Number Mp is of form 2kp + 1 | :$q = 2 k p + 1$
for some integer $k$. | Let $q \divides M_p$.
Then:
:$2^p \equiv 1 \pmod q$
From Integer to Power of Multiple of Order, the multiplicative order of $2 \pmod q$ divides $p$.
By Fermat's Little Theorem, the multiplicative order of $2 \pmod q$ also divides $q - 1$.
Hence:
:$q - 1 = 2 k p$
{{qed}} | :$q = 2 k p + 1$
for some [[Definition:Integer|integer]] $k$. | Let $q \divides M_p$.
Then:
:$2^p \equiv 1 \pmod q$
From [[Integer to Power of Multiple of Order]], the [[Definition:Multiplicative Order of Integer|multiplicative order]] of $2 \pmod q$ [[Definition:Divisor of Integer|divides]] $p$.
By [[Fermat's Little Theorem]], the [[Definition:Multiplicative Order of Integer|mu... | Factor of Mersenne Number Mp is of form 2kp + 1 | https://proofwiki.org/wiki/Factor_of_Mersenne_Number_Mp_is_of_form_2kp_+_1 | https://proofwiki.org/wiki/Factor_of_Mersenne_Number_Mp_is_of_form_2kp_+_1 | [
"Mersenne Numbers"
] | [
"Definition:Integer"
] | [
"Integer to Power of Multiple of Order",
"Definition:Multiplicative Order of Integer",
"Definition:Divisor (Algebra)/Integer",
"Fermat's Little Theorem",
"Definition:Multiplicative Order of Integer",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-13176 | Factor of Mersenne Number Mp equivalent to 1 mod p | :$q \equiv 1 \pmod p$ | Let $q \divides M_p$.
From Factor of Mersenne Number $M_p$ is of form $2 k p + 1$:
:$q = 2 k p + 1$
and so by definition of congruence modulo an integer:
:$q \equiv 1 \pmod p$
immediately.
{{qed}} | :$q \equiv 1 \pmod p$ | Let $q \divides M_p$.
From [[Factor of Mersenne Number Mp is of form 2kp + 1|Factor of Mersenne Number $M_p$ is of form $2 k p + 1$]]:
:$q = 2 k p + 1$
and so by definition of [[Definition:Congruence Modulo Integer|congruence modulo an integer]]:
:$q \equiv 1 \pmod p$
immediately.
{{qed}} | Factor of Mersenne Number Mp equivalent to 1 mod p | https://proofwiki.org/wiki/Factor_of_Mersenne_Number_Mp_equivalent_to_1_mod_p | https://proofwiki.org/wiki/Factor_of_Mersenne_Number_Mp_equivalent_to_1_mod_p | [
"Mersenne Numbers"
] | [] | [
"Factor of Mersenne Number Mp is of form 2kp + 1",
"Definition:Congruence (Number Theory)/Integers"
] |
proofwiki-13177 | Factor of Mersenne Number equivalent to +-1 mod 8 | :$q \equiv \pm 1 \pmod 8$ | Suppose $q \divides M_p$, where $\divides$ denotes divisibility.
From Factor of Mersenne Number $M_p$ is of form $2 k p + 1$:
:$q - 1 = 2 k p$
From above:
:$2^{\paren {q - 1} / 2} \equiv 2 k p \equiv 1 \pmod q$
and so $2$ is a quadratic residue $\pmod q$.
From Second Supplement to Law of Quadratic Reciprocity:
:$q \eq... | :$q \equiv \pm 1 \pmod 8$ | Suppose $q \divides M_p$, where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
From [[Factor of Mersenne Number Mp is of form 2kp + 1|Factor of Mersenne Number $M_p$ is of form $2 k p + 1$]]:
:$q - 1 = 2 k p$
From above:
:$2^{\paren {q - 1} / 2} \equiv 2 k p \equiv 1 \pmod q$
and so $2$ is a [[... | Factor of Mersenne Number equivalent to +-1 mod 8 | https://proofwiki.org/wiki/Factor_of_Mersenne_Number_equivalent_to_+-1_mod_8 | https://proofwiki.org/wiki/Factor_of_Mersenne_Number_equivalent_to_+-1_mod_8 | [
"Mersenne Numbers"
] | [] | [
"Definition:Divisor (Algebra)/Integer",
"Factor of Mersenne Number Mp is of form 2kp + 1",
"Definition:Quadratic Residue",
"Second Supplement to Law of Quadratic Reciprocity"
] |
proofwiki-13178 | Numbers with 7 or more Prime Factors | The sequence of positive integers with $7$ or more prime factors (not necessarily distinct) begins:
:$128, 192, 256, 288, 320, 384, 432, 448, 480, 512, \ldots$
{{OEIS|A046307}} | {{begin-eqn}}
{{eqn | l = 128
| r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2
| c =
}}
{{eqn | l = 192
| r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3
| c =
}}
{{eqn | l = 256
| r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \left({\times 2}\right... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] with $7$ or more [[Definition:Prime Factor|prime factors]] (not necessarily [[Definition:Distinct|distinct]]) begins:
:$128, 192, 256, 288, 320, 384, 432, 448, 480, 512, \ldots$
{{OEIS|A046307}} | {{begin-eqn}}
{{eqn | l = 128
| r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2
| c =
}}
{{eqn | l = 192
| r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3
| c =
}}
{{eqn | l = 256
| r = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \left({\times 2}\right... | Numbers with 7 or more Prime Factors | https://proofwiki.org/wiki/Numbers_with_7_or_more_Prime_Factors | https://proofwiki.org/wiki/Numbers_with_7_or_more_Prime_Factors | [
"Prime Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Prime Factor",
"Definition:Distinct"
] | [] |
proofwiki-13179 | Numbers not Sum of Distinct Squares | The positive integers which are not the sum of $1$ or more distinct squares are:
:$2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128$
{{OEIS|A001422}} | {{finish|a) Demonstration that these cannot be so expressed, b) demonstration that all others below 324 can be so expressed}}
It will be proved that the largest integer which cannot be expressed as the sum of distinct squares is $128$.
The remaining integers in the sequence can be identified by inspection.
We prove thi... | The [[Definition:Positive Integer|positive integers]] which are not the [[Definition:Integer Addition|sum]] of $1$ or more [[Definition:Distinct|distinct]] [[Definition:Square Number|squares]] are:
:$2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128... | {{finish|a) Demonstration that these cannot be so expressed, b) demonstration that all others below 324 can be so expressed}}
It will be proved that the largest [[Definition:Integer|integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:... | Numbers not Sum of Distinct Squares | https://proofwiki.org/wiki/Numbers_not_Sum_of_Distinct_Squares | https://proofwiki.org/wiki/Numbers_not_Sum_of_Distinct_Squares | [
"Sums of Squares"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Square Number"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Distinct/Plural",
"Definition:Square Number",
"Definition:Integer",
"Definition:Integer Sequence",
"Second Principle of Mathematical Induction",
"Definition:Addition/Integers",
"Definition:Distinct/Plural",
"Definition:Square Number... |
proofwiki-13180 | 132 is Sum of all 2-Digit Numbers formed from its Digits | $132$ is the smallest sum of all the $2$-digit (positive) integers formed from its own digits. | It is necessary to postulate that such (positive) integers have $3$ digits or more, as the $2$ digit solution is trivial.
Let $n = \sqbrk {abc}$ be a $3$-digit number.
Let $\map s n$ denote the sum of all the $2$-digit (positive) integers formed from the digits of $n$.
Then:
{{begin-eqn}}
{{eqn | l = \map s n
| r... | $132$ is the smallest [[Definition:Integer Addition|sum]] of all the [[Definition:Digit|$2$-digit]] [[Definition:Positive Integer|(positive) integers]] formed from its own [[Definition:Digit|digits]]. | It is necessary to postulate that such [[Definition:Positive Integer|(positive) integers]] have $3$ [[Definition:Digit|digits]] or more, as the $2$ [[Definition:Digit|digit]] solution is trivial.
Let $n = \sqbrk {abc}$ be a $3$-[[Definition:Digit|digit]] number.
Let $\map s n$ denote the [[Definition:Integer Additio... | 132 is Sum of all 2-Digit Numbers formed from its Digits | https://proofwiki.org/wiki/132_is_Sum_of_all_2-Digit_Numbers_formed_from_its_Digits | https://proofwiki.org/wiki/132_is_Sum_of_all_2-Digit_Numbers_formed_from_its_Digits | [
"132"
] | [
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Digit"
] | [
"Definition:Positive/Integer",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Digit",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-13181 | Numbers which are Sum of Increasing Powers of Digits | The following integers are the sum of the increasing powers of their digits taken in order begin as follows:
:$0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2 \, 646 \, 798, 12 \, 157 \, 692 \, 622 \, 039 \, 623 \, 539$
{{OEIS|A032799}} | Single digit integers are trivial:
:$d^1 = d$
for all $d \in \Z$.
Then we have:
{{begin-eqn}}
{{eqn | l = 8^1 + 9^2
| r = 8 + 81
| rr= = 89
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 1^1 + 3^2 + 5^3
| r = 1 + 9 + 125
| rr= = 135
| c =
}}
{{eqn | l = 1^1 + 7^2 + 5^3
| r = ... | The following [[Definition:Integer|integers]] are the [[Definition:Integer Addition|sum]] of the increasing [[Definition:Integer Power|powers]] of their [[Definition:Digit|digits]] taken in order begin as follows:
:$0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2 \, 646 \, 798, 12 \, 157 \, 69... | Single [[Definition:Digit|digit]] [[Definition:Integer|integers]] are trivial:
:$d^1 = d$
for all $d \in \Z$.
Then we have:
{{begin-eqn}}
{{eqn | l = 8^1 + 9^2
| r = 8 + 81
| rr= = 89
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 1^1 + 3^2 + 5^3
| r = 1 + 9 + 125
| rr= = 135
|... | Numbers which are Sum of Increasing Powers of Digits | https://proofwiki.org/wiki/Numbers_which_are_Sum_of_Increasing_Powers_of_Digits | https://proofwiki.org/wiki/Numbers_which_are_Sum_of_Increasing_Powers_of_Digits | [
"Powers",
"Recreational Mathematics"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Digit"
] | [
"Definition:Digit",
"Definition:Integer"
] |
proofwiki-13182 | Prime Gaps of 10 | The following pairs of consecutive prime numbers are those whose difference is $10$:
:$\tuple {139, 149}, \tuple {181, 191}, \tuple {241, 251}, \tuple {283, 293}, \ldots$
{{OEIS|A031928|order = lower}} | Demonstrated by listing the prime gaps.
{{qed}} | The following [[Definition:Ordered Pair|pairs]] of consecutive [[Definition:Prime Number|prime numbers]] are those whose [[Definition:Integer Subtraction|difference]] is $10$:
:$\tuple {139, 149}, \tuple {181, 191}, \tuple {241, 251}, \tuple {283, 293}, \ldots$
{{OEIS|A031928|order = lower}} | Demonstrated by listing the [[Definition:Prime Gap|prime gaps]].
{{qed}} | Prime Gaps of 10 | https://proofwiki.org/wiki/Prime_Gaps_of_10 | https://proofwiki.org/wiki/Prime_Gaps_of_10 | [
"Prime Gaps"
] | [
"Definition:Ordered Pair",
"Definition:Prime Number",
"Definition:Subtraction/Integers"
] | [
"Definition:Prime Gap"
] |
proofwiki-13183 | Harmonic Mean of Divisors in terms of Divisor Count and Divisor Sum | Let $n \in \Z_{>0}$ be a positive integer.
The harmonic mean of the divisors of $n$ is given by:
:$\map H n = \dfrac {n \, \map {\sigma_0} n} {\map {\sigma_1} n}$
where:
:$\map {\sigma_0} n$ denotes the divisor count function: the number of divisors of $n$
:$\map {\sigma_1} n$ denotes the divisor sum function: the sum ... | {{begin-eqn}}
{{eqn | l = \frac 1 {\map H n}
| r = \frac 1 {\map {\sigma_0} n} \paren {\sum_{d \mathop \divides n} \frac 1 d}
| c = {{Defof|Harmonic Mean}}
}}
{{eqn | l = \sum_{d \mathop \divides n} \frac 1 d
| r = \frac {\map {\sigma_1} n} n
| c = Sum of Reciprocals of Divisors equals Abundancy... | Let $n \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]].
The [[Definition:Harmonic Mean|harmonic mean]] of the [[Definition:Divisor of Integer|divisors]] of $n$ is given by:
:$\map H n = \dfrac {n \, \map {\sigma_0} n} {\map {\sigma_1} n}$
where:
:$\map {\sigma_0} n$ denotes the [[Definition:Divisor ... | {{begin-eqn}}
{{eqn | l = \frac 1 {\map H n}
| r = \frac 1 {\map {\sigma_0} n} \paren {\sum_{d \mathop \divides n} \frac 1 d}
| c = {{Defof|Harmonic Mean}}
}}
{{eqn | l = \sum_{d \mathop \divides n} \frac 1 d
| r = \frac {\map {\sigma_1} n} n
| c = [[Sum of Reciprocals of Divisors equals Abundan... | Harmonic Mean of Divisors in terms of Divisor Count and Divisor Sum | https://proofwiki.org/wiki/Harmonic_Mean_of_Divisors_in_terms_of_Divisor_Count_and_Divisor_Sum | https://proofwiki.org/wiki/Harmonic_Mean_of_Divisors_in_terms_of_Divisor_Count_and_Divisor_Sum | [
"Harmonic Mean",
"Divisor Sum Function",
"Divisor Count Function"
] | [
"Definition:Positive/Integer",
"Definition:Harmonic Mean",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor Count Function",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor Sum Function",
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Sum of Reciprocals of Divisors equals Abundancy Index"
] |
proofwiki-13184 | Perfect Number is Ore Number | Let $n \in \Z_{>0}$ be a perfect number.
Then $n$ is an Ore number. | From Harmonic Mean of Divisors in terms of Divisor Count and Divisor Sum, the harmonic mean of the divisors of $n$ is given by:
:$\map H n = \dfrac {n \map {\sigma_0} n} {\map {\sigma_1} n}$
where:
:$\map {\sigma_0} n$ denotes the divisor count function: the number of divisors of $n$
:$\map {\sigma_1} n$ denotes the di... | Let $n \in \Z_{>0}$ be a [[Definition:Perfect Number|perfect number]].
Then $n$ is an [[Definition:Ore Number|Ore number]]. | From [[Harmonic Mean of Divisors in terms of Divisor Count and Divisor Sum]], the [[Definition:Harmonic Mean|harmonic mean]] of the [[Definition:Divisor of Integer|divisors]] of $n$ is given by:
:$\map H n = \dfrac {n \map {\sigma_0} n} {\map {\sigma_1} n}$
where:
:$\map {\sigma_0} n$ denotes the [[Definition:Divisor C... | Perfect Number is Ore Number | https://proofwiki.org/wiki/Perfect_Number_is_Ore_Number | https://proofwiki.org/wiki/Perfect_Number_is_Ore_Number | [
"Ore Numbers",
"Perfect Numbers"
] | [
"Definition:Perfect Number",
"Definition:Ore Number"
] | [
"Harmonic Mean of Divisors in terms of Divisor Count and Divisor Sum",
"Definition:Harmonic Mean",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor Count Function",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor Sum Function",
"Definition:Addition/Integers",
"Definition:Divisor... |
proofwiki-13185 | Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors | The following sequence of integers have the property that both the harmonic mean and arithmetic mean of their divisors are integers:
:$1, 6, 140, 270, 672, \ldots$
{{OEIS|A007340}} | Note the integers whose harmonic mean of their divisors are integers are the Ore numbers:
:$1, 6, 28, 140, 270, 496, 672, \ldots$
{{OEIS|A001599}}
It remains to calculate the arithmetic mean of their divisors.
Let $\map A n$ denote the arithmetic mean of the divisors of $n$.
Then we have:
:$\map A n = \dfrac {\map {\si... | The following [[Definition:Integer Sequence|sequence]] of [[Definition:Integer|integers]] have the property that both the [[Definition:Harmonic Mean|harmonic mean]] and [[Definition:Arithmetic Mean|arithmetic mean]] of their [[Definition:Divisor of Integer|divisors]] are [[Definition:Integer|integers]]:
:$1, 6, 140, 27... | Note the [[Definition:Integer|integers]] whose [[Definition:Harmonic Mean|harmonic mean]] of their [[Definition:Divisor of Integer|divisors]] are [[Definition:Integer|integers]] are the [[Definition:Ore Number|Ore numbers]]:
:$1, 6, 28, 140, 270, 496, 672, \ldots$
{{OEIS|A001599}}
It remains to calculate the [[Definit... | Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors | https://proofwiki.org/wiki/Sequence_of_Numbers_with_Integer_Arithmetic_and_Harmonic_Means_of_Divisors | https://proofwiki.org/wiki/Sequence_of_Numbers_with_Integer_Arithmetic_and_Harmonic_Means_of_Divisors | [
"Ore Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Integer",
"Definition:Harmonic Mean",
"Definition:Arithmetic Mean",
"Definition:Divisor (Algebra)/Integer",
"Definition:Integer"
] | [
"Definition:Integer",
"Definition:Harmonic Mean",
"Definition:Divisor (Algebra)/Integer",
"Definition:Integer",
"Definition:Ore Number",
"Definition:Arithmetic Mean",
"Definition:Divisor (Algebra)/Integer",
"Definition:Arithmetic Mean",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor C... |
proofwiki-13186 | Carmichael's Theorem | Let $n \in \Z$ such that $n > 12$.
Then the $n$th Fibonacci number $F_n$ has at least one prime factor which does not divide any smaller Fibonacci number.
The exceptions for $n \le 12$ are:
:$F_1 = 1, F_2 = 1$: neither have any prime factors
:$F_6 = 8$ whose only prime factor is $2$ which is $F_3$
:$F_{12} = 144$ whose... | We have that:
:$1$ has no prime factors.
Hence, vacuously, $1$ has no primitive prime factors.
:$8 = 2^3$
and $2 \divides 2 = F_3$
:$144 = 2^4 3^2$
and:
:$2 \divides 8 = F_6$
:$3 \divides 21 = F_8$
for example.
{{ProofWanted|It remains to be shown that these are the only examples.}} | Let $n \in \Z$ such that $n > 12$.
Then the $n$th [[Definition:Fibonacci Numbers|Fibonacci number]] $F_n$ has at least one [[Definition:Prime Factor|prime factor]] which does not [[Definition:Divisor of Integer|divide]] any smaller [[Definition:Fibonacci Numbers|Fibonacci number]].
The exceptions for $n \le 12$ are:
... | We have that:
:$1$ has no [[Definition:Prime Factor|prime factors]].
Hence, [[Definition:Vacuous Truth|vacuously]], $1$ has no [[Definition:Primitive Prime Factor|primitive prime factors]].
:$8 = 2^3$
and $2 \divides 2 = F_3$
:$144 = 2^4 3^2$
and:
:$2 \divides 8 = F_6$
:$3 \divides 21 = F_8$
for example.
{{Proof... | Carmichael's Theorem | https://proofwiki.org/wiki/Carmichael's_Theorem | https://proofwiki.org/wiki/Carmichael's_Theorem | [
"Fibonacci Numbers",
"Primitive Prime Factors"
] | [
"Definition:Fibonacci Number",
"Definition:Prime Factor",
"Definition:Divisor (Algebra)/Integer",
"Definition:Fibonacci Number",
"Definition:Prime Factor",
"Definition:Prime Factor",
"Definition:Prime Factor"
] | [
"Definition:Prime Factor",
"Definition:Vacuous Truth",
"Definition:Primitive Prime Factor"
] |
proofwiki-13187 | Weierstrass's Necessary Condition | Let $\mathbf y: \R \to \R^n$ be an $n$-dimensional vector-valued function such that $\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$.
Let $J$ be a functional such that:
:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
Let $\mathbf w$ be an $n$-dimensional vector such that $\mathbf w \... | {{ProofWanted}}
{{Namedfor|Karl Theodor Wilhelm Weierstrass|cat = Weierstrass}} | Let $\mathbf y: \R \to \R^n$ be an [[Definition:Dimension|$n$-dimensional]] [[Definition:Vector-Valued Function|vector-valued function]] such that $\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$.
Let $J$ be a [[Definition:Real Functional|functional]] such that:
:$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \... | {{ProofWanted}}
{{Namedfor|Karl Theodor Wilhelm Weierstrass|cat = Weierstrass}} | Weierstrass's Necessary Condition | https://proofwiki.org/wiki/Weierstrass's_Necessary_Condition | https://proofwiki.org/wiki/Weierstrass's_Necessary_Condition | [
"Calculus of Variations"
] | [
"Definition:Dimension",
"Definition:Vector-Valued Function",
"Definition:Functional/Real",
"Definition:Dimension",
"Definition:Vector",
"Definition:Minimum Value of Functional",
"Definition:Weierstrass E-Function"
] | [] |
proofwiki-13188 | Squares Ending in Repeated Digits | A square number $n^2$ can end in a repeated digit {{iff}} either:
:$(1): \quad n^2$ is a multiple of $100$, in which case $n$ is a multiple of $10$
:$(2): \quad n^2$ ends in $44$ and $n$ ends in $12, 38, 62$ or $88$. | Let $n \in \Z_{>0}$ end in $a b$.
By the Basis Representation Theorem, $n$ can be expressed as:
:$n = 100 k + 10 a + b$
for some $k \in \Z_{>0}$ and for $0 \le a < 10, 0 \le b < 10$.
Then:
{{begin-eqn}}
{{eqn | l = n^2
| r = \paren {100k + 10 a + b}^2
| c =
}}
{{eqn | r = 100^2 k^2 + 2000 k a + 200 k b + 1... | A [[Definition:Square Number|square number]] $n^2$ can end in a repeated [[Definition:Digit|digit]] {{iff}} either:
:$(1): \quad n^2$ is a [[Definition:Integer Multiple|multiple]] of $100$, in which case $n$ is a [[Definition:Integer Multiple|multiple]] of $10$
:$(2): \quad n^2$ ends in $44$ and $n$ ends in $12, 38, 6... | Let $n \in \Z_{>0}$ end in $a b$.
By the [[Basis Representation Theorem]], $n$ can be expressed as:
:$n = 100 k + 10 a + b$
for some $k \in \Z_{>0}$ and for $0 \le a < 10, 0 \le b < 10$.
Then:
{{begin-eqn}}
{{eqn | l = n^2
| r = \paren {100k + 10 a + b}^2
| c =
}}
{{eqn | r = 100^2 k^2 + 2000 k a + 200 ... | Squares Ending in Repeated Digits | https://proofwiki.org/wiki/Squares_Ending_in_Repeated_Digits | https://proofwiki.org/wiki/Squares_Ending_in_Repeated_Digits | [
"Square Numbers"
] | [
"Definition:Square Number",
"Definition:Digit",
"Definition:Integral Multiple/Real Numbers",
"Definition:Integral Multiple/Real Numbers"
] | [
"Basis Representation Theorem",
"Definition:Digit",
"Definition:Digit",
"Definition:Digit",
"Definition:Square/Function",
"Definition:Integral Multiple/Real Numbers",
"Definition:Integral Multiple/Real Numbers",
"Definition:Square/Function",
"Definition:Digit",
"Definition:Square/Function",
"Def... |
proofwiki-13189 | Magic Constant of Smallest Prime Magic Square with Consecutive Primes | The magic constant of the smallest prime magic square whose elements are consecutive odd primes is $4 \, 440 \, 084 \, 513$. | The smallest prime magic square whose elements are consecutive odd primes is:
{{:Prime Magic Square/Examples/Order 3/Smallest with Consecutive Primes}}
As can be seen by inspection, the sums of the elements in the rows, columns and Diagonal of Array is $4 \, 440 \, 084 \, 513$:
{{begin-eqn}}
{{eqn | l = 1 \, 480 \, 028... | The [[Definition:Magic Constant|magic constant]] of the smallest [[Definition:Prime Magic Square|prime magic square]] whose [[Definition:Element of Array|elements]] are consecutive [[Definition:Odd Prime|odd primes]] is $4 \, 440 \, 084 \, 513$. | The smallest [[Definition:Prime Magic Square|prime magic square]] whose [[Definition:Element of Array|elements]] are consecutive [[Definition:Odd Prime|odd primes]] is:
{{:Prime Magic Square/Examples/Order 3/Smallest with Consecutive Primes}}
As can be seen by inspection, the [[Definition:Integer Addition|sums]] of th... | Magic Constant of Smallest Prime Magic Square with Consecutive Primes | https://proofwiki.org/wiki/Magic_Constant_of_Smallest_Prime_Magic_Square_with_Consecutive_Primes | https://proofwiki.org/wiki/Magic_Constant_of_Smallest_Prime_Magic_Square_with_Consecutive_Primes | [
"Prime Magic Squares",
"4,440,084,513"
] | [
"Definition:Magic Square/Magic Constant",
"Definition:Prime Magic Square",
"Definition:Array/Element",
"Definition:Odd Prime"
] | [
"Definition:Prime Magic Square",
"Definition:Array/Element",
"Definition:Odd Prime",
"Definition:Addition/Integers",
"Definition:Array/Element",
"Definition:Array/Row",
"Definition:Array/Column",
"Definition:Array/Diagonal",
"Category:Prime Magic Squares",
"Category:4,440,084,513"
] |
proofwiki-13190 | Magic Constant of Smallest Prime Magic Square with Consecutive Primes from 3 | The magic constant of the smallest prime magic square whose elements are consecutive odd primes from $3$ upwards is $4514$. | The smallest prime magic square whose elements are the first consecutive odd primes is:
{{:Prime Magic Square/Examples/Order 12/Smallest with Consecutive Primes from 3}}
The fact that this is the smallest is proven here.
The sum of the first $144$ prime numbers can either be calculated or looked up: it is $54 \, 169$.
... | The [[Definition:Magic Constant|magic constant]] of the smallest [[Definition:Prime Magic Square|prime magic square]] whose [[Definition:Element of Array|elements]] are consecutive [[Definition:Odd Prime|odd primes]] from $3$ upwards is $4514$. | The smallest [[Definition:Prime Magic Square|prime magic square]] whose [[Definition:Element of Array|elements]] are the first consecutive [[Definition:Odd Prime|odd primes]] is:
{{:Prime Magic Square/Examples/Order 12/Smallest with Consecutive Primes from 3}}
The fact that this is the smallest is proven [[Prime Magic ... | Magic Constant of Smallest Prime Magic Square with Consecutive Primes from 3 | https://proofwiki.org/wiki/Magic_Constant_of_Smallest_Prime_Magic_Square_with_Consecutive_Primes_from_3 | https://proofwiki.org/wiki/Magic_Constant_of_Smallest_Prime_Magic_Square_with_Consecutive_Primes_from_3 | [
"Prime Magic Squares",
"4514"
] | [
"Definition:Magic Square/Magic Constant",
"Definition:Prime Magic Square",
"Definition:Array/Element",
"Definition:Odd Prime"
] | [
"Definition:Prime Magic Square",
"Definition:Array/Element",
"Definition:Odd Prime",
"Prime Magic Square/Examples/Order 12/Smallest with Consecutive Primes from 3",
"Definition:Addition/Integers",
"Definition:Prime Number",
"Definition:Prime Magic Square",
"Definition:Array/Element",
"Definition:Pri... |
proofwiki-13191 | Factorions Base 10 | The following positive integers are the only factorions base $10$:
:$1, 2, 145, 40 \, 585$ | From examples of factorials:
{{begin-eqn}}
{{eqn | l = 1
| r = 1!
| c =
}}
{{eqn | l = 2
| r = 2!
| c =
}}
{{eqn|| l = 145
| r = 1 + 24 + 120
| c =
}}
{{eqn | r = 1! + 4! + 5!
| c =
}}
{{eqn|| l = 40 \, 585
| r = 24 + 1 + 120 + 40 \, 320 + 120
| c =
}}
{{eqn | ... | The following [[Definition:Positive Integer|positive integers]] are the only [[Definition:Factorion|factorions base $10$]]:
:$1, 2, 145, 40 \, 585$ | From [[Factorial/Examples|examples of factorials]]:
{{begin-eqn}}
{{eqn | l = 1
| r = 1!
| c =
}}
{{eqn | l = 2
| r = 2!
| c =
}}
{{eqn|| l = 145
| r = 1 + 24 + 120
| c =
}}
{{eqn | r = 1! + 4! + 5!
| c =
}}
{{eqn|| l = 40 \, 585
| r = 24 + 1 + 120 + 40 \, 320 + 120
... | Factorions Base 10 | https://proofwiki.org/wiki/Factorions_Base_10 | https://proofwiki.org/wiki/Factorions_Base_10 | [
"Factorions"
] | [
"Definition:Positive/Integer",
"Definition:Factorion"
] | [
"Factorial/Examples",
"Definition:Digit",
"Definition:Addition/Integers",
"Definition:Factorial",
"Definition:Digit",
"Definition:Factorion",
"Definition:Digit",
"Definition:Addition/Integers",
"Definition:Factorial",
"Definition:Digit",
"Bernoulli's Inequality",
"Definition:Factorion"
] |
proofwiki-13192 | Sum of 2 Squares in 2 Distinct Ways/Examples/145 | $145$ can be expressed as the sum of two square numbers in two distinct ways:
{{begin-eqn}}
{{eqn | l = 145
| r = 12^2 + 1^2
}}
{{eqn | r = 9^2 + 8^2
}}
{{end-eqn}} | We have that:
:$145 = 5 \times 29$
Both $5$ and $29$ can be expressed as the sum of two distinct square numbers:
{{begin-eqn}}
{{eqn | l = 5
| r = 1^2 + 2^2
}}
{{eqn | l = 29
| r = 2^2 + 5^2
}}
{{end-eqn}}
Thus:
{{begin-eqn}}
{{eqn | r = \paren {1^2 + 2^2} \paren {2^2 + 5^2}
| c =
}}
{{eqn | r = \par... | $145$ can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|square numbers]] in two [[Definition:Distinct|distinct]] ways:
{{begin-eqn}}
{{eqn | l = 145
| r = 12^2 + 1^2
}}
{{eqn | r = 9^2 + 8^2
}}
{{end-eqn}} | We have that:
:$145 = 5 \times 29$
Both $5$ and $29$ can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Distinct|distinct]] [[Definition:Square Number|square numbers]]:
{{begin-eqn}}
{{eqn | l = 5
| r = 1^2 + 2^2
}}
{{eqn | l = 29
| r = 2^2 + 5^2
}}
{{end-eqn}}
Thus:
{{begi... | Sum of 2 Squares in 2 Distinct Ways/Examples/145 | https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways/Examples/145 | https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways/Examples/145 | [
"Sum of 2 Squares in 2 Distinct Ways",
"145"
] | [
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Distinct"
] | [
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Square Number",
"Brahmagupta-Fibonacci Identity"
] |
proofwiki-13193 | Representation of 1 as Sum of n Unit Fractions | Let $U \left({n}\right)$ denote the number of different ways of representing $1$ as the sum of $n$ unit fractions.
Then for various $n$, $U \left({n}\right)$ is given by the following table:
:{| border="1"
|-
! align="right" style = "padding: 2px 10px" | $n$
! align="right" style = "padding: 2px 10px" | $U \left({n}\r... | Trivially:
:$1 = \dfrac 1 1$
and it follows that:
$U \left({1}\right) = 1$
Also trivially:
:$1 = \dfrac 1 2 + \dfrac 1 2$
and it follows that:
$U \left({2}\right) = 1$
From Sum of 3 Unit Fractions that equals 1:
:$U \left({3}\right) = 3$
From Sum of 4 Unit Fractions that equals 1:
:$U \left({4}\right) = 14$
From Sum of... | Let $U \left({n}\right)$ denote the number of different ways of representing $1$ as the [[Definition:Rational Addition|sum]] of $n$ [[Definition:Unit Fraction|unit fractions]].
Then for various $n$, $U \left({n}\right)$ is given by the following table:
:{| border="1"
|-
! align="right" style = "padding: 2px 10px" | $... | Trivially:
:$1 = \dfrac 1 1$
and it follows that:
$U \left({1}\right) = 1$
Also trivially:
:$1 = \dfrac 1 2 + \dfrac 1 2$
and it follows that:
$U \left({2}\right) = 1$
From [[Sum of 3 Unit Fractions that equals 1]]:
:$U \left({3}\right) = 3$
From [[Sum of 4 Unit Fractions that equals 1]]:
:$U \left({4}\right) = 1... | Representation of 1 as Sum of n Unit Fractions | https://proofwiki.org/wiki/Representation_of_1_as_Sum_of_n_Unit_Fractions | https://proofwiki.org/wiki/Representation_of_1_as_Sum_of_n_Unit_Fractions | [
"Unit Fractions"
] | [
"Definition:Addition/Rational Numbers",
"Definition:Unit Fraction"
] | [
"Sum of 3 Unit Fractions that equals 1",
"Sum of 4 Unit Fractions that equals 1",
"Sum of 5 Unit Fractions that equals 1"
] |
proofwiki-13194 | Sum of 3 Unit Fractions that equals 1 | There are $3$ ways to represent $1$ as the sum of exactly $3$ unit fractions. | Let:
:$1 = \dfrac 1 a + \dfrac 1 b + \dfrac 1 c$
where:
:$0 < a \le b \le c$
and:
{{AimForCont}} $a = 1$.
Then:
:$1 = \dfrac 1 1 + \dfrac 1 b + \dfrac 1 c$
and so:
:$\dfrac 1 b + \dfrac 1 c = 0$
which contradicts the stipulation that $b, c > 0$.
So there is no solution possible when $a = 1$.
Therefore $a \ge 2$. | There are $3$ ways to represent $1$ as the sum of exactly $3$ [[Definition:Unit Fraction|unit fractions]]. | Let:
:$1 = \dfrac 1 a + \dfrac 1 b + \dfrac 1 c$
where:
:$0 < a \le b \le c$
and:
{{AimForCont}} $a = 1$.
Then:
:$1 = \dfrac 1 1 + \dfrac 1 b + \dfrac 1 c$
and so:
:$\dfrac 1 b + \dfrac 1 c = 0$
which [[Definition:Contradiction|contradicts]] the stipulation that $b, c > 0$.
So there is no solution possible when $a ... | Sum of 3 Unit Fractions that equals 1 | https://proofwiki.org/wiki/Sum_of_3_Unit_Fractions_that_equals_1 | https://proofwiki.org/wiki/Sum_of_3_Unit_Fractions_that_equals_1 | [
"Unit Fractions",
"Recreational Mathematics"
] | [
"Definition:Unit Fraction"
] | [
"Definition:Contradiction"
] |
proofwiki-13195 | Necessary Condition for Integral Functional to have Extremum for given function/Dependent on n Variables | Let $\mathbf x$ be an $n$-dimensional vector.
Let $\map u {\mathbf x}$ be a real function.
Let $R$ be a fixed region.
Let $J$ be a functional such that
:$\ds J \sqbrk u = \idotsint_R \map F {\mathbf x, u, u_{\mathbf x} } \rd x_1 \cdots \rd x_n$
Then a necessary condition for $J \sqbrk u$ to have an extremum (strong or ... | By definition of increment of the functional:
:$\ds J \sqbrk {u + h} - J \sqbrk u = \idotsint_R \paren {F \sqbrk {x, u + h, u_{\mathbf x} + h_{\mathbf x} } - F \sqbrk {x, u, u_{\mathbf x} } } \rd x_1 \cdots \rd x_n$
Use multivariate Taylor's Theorem on $F$ around the point $\tuple {\mathbf x, u, u_{\mathbf x} }$:
:$F \... | Let $\mathbf x$ be an $n$-dimensional vector.
Let $\map u {\mathbf x}$ be a real function.
Let $R$ be a fixed [[Definition:Region|region]].
Let $J$ be a [[Definition:Real Functional|functional]] such that
:$\ds J \sqbrk u = \idotsint_R \map F {\mathbf x, u, u_{\mathbf x} } \rd x_1 \cdots \rd x_n$
Then a [[Definit... | By definition of [[Definition:Increment of Functional|increment of the functional]]:
:$\ds J \sqbrk {u + h} - J \sqbrk u = \idotsint_R \paren {F \sqbrk {x, u + h, u_{\mathbf x} + h_{\mathbf x} } - F \sqbrk {x, u, u_{\mathbf x} } } \rd x_1 \cdots \rd x_n$
Use multivariate [[Taylor's Theorem]] on $F$ around the point ... | Necessary Condition for Integral Functional to have Extremum for given function/Dependent on n Variables | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum_for_given_function/Dependent_on_n_Variables | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum_for_given_function/Dependent_on_n_Variables | [
"Calculus of Variations"
] | [
"Definition:Region",
"Definition:Functional/Real",
"Definition:Conditional/Necessary Condition",
"Definition:Extremum/Functional",
"Definition:Mapping",
"Definition:Euler's Equation for Vanishing Variation"
] | [
"Definition:Increment/Functional",
"Taylor's Theorem",
"Definition:Big-O Notation",
"Definition:Differentiable Functional",
"Green's Theorem",
"Definition:Boundary (Topology)",
"Definition:Region",
"Definition:Boundary (Topology)",
"Definition:Point",
"Definition:Boundary (Topology)",
"Definitio... |
proofwiki-13196 | Repeated Sum of Cubes of Digits of Multiple of 3 | Let $k \in \Z_{>0}$ be a positive integer.
Let $f: \Z_{>0} \to \Z_{>0}$ be the mapping defined as:
:$\forall m \in \Z_{>0}: \map f m = $ the sum of the cubes of the digits of $n$.
Let $n_0 \in \Z_{>0}$ be a (strictly) positive integer which is a multiple of $3$.
Consider the sequence:
:$s_n = \begin{cases} n_0 & : n = ... | We verify by brute force:
:Starting on $n_0 \le 2916 = 4 \times 9^3$, we will end on $153$.
{{qed|lemma}}
First we prove that if $3 \divides n_0$, then $3 \divides \map f {n_0}$.
From Divisibility by 3:
The sum of digits of $n_0$ is divisible by $3$.
By Fermat's Little Theorem:
:$\forall x \in \Z: x^3 \equiv x \pmod 3$... | Let $k \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]].
Let $f: \Z_{>0} \to \Z_{>0}$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall m \in \Z_{>0}: \map f m = $ the [[Definition:Integer Addition|sum]] of the [[Definition:Cube (Algebra)|cubes]] of the [[Definition:Digit|digits]] of $n$.
... | We verify by brute force:
:Starting on $n_0 \le 2916 = 4 \times 9^3$, we will end on $153$.
{{qed|lemma}}
First we prove that if $3 \divides n_0$, then $3 \divides \map f {n_0}$.
From [[Divisibility by 3]]:
The [[Definition:Integer Addition|sum]] of [[Definition:Digit|digits]] of $n_0$ is [[Definition:Divisor of ... | Repeated Sum of Cubes of Digits of Multiple of 3 | https://proofwiki.org/wiki/Repeated_Sum_of_Cubes_of_Digits_of_Multiple_of_3 | https://proofwiki.org/wiki/Repeated_Sum_of_Cubes_of_Digits_of_Multiple_of_3 | [
"Recreational Mathematics",
"Pluperfect Digital Invariants",
"Proofs by Induction"
] | [
"Definition:Positive/Integer",
"Definition:Mapping",
"Definition:Addition/Integers",
"Definition:Cube/Algebra",
"Definition:Digit",
"Definition:Strictly Positive/Integer",
"Definition:Multiple/Integer",
"Definition:Multiple/Integer",
"Definition:Pluperfect Digital Invariant"
] | [
"Divisibility by 9/Corollary",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Divisor (Algebra)/Integer",
"Fermat's Little Theorem",
"Definition:Addition/Integers",
"Definition:Cube",
"Definition:Digit",
"Definition:Divisor (Algebra)/Integer",
"Definition:Digit",
"Definition:Dig... |
proofwiki-13197 | Numbers for which Euler Phi Function of 2n + 1 is less than that of 2n | The sequence of positive integers for which:
:$\map \phi {2 n + 1} < \map \phi {2 n}$
begins:
:$157, 262, 367, 412, \ldots$
{{OEIS|A001837}} | {{begin-eqn}}
{{eqn | l = 2 \times 157
| r = 314
| c =
}}
{{eqn | ll= \leadsto
| l = \map \phi {2 \times 157}
| r = 156
| c = {{EulerPhiLink|314}}
}}
{{eqn | l = 2 \times 157 + 1
| r = 315
| c =
}}
{{eqn | ll= \leadsto
| l = \map \phi {2 \times 157 + 1}
| r = 144
... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] for which:
:$\map \phi {2 n + 1} < \map \phi {2 n}$
begins:
:$157, 262, 367, 412, \ldots$
{{OEIS|A001837}} | {{begin-eqn}}
{{eqn | l = 2 \times 157
| r = 314
| c =
}}
{{eqn | ll= \leadsto
| l = \map \phi {2 \times 157}
| r = 156
| c = {{EulerPhiLink|314}}
}}
{{eqn | l = 2 \times 157 + 1
| r = 315
| c =
}}
{{eqn | ll= \leadsto
| l = \map \phi {2 \times 157 + 1}
| r = 144
... | Numbers for which Euler Phi Function of 2n + 1 is less than that of 2n | https://proofwiki.org/wiki/Numbers_for_which_Euler_Phi_Function_of_2n_+_1_is_less_than_that_of_2n | https://proofwiki.org/wiki/Numbers_for_which_Euler_Phi_Function_of_2n_+_1_is_less_than_that_of_2n | [
"Euler Phi Function"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer"
] | [] |
proofwiki-13198 | Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares | The largest odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers with greatest common divisor $1$ is $157$.
The full sequence of such odd positive integers which cannot be so expressed is:
:$1, \ldots, 37, 41, 43, 45, 47, 49, 53, 55, 59, 61, 67, 69, 73, 77, 83, 89, 97... | The statement of the result was taken from the paper cited below by {{AuthorRef|Paul T. Bateman}} and others:
:''The largest odd integer not expressible as a sum of $4$ distinct non-zero squares with greatest common divisor $1$ is $157$.''
By a brute force exercise, we assemble all the sets of $4$ distinct non-zero int... | The largest [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive integer]] that cannot be expressed as the [[Definition:Integer Addition|sum]] of exactly $4$ [[Definition:Distinct Elements|distinct]] non-zero [[Definition:Square Number|square numbers]] with [[Definition:Greatest Common Divisor of Set o... | The statement of the result was taken from the paper cited below by {{AuthorRef|Paul T. Bateman}} and others:
:''The largest [[Definition:Odd Integer|odd integer]] not expressible as a [[Definition:Integer Addition|sum]] of $4$ [[Definition:Distinct Elements|distinct]] non-zero [[Definition:Square Number|squares]] with... | Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares | https://proofwiki.org/wiki/Odd_Numbers_Not_Expressible_as_Sum_of_4_Distinct_Non-Zero_Coprime_Squares | https://proofwiki.org/wiki/Odd_Numbers_Not_Expressible_as_Sum_of_4_Distinct_Non-Zero_Coprime_Squares | [
"Sums of Squares",
"157"
] | [
"Definition:Odd Integer",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Distinct/Plural",
"Definition:Square Number",
"Definition:Greatest Common Divisor/Integers/General Definition",
"Definition:Integer Sequence",
"Definition:Odd Integer",
"Definition:Positive/Integer",... | [
"Definition:Odd Integer",
"Definition:Addition/Integers",
"Definition:Distinct/Plural",
"Definition:Square Number",
"Definition:Greatest Common Divisor/Integers/General Definition",
"Definition:Set",
"Definition:Distinct/Plural",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Sq... |
proofwiki-13199 | 159 is not Expressible as Sum of Fewer than 19 Fourth Powers | :$159 = 14 \times 1^4 + 4 \times 2^4 + 3^4$ | We have:
:$4^4 = 256 > 159$
:$3^4 = 81$
:$2^4 = 16$
:$1^4 = 1$
Let us attempt to construct an expression of $159$ as the sum of fewer than $19$ fourth powers:
If no $3^4$ is used in our sum, the sum consists only of $2^4$ and $1^4$.
Using $2^4$ is more efficient than using $1^4$, since $2^4$ can replace $16 \times 1^4$... | :$159 = 14 \times 1^4 + 4 \times 2^4 + 3^4$ | We have:
:$4^4 = 256 > 159$
:$3^4 = 81$
:$2^4 = 16$
:$1^4 = 1$
Let us attempt to construct an expression of $159$ as the [[Definition:Integer Addition|sum]] of fewer than $19$ [[Definition:Fourth Power|fourth powers]]:
If no $3^4$ is used in our [[Definition:Integer Addition|sum]], the [[Definition:Integer Addition... | 159 is not Expressible as Sum of Fewer than 19 Fourth Powers | https://proofwiki.org/wiki/159_is_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | https://proofwiki.org/wiki/159_is_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | [
"Fourth Powers",
"Hilbert-Waring Theorem",
"159"
] | [] | [
"Definition:Addition/Integers",
"Definition:Fourth Power",
"Definition:Addition/Integers",
"Definition:Addition/Integers",
"Definition:Fourth Power",
"Definition:Addition/Integers",
"Definition:Fourth Power"
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.