hcju/proofwikips huggingface dataset:
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 $b - 1$: :$\dfrac 1 n = \sqbrk {0.012 \ldots cd012 \ldots}_b$ where: :$c = b - 3$ :$d = b - 1$
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 $c = b - 2$.
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.
Closed Form for Heptagonal Numbers
The closed-form expression for the $n$th heptagonal number is: :$H_n = \dfrac {n \paren {5 n - 3} } 2$
Square Numbers whose Sigma is Square
The sequence of square numbers whose $\sigma$ value is square starts as follows: {{begin-eqn}} {{eqn | l = \map \sigma {1^2}       | r = 1^2       | c =  }} {{eqn | l = \map \sigma {9^2}       | r = 11^2       | c =  }} {{eqn | l = \map \sigma {20^2}       | r = 31^2       | c =  }} {{eqn | l = \map \sigma {180^2}       | r = 341^2       | c =  }} {{eqn | l = \map \sigma {1306^2}       | r = 1729^2       | c =  }} {{end-eqn}} {{OEIS|A008847}}
Integers for which Sigma of Phi equals Sigma
The following positive integers have the property that the $\sigma$ value of their Euler $\phi$ value equals their $\sigma$ value: :$\map \sigma {\map \phi n} = \map \sigma n$ :$1, 87, 362, 1257, 1798, 5002, 9374, \ldots$ {{OEIS|A033631}}
Consecutive Triplets not Sum of Pentagonal Numbers
The following triplets of consecutive positive integers are such that none is the sum of $3$ pentagonal numbers: :$\tuple {19, 20, 21}$ :$\tuple {88, 89, 90}$ :$\tuple {99, 100, 101}$ :$\tuple {111, 112, 113}$
Prime Gaps of 8
The following pairs of consecutive prime numbers are those whose difference is $8$: :$\left({89, 97}\right), \left({359, 367}\right), \left({389, 397}\right), \left({401, 409}\right), \ldots$ {{OEIS|A031926}}
Smallest Cunningham Chain of the First Kind of Length 6
The smallest Cunningham chain of the first kind of length $6$ is: :$\tuple {89, 179, 359, 719, 1439, 2879}$
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: :$n_k = \begin{cases} m & : n = 1 \\ \map f {n_{k - 1} } & : n > 1 \end{cases}$ Then eventually either $\sequence {S_m}$ sticks at $1$, or goes into the cycle: :$\ldots, 4, 16, 37, 58, 89, 145, 42, 20, 4, \ldots$
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 $24$ iterations.
Cauchy Sequence is Bounded/Real Numbers
Every Cauchy sequence in $\R$ is bounded.
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: :$91 = 7 \times 13$
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.}}
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 $J$.
Continued Fraction Expansion of Fourth Power of Pi
The continued fraction expansion of the $4$th power of $\pi$ begins: :$\pi^4 \approx 97 + \cfrac 1 {2 + \cfrac 1 {2 + \cfrac 1 {3 + \cfrac 1 {1 + \cfrac 1 {16 \, 539} } } } }$ {{OEIS|A058286}}
Decimal Expansion of Fourth Power of Pi
The decimal expansion of the $4$th power of $\pi$ begins: :$97 \cdotp 40909 \, 10340 \, 0 \ldots$ {{OEIS|A092425}}
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}}
Repdigit Number consisting of Instances of 9 is Kaprekar
A repdigit number that consists entirely of the digit $9$ is a Kaprekar number.
Numbers equal to Sum of Primes not Greater than its Prime Counting Function Value
Let $\pi \left({n}\right): \Z_{\ge 0} \to \Z_{\ge 0}$ denote the prime-counting function: :$\pi \left({n}\right) =$ the count of the primes less than $n$ Consider the equation: :$n = \displaystyle \sum_{p \mathop \le \pi \left({n}\right)} p$ where $p \le \pi \left({n}\right)$ denotes the primes not greater than $\pi \left({n}\right)$. Then $n$ is one of: :$5, 17, 41, 77, 100$ {{OEIS|A091864}}
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 $102^7$: :$102^7 = 12^7 + 35^7 + 53^7 + 58^7 + 64^7 + 83^7 + 85^7 + 90^7$
Consecutive Integers with Same Euler Phi Value
Let $\phi: \Z_{>0} \to \Z_{>0}$ be the Euler $\phi$ function, defined on the strictly positive integers. The equation: :$\map \phi n = \map \phi {n + 1}$ is satisfied by integers in the sequence: :$1, 3, 15, 104, 164, 194, 255, 495, 584, 975, 2204, \ldots$ {{OEIS|A001274}}
Sequences of 3 Consecutive Integers with Rising Phi
The following ordered triples of consecutive integers have $\phi$ values which are strictly increasing: :$105, 106, 107$ :$165, 166, 167$ :$315, 316, 317$
Integers whose Sigma equals Half Phi times Tau
The following positive integers $n$ have the property where: :$\sigma \left({n}\right) = \dfrac {\phi \left({n}\right) \times \tau \left({n}\right)} 2$ where: :$\sigma \left({n}\right)$ denotes the $\sigma$ function: the sum of the divisors of $n$ :$\phi \left({n}\right)$ denotes the Euler $\phi$ function: the count of positive integers smaller than of $n$ which are coprime to $n$ :$\tau \left({n}\right)$ denotes the $\tau$ function: the count of the divisors of $n$: These positive integers are: :$35, 105, \ldots$
Reciprocals of Odd Numbers adding to 1
$105$ is the smallest positive integer $n$ such that $1$ can be expressed as the sum of reciprocals of distinct odd integers such that none are less than $\dfrac 1 n$: :$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {33} + \dfrac 1 {35} + \dfrac 1 {45} + \dfrac 1 {55} + \dfrac 1 {77} + \dfrac 1 {105}$ {{OEIS|A238795}} There are $5$ ways of expressing $1$ as the sum of reciprocals of only $9$ distinct odd integers, but then the least term is less than $\dfrac 1 {105}$. The $9$-term solutions are as follows: :$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {35} + \dfrac 1 {45} + \dfrac 1 {231}$ {{OEIS|A201644}} :$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {21} + \dfrac 1 {231} + \dfrac 1 {315}$ {{OEIS|A201648}} :$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {33} + \dfrac 1 {45} + \dfrac 1 {385}$ {{OEIS|A201649}} :$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {21} + \dfrac 1 {165} + \dfrac 1 {693}$ {{OEIS|A201647}} :$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {21} + \dfrac 1 {135} + \dfrac 1 {10 \, 395}$ {{OEIS|A201646}}
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
Integers whose Sigma Value is Cube
The following positive integers are those whose $\sigma$ value is a cube: :$1, 7, 102, 110, 142, 159, 187, 381, 690, 714, 770, 994, 1034, \ldots$ {{OEIS|A020477}}
Magic Constant of Smallest Prime Magic Square
The magic constant of the smallest prime magic square is $111$.
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}}
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: :$\forall x \in \closedint a b: \det \paren {F_{\mathbf y' \mathbf y'} } \ne 0$ Suppose no points conjugate to $\paren {a, \map {\mathbf y} a}$ lie on $\gamma$. Then $\gamma$ can be embedded in a field.
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 $370 \, 372$.
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 = 56^2 - 55^2       | r = 111       | c =  }} {{eqn | l = 156^2 - 115^2       | r = 11 \, 111       | c =  }} {{eqn | l = 340^2 - 67^2       | r = 111 \, 111       | c =  }} {{eqn | l = 344^2 - 65^2       | r = 111 \, 111       | c =  }} {{eqn | l = 356^2 - 125^2       | r = 111 \, 111       | c =  }} {{end-eqn}}
Smallest Equilateral Triangle with Internal Point at Integer Distances from Vertices
The smallest equilateral triangle with sides of integer length which contains a point which is an integer distance from each vertex has a side length $112$: :500px There exists a point inside it which is $57$, $65$ and $73$ away from the three vertices.
Smallest 3-Digit Permutable Prime
The smallest $3$-digit permutable prime is $113$.
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.
Group is Finite iff Finite Number of Subgroups
Let $\struct {G, \circ}$ be a group. Then $G$ is finite  $\struct {G, \circ}$ has a  finite number of subgroups.
3-Digit Permutable Primes
The $3$-digit permutable primes are: :$311, 199, 337$ and their anagrams, and no other.
Digits of Permutable Prime
Let $p$ be a permutable prime with more than $1$ digit. Then $p$ contains only digits from the set: :$\left\{ {1, 3, 7, 9}\right\}$
Divisibility by 5
An integer $N$ expressed in decimal notation is divisible by $5$  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 $5$  : :$a_0$ is divisible by $5$.
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 $b$).
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}}
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 $18$.
Number of Different Ways to Colour the Faces of Cube with 3 Colours
The number of different ways to colour the faces of a cube with $3$ given colours, one colour per face, is $114$.
Number of Different Ways to Colour the Faces of Cube with 4 Colours
The number of different ways to colour the faces of a cube with $4$ given colours, one colour per face, is $2652$.
Number of Different Ways to Colour the Faces of Cube with 5 Colours
The number of different ways to colour the faces of a cube with $5$ given colours, one colour per face, is $29 \, 660$.
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}}
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: :$4 m - 1, 4 m + 1, 6 m - 1, 6 m + 1$
Sum of Cubes of 5 Consecutive Integers which is Square
The following sequences of $5$ consecutive positive integers have cubes that sum to squares: :$1, 2, 3, 4, 5$ :$96, 96, 98, 99, 100$ :$118, 119, 120, 121, 122$ No other such sequence of $5$ consecutive positive integers has the same property.
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 $120$.
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 $20$.
Smallest Number with 16 Divisors
The smallest positive integer with $16$ divisors is $120$.
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 $n \in \Z$.
Fermat Set cannot be Extended to Diophantine Quintuple
The Fermat set $F = \left\{{1, 3, 8, 120}\right\}$ cannot be extended to a Diophantine quintuple.
Ratio of 360 to Aliquot Sum
$360$ has the property that its ratio to its aliquot sum is $4 : 9$.
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 $10^{108}$.
Are All Triperfect Numbers Even?/Progress/Prime Factors
An odd triperfect number has: :at least $11$ distinct prime factors :at least $32$ distinct prime factors if $3$ is not one of them.
Are All Triperfect Numbers Even?/Progress/Form
An odd triperfect number is square.
Squares which are 4 Less than Cubes
The only two square numbers which are $4$ less than a cube are: :$2^2 + 4 = 2^3$ :$11^2 + 4 = 5^3$
Integer Greater than 121 is Sum of Distinct Primes of form 4 n + 1
Let $n$ be an integer greater than $121$. Then $n$ can be expressed as the sum of distinct prime numbers of the form $4 n + 1$.
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}}
Square Number which is Sum of Consecutive Powers
The only square number which is the sum of consecutive powers of a positive integer is $121$: :$121 = 3^0 + 3^1 + 3^2 + 3^3 + 3^4$
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: :$\left\vert {5^3 - 2^7}\right\vert = 5 - 2$ :$\left\vert {13^3 - 3^7}\right\vert = 13 - 3$
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: :$6, 84, 126, 156$
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}}
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 $J$: :$\mathbf y' = \map {\boldsymbol \psi} {x, \mathbf y}$ Let $R$ be an open region containing $\gamma$ and have the field $\boldsymbol \psi$ defined as $\forall \paren {x, \mathbf y} \in R$. Let $\mathbf w$ be a finite vector. Suppose that: :$\forall \paren {x, \mathbf y} \in R: \map E {x, \mathbf y, \boldsymbol \psi, \mathbf w}\ge 0$ where $E$ is Weierstrass E-Function. Then $J$ has a strong minimum for $\gamma$. {{explain|What is a strong minimum?}}
Factor of Mersenne Number Mp is of form 2kp + 1
:$q = 2 k p + 1$ for some integer $k$.
Factor of Mersenne Number Mp equivalent to 1 mod p
:$q \equiv 1 \pmod p$
Factor of Mersenne Number equivalent to +-1 mod 8
:$q \equiv \pm 1 \pmod 8$
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}}
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}}
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.
Harmonic Mean of Divisors in terms of Tau and Sigma
Let $n \in \Z_{>0}$ be a positive integer. The harmonic mean of the divisors of $n$ is given by: :$H \left({n}\right) = \dfrac {n \, \tau \left({n}\right)} {\sigma \left({n}\right)}$ where: :$\tau \left({n}\right)$ denotes the $\tau$ (tau) function: the number of divisors of $n$ :$\sigma \left({n}\right)$ denotes the $\sigma$ (sigma) function: the sum of the divisors of $n$.
Perfect Number is Ore Number
Let $n \in \Z_{>0}$ be a perfect number. Then $n$ is an Ore number.
Square Fibonacci Number
After $1$, there exists exactly one Fibonacci number which is also square: :$F_{12} = 144 = 12^2$ which is also coincidentally the square of its index.
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 only prime factors are $2$ (which is $F_3$) and $3$ (which is $F_4$).
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 $\mathbf w \in \R^n$. Let $\gamma$ be a strong minimum of $J$. {{explain|exactly what a ''strong'' minimum is, by means of a link to a definition page}} Then along $\gamma$ and for every $\mathbf w$: :$\map E {x, \mathbf y, \mathbf y', \mathbf w} \ge 0$ {{explain|exactly what ''along'' $\gamma$ means, by means of a link to a definition page}} where $E$ stands for the Weierstrass E-Function.
Squares Ending in Repeated Digits
A square number $n^2$ can end in a repeated digit  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 $88$.
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 $4 \, 440 \, 084 \, 513$.
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 $4514$.
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}}
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" | $U \left({n}\right)$  |- | align="right" style = "padding: 2px 10px" | $1$ | align="right" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $2$ | align="right" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $3$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $14$ |- | align="right" style = "padding: 2px 10px" | $5$ | align="right" style = "padding: 2px 10px" | $147$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="right" style = "padding: 2px 10px" | $3462$ |} {{OEIS|A002966}}
Sum of 3 Unit Fractions that equals 1
There are $3$ ways to represent $1$ as the sum of exactly $3$ unit fractions.
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 weak) for a given mapping $\map u {\mathbf x}$ is that $\map u {\mathbf x}$ satisfies Euler's equation: :$F_u - \dfrac {\partial} {\partial \mathbf x} F_{u_{\mathbf x} } = 0$
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: :$s_n = \begin{cases} n_0 & : n = 0 \\ \map f {s_{n - 1} } & : n > 0 \end{cases}$ Then: :$\exists r \in \N_{>0}: s_r = 153$ That is, by performing $f$ repeatedly on a multiple of $3$ eventually results in the pluperfect digital invariant $153$.
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}}
Odd Numbers Not Expressible as Sum of 5 Distinct Non-Zero Coprime Squares
The largest odd positive integer that cannot be expressed as the sum of exactly $5$ non-zero square numbers all of which are coprime is $245$. {{expand|The full list of those numbers is to be investigated.}}
159 is not Expressible as Sum of Fewer than 19 Fourth Powers
:$159 = 14 \times 1^4 + 4 \times 2^4 + 3^4$
Numbers not Expressible as Sum of Distinct Pentagonal Numbers
The positive integer which cannot be expressed as the sum of distinct pentagonal numbers are: :$2, 3, 4, 7, 8, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30,$ :$31, 32, 33, 37, 38, 42, 43, 44, 45, 46, 49, 50, 54, 55, 59, 60, 61, 65,$ :$66, 67, 72, 77, 80, 81, 84, 89, 94, 95, 96, 100, 101, 102, 107, 112, 116,$ :$124, 136, 137, 141, 142, 147, 159$
Sum of Distinct Primes of form 6n-1
$161$ is the largest integer that cannot be expressed as the sum of distinct primes of the form $6 n - 1$.