hcju/proofwikips huggingface dataset:
Divisibility Test for 7, 11 and 13
Mark off the integer $N$ being tested into groups of $3$ digits. Because of the standard way of presenting integers, this may already be done, for example: :$N = 22 \, 846 \, 293 \, 462 \, 733 \, 356$ Number the groups of $3$ from the right:  :$N = \underbrace{22}_6 \, \underbrace{846}_5 \, \underbrace{293}_4 \, \underbrace{462}_3 \, \underbrace{733}_2 \, \underbrace{356}_1$ Considering each group a $3$-digit integer, add the even numbered groups together, and subtract the odd numbered groups: :$22 - 846 + 293 - 462 + 733 - 356 = -616$ where the sign is irrelevant. If the result is divisible by $7$, $11$ or $13$, then so is $N$. In this case: :$616 = 2^3 \times 7 \times 11$ and so $N$ is divisible by $7$ and $11$ but not $13$.
Solutions to Diophantine Equation x (x + 1) = y (y + 5) (y + 10) (y + 15)
The Diophantine equation : $n = x \left({x + 1}\right) = y \left({y + 5}\right) \left({y + 10}\right) \left({y + 15}\right)$ has exactly $2$ solutions: {{begin-eqn}} {{eqn | l = 1056       | r = 32 \times 33 = 1 \times 6 \times 11 \times 16 }} {{eqn | l = 43 \, 056       | r = 207 \times 208 = 8 \times 13 \times 18 \times 23 }} {{end-eqn}}
Numbers Reversed when Multiplying by 9
Numbers of the form $\sqbrk {10 (9) 89}_{10}$ are reversed when they are multiplied by $9$: {{begin-eqn}} {{eqn | l = 1089 \times 9       | r = 9801 }} {{eqn | l = 10 \, 989 \times 9       | r = 98 \, 901 }} {{eqn | l = 109 \, 989 \times 9       | r = 989 \, 901 }} {{end-eqn}} and so on.
Square which is Difference between Square and Square of Reversal
$33^2 = 65^2 - 56^2$ This is the only square of a $2$-digit number which has this property.
Numbers for which Sixth Power plus 1091 is Composite
The number $1091$ has the property that: :$x^6 + 1091$ is composite for all integer values of $x$ from $1$ to $3905$.
Wieferich's Criterion
Suppose Fermat's equation: :$x^p + y^p = z^p$ has a solution in which $p$ is an odd prime that does not divide any of $x$, $y$ or $z$. Then $2^{p - 1} - 1$ is divisible by $p^2$.
1105 as Sum of Two Squares
$1105$ can be expressed as the sum of two squares in more ways than any smaller integer: {{begin-eqn}} {{eqn | l = 1105       | m = 1089 + 16       | mo= =       | r = 33^2 + 4^2       | c =  }} {{eqn | m = 1024 + 81       | mo= =       | r = 32^2 + 9^2       | c =  }} {{eqn | m = 961 + 144       | mo= =       | r = 31^2 + 12^2       | c =  }} {{eqn | m = 625 + 529       | mo= =       | r = 24^2 + 23^2       | c =  }} {{end-eqn}}
Integer as Difference between Two Squares
Let $n$ be a positive integer. Then $n$ can be expressed as: :$n = a^2 - b^2$  $n$ has at least two distinct divisors of the same parity.
Difference between Two Squares equal to Repunit/Corollary 1
{{begin-eqn}} {{eqn | l = 6^2 - 5^2       | r = 11       | c =  }} {{eqn | l = 56^2 - 45^2       | r = 1111       | c =  }} {{eqn | l = 556^2 - 445^2       | r = 111 \, 111       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}} and in general for integer $n$: :$R_{2 n} = {\underbrace {55 \ldots 56}_{n - 1 \ 5 \text{'s} } }^2 - {\underbrace {44 \ldots 45}_{n - 1 \ 4 \text{'s} } }^2$ that is: :$\displaystyle \sum_{k \mathop = 0}^{2 n - 1} 10^k = \paren {\sum_{k \mathop = 1}^{n - 1} 5 \times 10^k + 6}^2 - \paren {\sum_{k \mathop = 1}^{n - 1} 4 \times 10^k + 5}^2$
Difference between Two Squares equal to Repunit/Corollary 2
{{begin-eqn}} {{eqn | l = 6^2 - 5^2       | r = 11       | c =  }} {{eqn | l = 56^2 - 45^2       | r = 1111       | c =  }} {{eqn | l = 5056^2 - 5045^2       | r = 111 \, 111       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}} and in general for integer $n$: :$R_{2 n} = {\underbrace{5050 \ldots 56}_{n - 1 \ 5 \text{'s} } }^2 - {\underbrace{5050 \ldots 45}_{n - 1 \ 5 \text{'s} } }^2$ that is: :$\displaystyle \sum_{k \mathop = 0}^{2 n - 1} 10^k = \left({\sum_{k \mathop = 1}^{n - 1} 5 \times 10^{2 k - 1} + 6}\right)^2 - \left({\sum_{k \mathop = 1}^{n - 1} 5 \times 10^{2 k - 1} - 5}\right)^2$
Numbers whose Squares are Consecutive Odd or Even Integers Juxtaposed
Integers whose squares consists of $2$ consecutive odd or even integers juxtaposed include: :$1127^2 = 01 \, 270 \, 129$ :$8874^2 = 78 \, 747 \, 876$ Such integers come in pairs which add to $1$ more than a power of $10$: :$1127 + 8874 = 10 \, 001$ {{expand|A great deal more to be done here}}
Sixth Power as Sum of 7 Sixth Powers
The smallest known integer whose $6$th power can be expressed as the sum of $7$ smaller $6$th powers is $1141$: :$1141^6 = 74^6 + 234^6 + 402^6 + 474^6 + 702^6 + 894^6 + 1077^6$
Numbers Not Expressible as Sum of no more than 5 Squares of Composite Numbers
There are $256$ integers which cannot be expressed as the sum of no more than $5$ squares of composite numbers: :$1, 2, 3, \ldots, 1167$
Product of Injective Spaces is Injective
Let $I$ be a non-empty set. Let $\left({\left({S_i, \tau_i}\right)}\right)_{i \in I}$ be an indexed family of injective topological spaces. Then $\displaystyle \prod_{i \mathop \in I} \left({S_i, \tau_i}\right)$ is injective space.
Retract of Injective Space is Injective
Let $T = \left({S, \tau}\right)$ be an injective topological space. Let $R = \left({Z, \tau'}\right)$ be a retract of $T$. Then $R$ is injective.
Numbers equal to Sum of Squares of two Parts
Integers that can be split into two parts whose squares add up to it include: :$1233 = 12^2 + 33^2$ :$8833 = 88^2 + 33^2$ {{expand|Need to establish the parameters of this}}
Triples of Consecutive Sphenic Numbers
The sequence of triplets of consecutive sphenic numbers starts: :$\tuple {1309, 1310, 1311}, \tuple {1885, 1886, 1887}, \tuple {2013, 2014, 2015}, \ldots$ {{OEIS|A066509|order = first}} {{OEIS|A248202|order = middle}}
Smallest Quadruplet of Consecutive Integers Divisible by Cube
The smallest sequence of quadruplets of consecutive integers each of which is divisible by a cube greater than $1$ is: :$\tuple {22 \, 624, 22 \, 625, 22 \, 626, 22 \, 627}$
Riemann Zeta Function as a Multiple Integral
For $n \in \Z_{> 0}$, the Riemann zeta function is given by: :$\displaystyle \map \zeta n = \int_{\closedint 0 1^n} \frac 1 {1 - \prod_{i \mathop = 1}^n x_i} \prod_{i \mathop = 1}^n \rd x_i$ where $\closedint 0 1^n$ denotes the Cartesian $n$th power of the closed real interval $\closedint 0 1$.
Closed Form for Hexagonal Pyramidal Numbers
The closed-form expression for the $n$th hexagonal pyramidal number is: :$S_n = \dfrac {n \paren {n + 1} \paren {4 n - 1} } 6$
Tetrahedral and Triangular Numbers
The only positive integers which are simultaneously tetrahedral and triangular are: :$1, 10, 120, 1540, 7140$
Restriction of Composition is Composition of Restriction
Let $X, Y, Z$ be sets. Let $f: X \to Y$ and $g: Y \to Z$ be mappings. Let $S \subseteq X$. Then: : $\left({g \circ f}\right) \restriction S = g \circ \left({f \restriction S}\right)$
Smallest Fermat Pseudoprime to Bases 2, 3 and 5
The smallest Fermat pseudoprime to bases $2$, $3$ and $5$ is $1729$.
Numbers that Factorise into Sum of Digits and Reversal
The following positive integers can each be expressed as the product of the sum of its digits and the reversal of the sum of its digits: :$1, 81, 1458, 1729$ {{OEIS|A110921}}
1782 is 3 Times Sum of all 2-Digit Numbers from its Digits
$1782$ equals $3$ multiplied by the sum of all the $2$-digit integers that can be formed from its digits.
Triple of Consecutive Happy Numbers
The smallest triple of consecutive integers all of which are happy is: :$\left({1880, 1881, 1882}\right)$
Numbers whose Digits are Unchanged when Subtracting Reversal
The following sequence consists of the integers which have the property that subtraction of their reversals results in anagrams of them: :$954, 1980, 2961, 3870, 5823, 7641, 9108, 19980, 29880, 29961, 32760, \ldots$ {{OEIS|A121969}}
17 Consecutive Integers each with Common Factor with Product of other 16
The $17$ consecutive integers from $2184$ to $2200$ have the property that each one is not coprime with the product of the other $16$.
Lower and Upper Bounds for Sequences/Warning
Let $\sequence {x_n}$ be a sequence in $\R$. Let $x_n \to l$ as $n \to \infty$. Then it is '''not''' the case that: :$(1): \quad \forall n \in \N: x_n > a \implies l > a$ :$(2): \quad \forall n \in \N: x_n < b \implies l < b$
Smallest Fourth Power as Sum and Difference of Fourth Powers
The smallest $4$th power that can be expressed as the sum of $2$ $4$th powers minus a $3$rd is: :$2401 = 7^4 = 227^4 + 157^4 - 239^4$ with all numbers less than $10^4$.
Zsigmondy's Theorem for Sums
Let $a > b > 0$ be coprime positive integers. Let $n \ge 1$ be a (strictly) positive integer. Then there is a prime number $p$ such that :$p$ divides $a^n + b^n$ :$p$ does not divide $a^k + b^k$ for all $k < n$ with the following exception: :$n = 3$, $a = 2$, $b = 1$
Cyclotomic Polynomial of Index times Prime Power
Let $n, k \ge 1$ be natural numbers. Let $p$ be a prime number. Let $\Phi_n$ denote the $n$th cyclotomic polynomial. Then $\map {\Phi_{p^k n}} x = \begin{cases} \map {\Phi_n} {x^{p^k}} & \text{if } p \divides n\\ \dfrac {\map {\Phi_n} {x^{p^k}}} {\map {\Phi_n} {x^{p^{k - 1}}}} & \text{if } p \nmid n \end{cases}$
Multiplicative Order of Roots of Cyclotomic Polynomial Modulo Prime
Let $n\geq1$ be a natural number. Let $p$ be a prime number Let $n=p^\alpha q$ where $\alpha = \nu_p(n)$ is the valuation of $p$ in $n$. Let $a\in\Z$ with $\Phi_n(a)\equiv0\pmod p$. Then the order of $a$ modulo $p$ is $q$: :$\operatorname{ord}_p(a) = q$.
Lower Topology is Unique
Let $T_1 = \left({S, \preceq, \tau_1}\right)$ and $T_2 = \left({S, \preceq, \tau_2}\right)$ be relational structures with lower topologies. Then: : $\tau_1 = \tau_2$
Homogeneous Cyclotomic Polynomial is Symmetric
Let $n>1$ be a natural number. Let $\Phi_n(x,y)$ be the $n$th homogeneous cyclotomic polynomial. Then $\Phi_n(x,y) = \Phi_n(y,x)$, that is, $\Phi_n(x,y)$ is symmetric.
Cyclotomic Polynomial of Index Power of Two
Let $n \ge 1$ be a natural number. Then the $2^n$th cyclotomic polynomial is: :$\map {\Phi_{2^n} } x = x^{2^{n - 1} } + 1$
Trivial Estimate for Cyclotomic Polynomials
Let $n \ge 1$ be a natural number. Let $\Phi_n$ be the $n$th cyclotomic polynomial. Let $\phi$ be the Euler totient function. Let $z \in \C$ be a complex number. Then: :$\size {\size z - 1}^{\map \phi n} \le \size {\map {\Phi_n} z} \le \paren {\size z + 1}^{\map \phi n}$ where: :the first inequality becomes an equality only if: :::$n = 1$ and $z \in \R_{\ge 0}$ ::or: :::$n = 2$ and $z \in \R_{\le 0}$ :the second inequality becomes an equality only if: :::$n = 1$ and $z \in \R_{\le 0}$ ::or: :::$n = 2$ and $z \in \R_{\ge 0}$
Complete List of Special Highly Composite Numbers
There are exactly $6$ special highly composite numbers: :$1, 2, 6, 12, 60, 2520$ {{OEIS|A106037}}
Prime Decomposition of Highly Composite Number
Let $n$ be a highly composite number. Let the prime decomposition of $n$ be expressed as: :$n = \displaystyle \prod_{k \mathop \in \N} {p_k}^{r_k}$ where $p_k$ denotes the $k$th prime. Then the sequence $\left\langle{r_k}\right\rangle$ is decreasing. That is: :$\forall k \in \N: r_k \ge r_{k + 1}$
Complement of Upper Closure of Element is Open in Lower Topology
Let $T = \left({S, \preceq, \tau}\right)$ be a relational structure with lower topology. Let $x \in S$. Then $\complement_S\left({x^\succeq}\right)$ is open and $x^\succeq$ is closed.
Open Subset is Lower in Lower Topology
Let $T = \left({S, \preceq, \tau}\right)$ be a transitive relational structure with lower topology. Let $A \subseteq S$ such that :$A$ is open. Then $A$ is lower.
Ratio between Consecutive Highly Composite Numbers Greater than 2520 is Less than 2
The ratio between $2$ consecutive highly composite numbers both greater than $2520$ is less than $2$.
Modified Kaprekar Process on 4-Digit Number terminates in 2538
Let $n$ be a $4$-digit number. Let $n$ be operated on by the modified Kaprekar process. The eventual result is always $2538$.
Fibonacci Number equal to Sum of Sequence of Cubes
The following Fibonacci number can be expressed as the sum of a sequence of cubes: :$F_{18} = 2584 = 7^3 + 8^3 + 9^3 + 10^3$ {{expand|Any more?}}
Dudeney's Property of 2592
:$2592 = 2^5 \times 9^2$ It is the only number $n$ that has the property that: :$n = \sqbrk {abcd} = a^b \times c^d$ where $\sqbrk {abcd}$ denotes the decimal representation of $n$.
2601 as Sum of 3 Squares in 12 Different Ways
$2601$ can be expressed as the sum of $3$ squares in $12$ different ways.
Reversal of Number Multiplied by 11
Let $n \in \N$ be a number for which, when written in decimal notation, no two adjacent digits total to more than $9$. Let $n'$ denote the reversal of $n$. Then $n \times 11$ is the reversal of $n' \times 11$.
Reduction Formula for Integral of Power of Tangent
For all $n \in \Z_{> 1}$:   :$\displaystyle \int \map {\tan^n} x \rd x = \frac {\map {\tan^{n - 1} } x} {n - 1} - \int \map {\tan^{n - 2} } x \rd x$
Numbers Appearing 8 Times in Pascal's Triangle
The number $3003$ is the smallest integer to appear $8$ times in Pascal's triangle. No other number below $2^{23}$ appears as often.
3367 Multiplied by 2-Digit Number
In order to multiply $3367$ by a $2$-digit integer $\sqbrk {xy}$: :divide the $6$-digit integer $\sqbrk {xyxyxy}$ by $3$.
Integer as Sum of 2 Cubes in 3 Ways
$4104$ is the smallest natural number which can be expressed as the sum of $2$ cubes in $3$ different ways: {{begin-eqn}} {{eqn | l = 4104       | r = 16^3 + 2^3       | c =  }} {{eqn | r = 15^3 + 9^3       | c =  }} {{eqn | r = \paren {-12}^3 + 18^3       | c =  }} {{end-eqn}}
Composite Fibonacci Numbers with Prime Index
The sequence of composite Fibonacci numbers with a prime index begins: :$4181, 1 \, 346 \, 269, 24 \, 157 \, 817, 165 \, 580 \, 141, \ldots$ {{OEIS|A050937}} The corresponding sequence of prime indices begins: :$19, 31, 37, 41, 53, 59, 61, 67, 71, 73, 79, \ldots$ {{OEIS|A038672}}
Closed Subset is Upper in Lower Topology
Let $T = \left({S, \preceq, \tau}\right)$ be a transitive relational structure with lower topology. Let $A \subseteq S$ such that :$A$ is closed. Then $A$ is upper.
Square Pyramidal Number also Square
$4900$ is the only square pyramidal number which is also square: :$4900 = 70^2 = \displaystyle \sum_{k \mathop = 1}^{24} k^2 = \dfrac {24 \paren {24 + 1} \paren {2 \times 24 + 1} } 6$
Mapping Preserves Non-Empty Infima implies Mapping is Continuous in Lower Topological Lattice
Let $T = \left({S, \preceq, \tau}\right)$ and $Q = \left({X, \preceq', \tau'}\right)$ be complete topological lattices with lower topologies. Let $f: S \to X$ be a mapping such that :for all non-empty subsets $Y$ of $S$: $f$ preserves the infimum of $Y$. Then $f$ is continuous mapping.
5040 is Product of Consecutive Numbers in Two Ways
:$5040 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8 \times 7$
Products of Consecutive Integers in 2 Ways
The following integers are the product of consecutive integers in $2$ ways: :$-720, 720, 5040$
8 Mutually Non-Attacking Rooks on Chessboard
On a standard chessboard, it is possible to arrange a maximum of $8$ rooks so that no rook is attacking any other rook. There are $5282$ such arrangements, up to rotation and reflection.
Triangular Lucas Numbers
The only Lucas numbers which are also triangular are: :$1, 3, 5778$ {{OEIS|A248506}}
Subspace of Subspace is Subspace
Let $T = \struct{S, \tau}$ be a topological space. Let $H \subseteq S$ and $\tau_H$ be the subspace topology on $H$. Let $K\subseteq H$. Then the subspace topology on $K$ induced by $\tau$ equals the subspace topology on $K$ induced by $\tau_H$.
Positive Integer Sum of 3 Fourth Powers in 2 Ways
The smallest positive integer which can be expressed as the sum of $3$ fourth powers in $2$ different ways is $6578$: {{begin-eqn}} {{eqn | l = 1^4 + 2^4 + 9^4       | r = 1 + 16 + 6561       | c =  }} {{eqn | r = 6578       | c =  }} {{eqn | r = 81 + 2401 + 4096       | c =  }} {{eqn | r = 3^4 + 7^4 + 8^4       | c =  }} {{end-eqn}}
Mapping Preserves Infima implies Mapping is Continuous in Lower Topological Lattice
Let $T = \struct {S, \preceq, \tau}$ and $Q = \struct {X, \preceq', \tau'}$ be complete topological lattices with lower topologies. Let $f: S \to X$ be a mapping such that :$f$ preserves all infima. Then $f$ is continuous mapping.
Compact in Subspace is Compact in Topological Space
Let $T = \struct {S, \tau}$ be a topological space. Let $K \subseteq S$ be a subset. Let $\tau_K$ be the subspace topology on $K$. Let $T' = \struct {K, \tau_K}$ be the topological subspace of $T$ determined by $K$. Let $H \subseteq K$ be compact in $T'$. Then $H$ is compact in $T$.
If Infimum of Filtered Subset belongs to Element of Sub-Basis then Subset and Element Intersect implies Infimum of Subset belongs to Closure of Subset
Let $T = \left({S, \preceq, \tau}\right)$ be a complete topological lattice with lower topology. Let $B$ be an analytic sub-basis of $T$. Let $F$ be a filtered subset of $S$ such that :$\forall A \in B: \inf F \in A \implies F \cap A \ne \varnothing$ Then $\inf F \in F^-$ where $F^-$ denotes the topological closure of $F$.
Product with Repdigit can be Split into Parts which Add to Repdigit
Let $n$ be a positive integer with $d_1$ digits. Let $m$ be a repdigit number with $d_2$ digits such that $d_2 > d_1$. Let $r$ consist of the result when the rightmost $d_2$ digits of $m n$ is cut off and added to the remaining left hand portion. Then $r$ is a repdigit number.
Sum of Reciprocals of Squares of Odd Integers as Double Integral
:$\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^2} = \int_0^1 \int_0^1 \frac 1 {1 - x^2 y^2} \rd x \rd y$
Three Tri-Automorphic Numbers for each Number of Digits
Let $d \in \Z_{>0}$ be a (strictly) positive integer. Then there exist exactly $3$ tri-automorphic numbers with exactly $d$ digits. These tri-automorphic numbers all end in $2$, $5$ or $7$.
Fourier Series/x squared over Minus Pi to Pi
:$\displaystyle x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos n x}$
Integer and its Double forming Pandigital Pair
$6729$ and its double contain all the digits from $1$ to $9$ between them.
4-Digit Numbers forming Longest Reverse-and-Add Sequence
Let $m \in \Z_{>0}$ be a positive integer expressed in decimal notation. Let $r \left({m}\right)$ be the reverse-and-add process on $m$. Let $r$ be applied iteratively to $m$. The $4$-digit integers $m$ which need the largest number of iterations before reaching a palindromic number are: :$6999, 7998, 8997, 9996$ all of which need $20$ iterations.
Two Consecutive Integers each Product of Four Distinct Primes
The sequence of pairs of consecutive positive integers which are each the product of exactly $4$ distinct prime numbers begins: :$\tuple {7314, 7315}, \tuple {8294, 8295}, \tuple {8645, 8646}, \tuple {9009, 9010}, \ldots$ {{OEIS|A140078|order = first}}
Fourier Series/Identity Function over Minus Pi to Pi
For $x \in \openint {-\pi} \pi$: :$\displaystyle x = 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin n x$
Fourier Series/Fourth Power of x over Minus Pi to Pi
:$\displaystyle x^4 = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \cos n \pi \cos n x$
Fourier Series/Absolute Value of x over Minus Pi to Pi
For $x \in \openint {-\pi} \pi$: :$\displaystyle \size x = \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\map \cos {2 n - 1} x} {\paren {2 n - 1}^2}$
Fourier Series/Pi minus x over 0 to 2 Pi
:$\displaystyle \pi - x = 2 \sum_{n \mathop = 1}^\infty \frac {\sin n x} n$
Mapping is Continuous implies Mapping Preserves Filtered Infima in Lower Topological Lattice
Let $T = \left({S, \preceq, \tau}\right)$ and $Q = \left({X, \preceq', \tau'}\right)$ be complete topological lattices with lower topologies. Let $f: S \to X$ be a mapping such that :$f$ is a continuous mapping. Then $f$ preserves filtered infima.
3 Numbers in A.P. whose 4th Powers are Sum of Four 4th Powers
The following triplets of integers in arithmetic sequence with common difference of $60$ can all be expressed as the sum of four $4$th powers: :$\tuple {8373, 8433, 8493}, \tuple {8517, 8577, 8637}, \ldots$
Intersection of Chain of Prime Ideals of Commutative Ring is Prime Ideal
Let $R$ be a commutative ring. Let $\Spec R$ be the spectrum of $R$, ordered by inclusion. Let $\set {P_\alpha}_{\alpha \mathop \in A}$ be a non-empty chain of prime ideals of $\Spec R$. Let $\displaystyle P = \bigcap_{\alpha \mathop \in A} P_\alpha$ be their intersection. Then $P$ is a prime ideal of $R$.
Number of Different Ways to play First n Moves in Chess
The sequence formed from the number of ways to play the first $n$ moves in chess begins: :$20, 400, 8902, 197 \, 742, \ldots$ {{OEIS|A007545}} The count for the fourth move is already ambiguous, as it depends on whether only legal moves count, or whether all moves, legal or illegal, are included. The count as given here does include illegal moves in addition to legal ones.
Largest Product of Pandigital Factors
The largest integer that can be obtained by multiplying $2$ integers which between them use all the digits from $1$ to $9$ is: :$843 \, 973 \, 902 = 9642 \times 87531$
Smallest Pandigital Square
The smallest square number which contains all the digits from $1$ to $9$ is: :$11 \, 826^2 = 139 \, 854 \, 276$
Locally Compact Space is Weakly Locally Compact
Let $T = \struct{S, \tau}$ be a locally compact topological space. Then $T$ is weakly locally compact.
Space is Neighborhood of all its Points
Let $T = \struct {X, \tau}$ be a topological space. Let $x\in X$. Then $X$ is a neighborhood of $x$.
Local Compactness is Preserved under Open Continuous Surjection
Let $T_A = \left({S_A, \tau_A}\right)$ and $T_B = \left({S_B, \tau_B}\right)$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous mapping which is also an open mapping and a surjection. If $T_A$ is locally compact, then $T_B$ is also locally compact.
Equivalence of Definitions of Noetherian Module
Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module. {{TFAE|def = Noetherian Module}}
Largest Positive Integer not Sum of Distinct Cubes
$12 \, 758$ is the largest positive integer that cannot be expressed as the sum of distinct cubes.
Particular Point Space is Locally Compact
Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is locally compact.
Product of Summations is Summation Over Cartesian Product of Products
This is a generalization of the distributive law: :$\displaystyle \prod_{a \mathop \in A} \sum_{b \mathop \in B_a} t_{a, b} = \sum_{c \mathop \in \prod_{a \mathop \in A} B_a} \prod_{a \mathop \in A} t_{a, c_a}$ where the product of sets $\prod_{a \mathop \in A} B_a$ is taken to be a cartesian product. {{explain|In order to make this comprehensible, the precise nature of $A$ and $B$ need to be defined. Presumably $A$ is a set (of numbers?), and $B$ is a family of sets of numbers indexed by $A$? And the $t$s are elements of $B$ and are numbers? I'm having difficulty.}}
Largest Number not Sum of Squares of Distinct Primes
The largest positive integer which cannot be expressed as the sum of the squares of distinct prime numbers is $17 \, 163$.
Volume of Smallest Rational Tetrahedron
The only rational tetrahedron whose edge lengths are less than $157$ has: : edges of length $117$, $80$, $53$, $52$, $51$, $84$ : faces of area $1800$, $1890$, $2016$, $1170$ : volume of $18 \, 144$.
Square and Tetrahedral Numbers
The only positive integers which are simultaneously tetrahedral and square are: :$1, 4, 19 \, 600$
Smallest Pythagorean Quadrilateral with Integer Sides
The smallest Pythagorean quadrilateral in which the sides of the $4$ right triangles formed by its sides and perpendicular diagonals are all integers has an area of $21 \, 576$. The sides of the right triangles in question are: :$25, 60, 65$ :$91, 60, 109$ :$91, 312, 325$ :$25, 312, 313$
Integers whose Tau value equals Cube Root
There are $3$ positive integers whose $\tau$ value equals its cube root: {{begin-eqn}} {{eqn | l = 1 = 1^3       | o = :       | r = \map \tau 1 = 1       | c = {{TauLink|1}} }} {{eqn | l = 21 \, 952 = 28^3       | o = :       | r = \map \tau {21 \, 952} = 28       | c = {{TauLink|21,952|21 \, 952}} }} {{eqn | l = 64 \, 000 = 40^3       | o = :       | r = \map \tau {64 \, 000} = 40       | c = {{TauLink|64,000|64 \, 000}} }} {{end-eqn}} {{OEIS|A066693}}