hcju/proofwikips huggingface dataset:
Eight Convex Deltahedra
There exist exactly $8$ distinct convex deltahedra: :$4$ faces: regular tetrahedron :$6$ faces: triangular bipyramid :$8$ faces: regular octahedron :$10$ faces: pentagonal bipyramid :$12$ faces: snub disphenoid (split a regular tetrahedron into two wedges and join them with a band of $8$ equilateral triangles) :$14$ faces: triaugmented triangular prism (attach $3$ square pyramids to a triangular prism) :$16$ faces: gyroelongated square bipyramid (attach $2$ square pyramids to a square antiprism) :$20$ faces: regular icosahedron.
Way Below Relation is Multiplicative implies Pseudoprime Element is Prime
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice such that :$\ll$ is multiplicative relation where $\ll$ denotes the way below relation of $L$. Let $p \in S$. Then $p$ is a pseudoprime element is a prime element.
If Every Element Pseudoprime is Prime then Way Below Relation is Multiplicative
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous distributive lattice. Let every element $p \in S$: $p$ is pseudoprime $\implies p$ is prime. Then $\ll$ is multiplicative where $\ll$ denotes the way below relation of $L$.
Upper Closure of Element is Way Below Open Filter iff Element is Compact
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $x \in S$. Then : $x^\succeq$ is a way below open filter on $L$  :$x$ is compact
9 is Only Square which is Sum of 2 Consecutive Positive Cubes
Discounting the trivial solution: :$1^2 = 1 = 0^3 + 1^3$ $9$ is the only square number which is the sum of $2$ consecutive positive cube numbers: :$3^2 = 9 = 1^3 + 2^3$
If Compact Between then Way Below
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $x, k, y \in S$ such that :$x \preceq k$ and $k \preceq y$ and $k \in K\left({L}\right)$ where $K\left({L}\right)$ denotes the compact subset of $L$. Then $x \ll y$ where $\ll$ denotes the way below relation.
Nine Regular Polyhedra
There exist $9$ regular polyhedra.
Way Below is Congruent for Join
Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice. Then $\ll$ is congruence relation for $\vee$: :$\forall a, b, x, y \in S: a \ll x \land b \ll y \implies a \vee b \ll x \vee y$ where $\ll$ denotes the way below relation.
Lifting The Exponent Lemma
Let $x, y \in \Z$ be distinct integers. Let $n \geq1$ be a natural number. Let $p$ be an odd prime. Let: :$p \mathrel \backslash x - y$ and: :$p \nmid x y$ where $\backslash$ and $\nmid$ denote divisibility and non-divisibility respectively. Then :$\nu_p \left({x^n - y^n}\right) = \nu_p \left({x - y}\right) + \nu_p \left({n}\right)$ where $\nu_p$ denotes $p$-adic valuation.
Zsigmondy's Theorem
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 exceptions: :$n = 1$ and $a - b = 1$ :$n = 2$ and $a + b$ is a power of $2$ :$n = 6$, $a = 2$, $b = 1$
Compact Subset is Join Subsemilattice
Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice. Let $\map K L$ be a compact subset of $L$. Then $\map K L$ is join subsemilattice: :$\forall x, y \in \map K L: x \vee y \in \map K L$
Totally Ordered Ring Zero Precedes Element or its Inverse
Let $\struct {R, +, \circ, \preceq}$ be an ordered ring. From the definition of ordered ring, $\preceq$ is compatible with $+$. Let $0_R$ be the zero element of $R$. Let $x \ne 0_R$ be a non-zero element of $R$. Let $-x$ be the ring negative of $x$. Then: :$0_R \prec x \lor 0_R \prec -x$ but not both.
Bottom in Compact Subset
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Then $\bot \in K\left({L}\right)$ where $\bot$ denotes the smallest element of $L$, :$K\left({L}\right)$ denotes the compact subset of $L$.
Only 1 by 1 Matrices Generally Commute Under Multiplication
Let $R$ be a commutative ring. Let $\MM_{m \times n}$ be the set of all $m \times n$ matrices over $R$. Then under conventional matrix multiplication: :$\mathbf {A B} = \mathbf {B A}$ for all $\mathbf A \in \MM_{m \times n}, \; \mathbf B \in \MM_{n \times p}$  $p = m = n = 1$.
Composition of Relations is not Commutative
Composition of relations is, in general, not commutative: That is, it is usually the case that: :$\mathcal R_1 \circ \mathcal R_2 \ne \mathcal R_2 \circ \mathcal R_1$ for relations $\mathcal R_1$ and $\mathcal R_2$.
Compact Closure is Intersection of Lower Closure and Compact Subset
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $x \in S$. Then $x^{\mathrm{compact} } = x^\preceq \cap K\left({L}\right)$ where :$x^{\mathrm{compact} }$ denotes the compact closure of $x$, :$x^\preceq$ denotes the lower closure of $x$, :$K\left({L}\right)$ denotes the compact subset of $L$.
Compact Closure is Subset of Way Below Closure
Let $L = \struct {S, \preceq}$ be an ordered set. Let $x \in S$. Then $x^{\mathrm {compact} } \subseteq x^\ll$ where :$x^{\mathrm {compact} }$ denotes the compact closure of $x$, :$x^\ll$ denotes the way below closure of $x$.
Algebraic iff Continuous and For Every Way Below Exists Compact Between
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a lattice. Then $L$ is algebraic : :$L$ is continuous and :$\forall x, y \in S: x \ll y \implies \exists k \in \map K L: x \preceq k \preceq y$ where :$\ll$ denotes the way below relation, :$\map K L$ denotes the compact subset of $L$.
Non-Empty Way Below Closure is Directed in Join Semilattice
Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice. Let $x \in S$ such that :$x^\ll \ne \varnothing$ where $x^\ll$ denotes the way below closure os $x$. Then $x^\ll$ is directed.
Image under Subset of Relation is Subset of Image under Relation
Let $S$ and $T$ be sets. Let $\RR_1 \subseteq S \times T$ be a relation in $S \times T$. Let $\RR_2 \subseteq \RR_1$. Let $A \subseteq S$. Then: :$\RR_2 \sqbrk A \subseteq \RR_1 \sqbrk A$ where $\RR_1 \sqbrk A$ denotes the image of $A$ under $\RR_1$.
Intersection of Transitive Relations is Transitive/General Result
Let $\left\{ {\mathcal R_i: i \mathop \in I}\right\}$ be an $I$-indexed collection of transitive relations on a set $S$. Then their intersection $\displaystyle \bigcap_{i \mathop \in I} \mathcal R_i$ is also a transitive relation on $S$.
Upper Bounds are Equivalent implies Suprema are equal
Let $L = \struct {S, \preceq}$ be an ordered set. Let $X, Y$ be subsets of $S$. Assume that :$X$ admits a supremum and :$\forall x \in S: x$ is upper bound for $X \iff x$ is upper bound for $Y$ Then $\sup X = \sup Y$
Inverse of Subset of Relation is Subset of Inverse
Let $S$ and $T$ be sets Let $\RR_1 = S \times T$ be a relation on $S \times T$. Let $\RR_2 \subseteq \RR_1$. Then: :$\RR_2^{-1} \subseteq \RR_1^{-1}$ where $\RR_1^{-1}$ denotes the inverse of $\RR_1$.
Preceding implies if Less Upper Bound then Greater Upper Bound
Let $L = \struct {S, \preceq}$ be an ordered set. Let $x, y \in S$ such that :$x \preceq y$ Let $X \subseteq S$. Then :$x$ is upper bound for $X \implies y$ is upper bound for $X$ and :$y$ is lower bound for $X \implies x$ is lower bound for $X$.
Three Regular Tessellations/Squares
Squares form a regular tessellation: :400px
Three Regular Tessellations/Triangles
Equilateral triangles form a regular tessellation: :400px
Arithmetic iff Way Below Relation is Multiplicative in Algebraic Lattice
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below algebraic lattice. Then $L$ is arithmetic  :$\ll$ is a multiplicative relation where $\ll$ denotes the way below relation of $L$.
Endomorphism Ring of Abelian Group is Ring with Unity
Let $\struct {G, +}$ be an abelian group. Let $\struct {\map {\mathrm {End} } G, +, \circ}$ be its endomorphism ring. Then $\struct {\map {\mathrm {End} } G, +, \circ}$ is a ring with unity $I_G$, where $I_G$ is the identity mapping on $G$.
Elements with Support in Ideal form Submagma of Direct Product
Let $\left({S_i, \circ_i}\right)_{i \mathop \in I}$ be a family of magmas with identity. Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be their direct product. Let $J \subset I$ be an ideal of $I$. {{explain|The link to ideal suggests that $I$ would need to be an ordered set, which does not appear to be the case here.}} Let $T = \left\{ {s \in S: \operatorname{supp} \left({s}\right) \in J}\right\}$ where $\operatorname{supp}$ denotes support. Then $T$ is a submagma of $S$.
Elements of Finite Support form Submagma of Direct Product
Let $\struct {S_i, \circ_i}_{i \mathop \in I}$ be a family of magmas with identity. Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be their direct product. Let $T$ be the subset of elements of $S$ whose support is finite: :$T = \set {s \in S: \map {\operatorname {supp} } s \text{ is finite} }$ Then $T$ is a submagma of $S$.
Direct Sum of Modules is Module
Let $A$ be a commutative ring with unity. Let $\left\{ {M_i}\right\}_{i \in I}$ be a family of $A$-modules indexed by $I$. Let $\displaystyle M = \bigoplus_{i \mathop \in I} M_i$ be their direct sum. Then $M$ is a module.
Möbius Inversion Formula/Abelian Group
Let $G$ be an abelian group. Let $f, g: \N \to G$ be mappings. Then :$\displaystyle \map f n = \prod_{d \mathop \divides n} \map g d$ : :$\displaystyle \map g n = \prod_{d \mathop \divides n} \map f d^{\mu \paren {\frac n d} }$
Möbius Inversion Formula for Cyclotomic Polynomials
Let $n > 0$ be a (strictly) positive integer. Let $\Phi_n$ be the $n$th cyclotomic polynomial. Then: :$\map {\Phi_n} x = \displaystyle \prod_{d \mathop \divides n} \paren {x^d - 1}^{\map \mu {n / d} }$ where: :the product runs over all divisors of $n$. :$\mu$ is the Möbius function.
Composite of Inverse of Mapping with Mapping
Let $f: S \to T$ be a mapping. Then: :$f \circ f^{-1} = I_{\Img f}$ where: : $f \circ f^{-1}$ is the composite of $f$ and $f^{-1}$ : $f^{-1}$ is the inverse of $f$ : $I_{\Img f}$ is the identity mapping on the image set of $f$.
Multilinear Mapping from Free Modules is Determined by Bases
Let $R$ be a commutative ring with unity. Let $M_1,\ldots,M_n$ be free $R$-modules. Let $B_1,\ldots,B_n$ be bases of $M_1,\ldots,M_n$. Let $N$ be an $R$-module. Let $f:B_1\times\cdots\times B_n\to N$ be a function. Then there exists a unique multilinear map $\phi:M_1\times\cdots\times M_n\to N$ such that $\phi(b)=f(b)$ for all $b\in B_1\times\cdots\times B_n$.
Equivalence of Definitions of Order Complete Set
Let $\struct {S, \preceq}$ be an ordered set. {{TFAE|def = Order Complete Set}}
Condition for Uniqueness of Increasing Mappings between Tosets
Let $\struct {S, \preceq}$ and $\struct {T, \preccurlyeq}$ be tosets. Let $f: S \to T$ and $g: S \to T$ be increasing mappings from $S$ to $T$. Let $H \subseteq S$ be a subset of $S$. Let $f$ and $g$ agree on $H$. Let $K = f \sqbrk H$ be the image set of $H$ under $f$. Let the intersection of $K$ with every set of the form: :$\set {y \in T: u < y < v: u, v \in T, u < v}$ be non-empty. Then $f = g$.
Arithmetic iff Compact Subset form Lattice in Algebraic Lattice
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below algebraic lattice. Then $L$ is arithmetic  $\left({K\left({L}\right), \precsim}\right)$ is a lattice, where $K\left({L}\right)$ denotes the compact subset of $L$, :$\mathord \precsim = \mathord \preceq \cap \left({K\left({L}\right) \times K\left({L}\right)}\right)$
Four Kepler-Poinsot Polyhedra
There exist exactly four Kepler-Poinsot polyhedra: : $(1): \quad$ the small stellated dodecahedron : $(2): \quad$ the great stellated dodecahedron : $(3): \quad$ the great dodecahedron : $(4): \quad$ the great icosahedron.
Dissection of Rectangle into 9 Distinct Integral Squares
Let $R$ be a rectangle. Let $R$ be divided into $n$ squares which all have different lengths of sides. Then $n \ge 9$. The smallest rectangle with integer sides that can be so divided into squares with integer sides is $32 \times 33$. :600px
Nine Point Circle Theorem
Let $\triangle ABC$ be a triangle. These $9$ points: :the feet of the altitudes of $\triangle ABC$ :the midpoints of the sides of $\triangle ABC$ :the midpoints of the lines from the vertices of $\triangle ABC$ to the orthocenter $H$ of $\triangle ABC$ all lie on the circumference of a circle. The center $M$ lies on the Euler line of $\triangle ABC$, at the midpoint between the orthocenter $H$ and the circumcenter $O$. :500px
Feuerbach's Theorem
Let $\triangle ABC$ be a triangle. The Feuerbach circle of $\triangle ABC$ is tangent to: :the incircle of $\triangle ABC$ and: :the $3$ excircles of $\triangle ABC$. :800px
Round Peg fits in Square Hole better than Square Peg fits in Round Hole
A round peg fits better in a square hole than a square peg fits in a round hole. :600px
Ratios of Sizes of Mutually Inscribed Multidimensional Cubes and Spheres
Consider: : a cube $C_n$ of $n$ dimensions inscribed within a sphere $S_n$ of $n$ dimensions : a sphere $S'_n$ of $n$ dimensions  inscribed within a cube $C'_n$ of $n$ dimensions. Let: : $A_{cn}$ be the $n$ dimensional volume of $C_n$ : $A_{sn}$ be the $n$ dimensional volume of $S_n$ : $A'_{cn}$ be the $n$ dimensional volume of $C'_n$ : $A'_{sn}$ be the $n$ dimensional volume of $S'_n$. For $n < 9$: :$\dfrac {S_n} {C_n} > \dfrac {C'_n} {S'_n}$ but for $n \ge 9$: :$\dfrac {S_n} {C_n} < \dfrac {C'_n} {S'_n}$ That is, for dimension $n$ less than $9$, the $n$ dimensional round peg fits better into an $n$ dimensional square hole than an $n$ dimensional square peg fits into an $n$ dimensional round hole, but for $9$ and higher dimensions, the situation is reversed.
Pseudoprime Element is Prime in Arithmetic Lattice
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below arithmetic lattice. Let $p \in S$. Then if $p$ is pseudoprime element, then $p$ is prime element.
Every Pseudoprime Element is Prime implies Lattice is Arithmetic
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below algebraic distributive lattice. Assume that :for every element $p$ of $S$ if $p$ is pseudoprime element, then $p$ is prime element. Then $L$ is arithmetic.
General Periodicity Property/Corollary
Let $f: \R \to \R$ be a real function. Then $L$ is a periodic element of $f$ : :$\forall x \in \R: \map f {x \bmod L} = \map f x$ where $x \bmod L$ is the modulo operation.
No Configurations of 7 or 8 Lines with 3 Intersection Points on each Line
There exist no configurations of $7$ or $8$ straight lines each of which has exactly $3$ points of intersection.
3 Configurations of 9 Lines with 3 Intersection Points on each Line
There exist exactly $3$ essentially different configurations of $9$ straight lines each of which has exactly $3$ points of intersection. This is one: there are two others. :400px
Power of Positive Real Number is Positive/Real Number
Let $x \in \R_{>0}$ be a (strictly) positive real number. Let $r \in \R$ be a real number. Then: :$x^r > 0$ where $x^r$ denotes the $x$ to the power of $r$.
Element is Finite iff Element is Compact in Lattice of Power Set
Let $X$ be a set. Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be a lattice of power set. Let $x \in \mathcal P\left({X}\right)$. Then $x$ is a finite set  $x$ is a compact element.
Function with Limit at Infinity of Exponential Order Zero
Let $f: \hointr 0 \to \to \R$ be a real function. Let $f$ be continuous everywhere on their domains, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\hointr 0 \to$. {{explain|Establish whether it is "finite subinterval" that is needed here, or what we have already defined as "Definition:Finite Subdivision". Also get the correct instance of "continuous".}} Let $f$ have a (finite) limit at infinity. Then $f$ is of exponential order $0$.
Bounded Function is of Exponential Order Zero
Let $f: \hointr 0 \to \to \mathbb F$ be a function, where $\mathbb F \in \set {\R, \C}$. Let $f$ be continuous everywhere on its domain, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\hointr 0 \to$. {{explain|Establish whether it is "finite subinterval" that is needed here, or what we have already defined as "Definition:Finite Subdivision".}} {{explain|Pin down a specific page on which the relevant definition of "Continuous" can be found for this context.}} Let $f$ be bounded. Then $f$ is of exponential order $0$.
Arctangent is of Exponential Order Zero
Let $\arctan: \R \to \openint {-\dfrac \pi 2} {\dfrac \pi 2}$ be the real arctangent. Then $\arctan$ is of exponential order $0$.
Arccotangent is of Exponential Order Zero
Let $\arccot: \R \to \openint 0 \pi$ be the real arccotangent. Then $\arccot$ is of exponential order $0$.
Limit at Infinity of Sine Integral Function
Let $\Si: \R \to \R$ denote the sine integral function. Then $\Si$ has a (finite) limit at infinity: :$\displaystyle \lim_{x \mathop \to +\infty} \map \Si x = \frac \pi 2$
Natural Number Power is of Exponential Order Epsilon
Let $n \in \N$ be a natural number. Then:  :$t \mapsto t^n$ is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude.
Raising Exponential Order
Let $\map f t: \R \to \mathbb F$ a function, where $\mathbb F \in \set {\R, \C}$. Let $f$ be continuous on the real interval $\hointr 0 \to$, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\hointr 0 \to$. {{explain|Establish whether it is "finite subinterval" that is needed here, or what we have already defined as "Definition:Finite Subdivision".}} {{explain|Pin down a specific page on which the relevant definition of "Continuous" can be found for this context.}} Let $f$ be of exponential order $a$. Let $b > a$. Then $f$ is of exponential order $b$.
Factorial is not of Exponential Order
Let $\Gamma$ denote the gamma function. Let $f \left({t}\right) = \Gamma \left({t+1}\right) = t!$. Then: :$f$ is not of exponential order. That is, it grows faster than any exponential.
X to the x is not of Exponential Order
Let $f: \R \to \R$ be defined on $\hointr 0 \to$ with $\map f x = x^x$. Then: :$f$ is not of exponential order. That is, it grows faster than any exponential.
Empty is Bottom of Lattice of Power Set
Let $X$ be a set. Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be the lattice of the power set of $X$. Then: :$\O = \bot$ where $\bot$ denotes the bottom of $L$.
Hermite-Lindemann-Weierstrass Theorem
Let $a_1, \cdots, a_n$ be algebraic numbers (possibly complex) that are linearly independent over the rational numbers $\Q$. Then: :$e^{a_1}, \cdots, e^{a_n}$ are algebraically independent. where $e$ is Euler's number.
Hermite-Lindemann-Weierstrass Theorem/Weaker
Let $a$ be a non-zero algebraic number (possibly complex). Then: :$e^a$ is transcendental where $e$ is Euler's number.
Schanuel's Conjecture Implies Transcendence of Pi by Euler's Number
Let Schanuel's Conjecture be true. Then $\pi \times e$ is transcendental.
Schanuel's Conjecture Implies Transcendence of Pi plus Euler's Number
Let Schanuel's Conjecture be true. Then $\pi + e$ is transcendental.
Pascal's Mystic Hexagram
Let $ABCDEF$ be a hexagram whose $6$ vertices lie on an ellipse such that the order of vertices along the ellipse is $AECFBD$. :500px Then the points of intersection of the sides of $ABCDEF$ lie on a straight line.
Transcendence of Sum or Product of Transcendentals
Let $a$ and $b$ be two transcendental numbers. Then at least one of $a + b$ and $a \times b$ is transcendental.
Schanuel's Conjecture Implies Algebraic Independence of Pi and Euler's Number over the Rationals
Let Schanuel's Conjecture be true. Then $\pi$ (pi) and $e$ (Euler's number) are algebraically independent over the rational numbers $\Q$.
Image of Mapping from Finite Set is Finite
Let $X, Y$ be sets. Let $f: X \to Y$ be a mapping. Let $X$ be a finite set. Then $f \sqbrk X$ is a finite set.
Magic Constant of Order 3 Magic Square
The magic constant of the order $3$ magic square is $15$.
Sum of Terms of Magic Square
The total of all the entries in a magic square of order $n$ is given by: :$T_n = \dfrac {n^2 \paren {n^2 + 1} } 2$
Schanuel's Conjecture Implies Transcendence of 2 to the power of Euler's Number
Let Schanuel's Conjecture be true. Then $2$ to the power of Euler's number $e$: :$2^e$ is transcendental, where $e$ is Euler's number.
Schanuel's Conjecture Implies Transcendence of Pi to the power of Euler's Number
Let Schanuel's Conjecture be true. Then $\pi$(pi) to the power of Euler's number $e$: :$\pi^e$ is transcendental.
Schanuel's Conjecture Implies Transcendence of Euler's Number to the power of Euler's Number
Let Schanuel's Conjecture be true. Then Euler's number $e$ to the power of itself: :$e^e$ is transcendental.
Magic Constant of Magic Square
The magic constant of a magic square of order $n$ is given by: :$S_n = \dfrac {n \paren {n^2 + 1} } 2$
Magic Square of Order 3 is Unique
Up to rotations and reflections, the magic square of order $3$ is unique: {{:Magic Square/Examples/Order 3}}
Wholly Real Number and Wholly Imaginary Number are Linearly Independent over the Rationals
Let $z_1$ be a non-zero wholly real number. Let $z_2$ be a non-zero wholly imaginary number. Then, $z_1$ and $z_2$ are linearly independent over the rational numbers $\Q$, where the group is the complex numbers $\C$.
Schanuel's Conjecture Implies Algebraic Independence of Pi and Log of Pi over the Rationals
Let Schanuel's Conjecture be true. Then $\pi$ (pi) and the logarithm of $\pi$ (pi): :$\ln \pi$ are algebraically independent over the rational numbers $\Q$.
Hermite-Lindemann-Weierstrass Theorem/Weaker/Corollary
Let $a$ be a algebraic number (possibly complex) which is neither $0$ nor $1$. Then: :any value of $\ln a$ is transcendental where $\ln$ denotes complex natural logarithm.
Sums of Squares in Lines of Order 3 Magic Square
Consider the order 3 magic square: {{:Magic Square/Examples/Order 3}} : The sums of the squares of the top and bottom rows are equal, and differ by $18$ from the sums of the squares of the middle row : The sums of the squares of the left and right columns are equal , and differ by $18$ from the sums of the squares of the middle column.
Lambert W of Zero is Zero
Let $W_0$ denote principal branch of the Lambert W function. Then: :$W_0 \left({0}\right) = 0$
Sums of Squares of Lines of Order 3 Magic Square
Consider the order 3 magic square: {{:Magic Square/Examples/Order 3}} : The sums of the squares of the rows, when expressed as $3$-digit decimal numbers, are equal to the sums of the squares of those same rows of that same order 3 magic square when reflected in a vertical axis: :$\begin{array}{|c|c|c|} \hline 6 & 7 & 2 \\ \hline 1 & 5 & 9 \\ \hline 8 & 3 & 4 \\ \hline \end{array}$ Similarly: : The sums of the squares of the columns, when expressed as $3$-digit decimal numbers, are equal to the sums of the squares of those same columns of that same order 3 magic square when reflected in a horizontal axis: :$\begin{array}{|c|c|c|} \hline 4 & 3 & 8 \\ \hline 9 & 5 & 1 \\ \hline 2 & 7 & 6 \\ \hline \end{array}$
123456789 x 8 + 9 = 987654321
{{begin-eqn}} {{eqn | l = 1 \times 8 + 1       | r = 9 }} {{eqn | l = 12 \times 8 + 2       | r = 98 }} {{eqn | l = 123 \times 8 + 3       | r = 987 }} {{eqn | l = 1234 \times 8 + 4       | r = 9876 }} {{eqn | l = 12345 \times 8 + 5       | r = 98765 }} {{eqn | l = 123456 \times 8 + 6       | r = 987654 }} {{eqn | l = 1234567 \times 8 + 7       | r = 9876543 }} {{eqn | l = 12345678 \times 8 + 8       | r = 98765432 }} {{eqn | l = 123456789 \times 8 + 9       | r = 987654321 }} {{end-eqn}} The above pattern is an instance of the identity: :$\displaystyle \left({b - 2}\right) \sum_{j \mathop = 1}^n j b^{n - j} + n = \sum_{j \mathop = 1}^n \left({b - j}\right) b^{n - j}$ where $b = 10$ and $n$ goes from $1$ to $9$.