hcju/proofwikips huggingface dataset:
Fibonacci Number with Prime Index is not necessarily Prime
Let $p \in \Z_{>0}$ be a prime number. Let $F_p$ be the $p$th Fibonacci number. Then $F_p$ is not itself necessarily prime.
Definition:Fibonacci Prime Pair
A '''Fibonacci prime pair''' is a pair of Fibonacci numbers $F_p$ and $F_{p + 2}$ such that: :$p$ and $p + 2$ are both prime numbers :$F_p$ and $F_{p + 2}$ are both prime numbers.
Complex Riemann Integral is Contour Integral
Let $f: \R \to \C$ be a complex Riemann integrable function over some closed real interval $\left[{a \,.\,.\, b}\right]$. Then: :$\displaystyle \int_a^b f \left({t}\right) \rd t = \int_\mathcal C f \left({t}\right) \rd t$ where: : the integral on the  is a complex Riemann integral : the integral on the  is a contour integral : $\mathcal C$ is a straight line segment along the real axis, connecting $a$ to $b$.
Fibonacci Number is not Product of Two Smaller Fibonacci Numbers
Let $m, n \in \Z$ be integers. Suppose $\size m, \size n \ge 3$. Let $F_m$ and $F_n$ be the $m$th and $n$th Fibonacci numbers. Then $F_m \times F_n$ is not a Fibonacci number.
Sequence of Fibonacci Numbers ending in Index
Let $F_k$ denote the $k$th Fibonacci number. For all $k \in \Z$, let $F_k$ be expressed in decimal notation. The sequence of integers $\sequence n$ such that $F_n$ ends in $n$ starts: :$0, 1, 5, 25, 29, 41, 49, 61, 65, 85, 89, 101, 125, 145, 149, 245, 265, 365, 385, 485, 505, 601, \ldots$ {{OEIS|A000350}}
Transformation of P-Norm
Let $p, q \ge 1$ be real numbers. Let $\ell^p$ denote the $p$-sequence space. Let $\norm {\mathbf x}_p$ denote the $p$-norm. Let $\mathbf x = \sequence {x_n} \in \ell^{p q}$. Suppose further that $\mathbf x^p = \sequence { {x_n}^p} \in \ell^q$. Then: :$\norm {\mathbf x^p}_q = \norm {\mathbf x}_{p q}^p$
Set is Subset of Finite Suprema Set
Let $\left({S, \preceq}\right)$ be an ordered set. Let $X$ be a subset of $S$. Then $X \subseteq \operatorname{finsups}\left({X}\right)$ where $\operatorname{finsups}\left({X}\right)$ denotes finite suprema set of $X$.
Lower Closure of Subset is Subset of Lower Closure
Let $\struct {S, \preceq}$ be an ordered set. Let $X, Y$ be subsets of $S$. Then :$X \subseteq Y \implies X^\preceq \subseteq Y^\preceq$ where $X^\preceq$ is the lower closure of $X$.
Finite Suprema Set and Lower Closure is Smallest Ideal
Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice. Let $X$ be a subset of $S$. Then $X \subseteq \operatorname{finsups}\left({X}\right)^\preceq$ and :for every ideal $I$ in $L$: $X \subseteq I \implies \operatorname{finsups}\left({X}\right)^\preceq \subseteq I$ where :$\operatorname{finsups}\left({X}\right)$ denotes the finite suprema set of $X$, :$X^\preceq$ denotes the lower closure of $X$.
Set is Subset of Lower Closure
Let $\left({S, \preceq}\right)$ be an ordered set. Let $X$ be a subset of $S$. Then $X \subseteq X^\preceq$ where $X^\preceq$ denotes the lower closure of $X$.
Algorithm to determine whether Polynomial Diophantine Equation has Integer Solution
There is no algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.
Maximum Volume of Unit Radius Sphere in Fractional Dimensions
The volume of a unit sphere in $x$-dimensional Euclidean space for real $x$ occurs when $x$ is given as: :$x = 5 \cdotp 25694 \, 64048 \, 60 \ldots$ {{OEIS|A074455}} The corresponding volume at that dimension is given by: :$V = 5 \cdotp 27776 \, 80211 \, 13400 \, 997 \ldots$ {{OEIS|A074454}}
Perfect Number is Sum of Successive Odd Cubes except 6
Let $n$ be an even perfect number such that $n \ne 6$. Then: :$\displaystyle n = \sum_{k \mathop = 1}^m \paren {2 k - 1}^3 = 1^3 + 3^3 + \cdots + \paren {2 m - 1}^3$ for some $m \in \Z_{>0}$. That is, every even perfect number is the sum of the sequence of the first $r$ odd cubes, for some $r$.
Sum of Sequence of Odd Cubes
:$\displaystyle \sum_{j \mathop = 1}^n \paren {2 j - 1}^3 = 1^3 + 3^3 + 5^3 + \dotsb + \paren {2 n − 1}^3 = n^2 \paren {2 n^2 − 1}$
Bottom in Ideal
Let $\left({S, \preceq}\right)$ be a bounded below ordered set. Let $I$ be a ideal in $S$. Then $\bot \in I$ where $\bot$ denotes the smallest element of $S$.
Real Power is of Exponential Order Epsilon
Let:  :$f: \hointr 0 \to \to \R: t \mapsto t^r$ be $t$ to the power of $r$, for $r \in \R, r > -1$. Then $f$ is of exponential order $\epsilon$ for any  $\epsilon > 0$ arbitrarily small in magnitude.
Finite Suprema Set and Lower Closure is Ideal
Let $P = \left({S, \vee, \preceq}\right)$ be a join semilattice. Let $X$ be a non-empty subset of $S$. Then :${\operatorname{finsups}\left({X}\right)}^\preceq$ is ideal in $P$. where :$\operatorname{finsups}\left({X}\right)$ denotes the finite suprema set of $X$, :$X^\preceq$ denotes the lower closure of $X$.
Only Number which is Sum of 3 Factors is 6
The only positive integer which is the sum of exactly $3$ of its distinct coprime divisors is $6$.
Consecutive Integers whose Sums of Squares of Divisors are Equal
The only two consecutive positive integers whose sums of the squares of their divisors are equal are $6$ and $7$.
Local Maxima of Number of Goldbach Decompositions
Let $\mathbb E$ be the set of even positive integers. Let $G: \mathbb E \to \N$ be the mapping defined as: :$\forall n \in \mathbb E: \map G n =$ the number of Goldbach decompositions of $n$. Then $G$ has local maxima when $n$ is a multiple of $6$.
Digital Root of 3 Consecutive Numbers ending in Multiple of 3
Let $n$, $n + 1$ and $n + 2$ be positive integers such that $n + 2$ is a multiple of $3$. Let $m = n + \paren {n + 1} + \paren {n + 2}$. Then the digital root of $m$ is $6$.
Six Regular 4-Dimensional Polytopes
There exist exactly six $4$-dimensional regular polytopes: :the pentatope (also known as the $4$-simplex) :the tesseract :the $16$-cell :the $24$-cell :the $120$-cell :the $600$-cell
Characteristics of Regular 4-Dimensional Polytopes
The $4$-dimensional regular polytopes have the following characteristics: {| border="1" |- ! Name ! No. of cells ! No. of faces ! No. of edges ! No. of vertices ! Dual |- | align="right" | Pentatope | align="right" | $5$ | align="right" | $10$ | align="right" | $10$ | align="right" | $5$ | align="right" | Self-dual |- | align="right" | Tesseract | align="right" | $8$ | align="right" | $24$ | align="right" | $32$ | align="right" | $16$ | align="right" | $16$-cell |- | align="right" | $16$-cell | align="right" | $16$ | align="right" | $32$ | align="right" | $24$ | align="right" | $8$ | align="right" | Tesseract |- | align="right" | $24$-cell | align="right" | $24$ | align="right" | $96$ | align="right" | $96$ | align="right" | $24$ | align="right" | Self-dual |- | align="right" | $120$-cell | align="right" | $120$ | align="right" | $720$ | align="right" | $1200$ | align="right" | $600$ | align="right" | $600$-cell |- | align="right" | $600$-cell | align="right" | $600$ | align="right" | $1200$ | align="right" | $720$ | align="right" | $120$ | align="right" | $120$-cell |}
Six Equal Circles Tangent to Equal Circle
It is possible to arrange $6$ equal circles around another circles equal in area to each of the $6$: :500px
Complex Power is of Exponential Order Epsilon
Let:  :$f: \hointr 0 \to \to \C: t \mapsto t^\phi$ be $t$ to the power of $\phi$, for $\phi \in \C$, defined on its principal branch. Let $\map \Re \phi > -1$. Then $f$ is of exponential order $\epsilon$ for any  $\epsilon > 0$ arbitrarily small in magnitude.
Three Regular Tessellations/Hexagons
Regular hexagons form a regular tessellation: :400px
Limiting Area of Polygon with given Perimeter
Let $\mathcal P$ be the set of plane geometric figures with perimeter $L$. The element of $P$ with the largest area is the circle of radius $\dfrac L {2 \pi}$ which has area $\dfrac {L^2} {4 \pi}$.
Brianchon's Theorem
Let tangents to $6$ points on a conic section $K$ form a hexagon $H$ to circumscribe the $K$. Then the main diagonals of $H$ meet at a single point. :400px
Brianchon's Theorem is Projective Dual to Pascal's Theorem
Brianchon's Theorem is the projective dual of Pascal's Theorem.
Steiner Tree of Unit Cube
The length of the Steiner tree for the vertices of the unit cube is given approximately by: :$L = 6 \cdotp 196 \ldots$
Integer as Sum of Three Squares/Sequence
The sequence of positive integers that cannot be expressed as the sum of at most $3$ squares begins: :$7, 15, 23, 28, 31, 39, 47, 55, 60, \ldots$
Finite Subset Bounds Element of Finite Suprema Set and Lower Closure
Let $L = \left({S, \vee, \preceq}\right)$ be join semilattice. Let $I$ be ideal in $L$. Let $X$ be non empty finite subset of $S$. Let $x \in S$ such that : $x \in \left({\operatorname{finsups}\left({F \cup X}\right)}\right)^\preceq$ where :$\operatorname{finsups}$ denotes the finite suprema set, :$X^\preceq$ denotes the lower closure of $X$. Then there exists $a \in S$: $a \in I \land x \preceq a \vee \sup X$
Divisibility of Elements of Pythagorean Triple by 7
Let $\tuple {a, b, c}$ be a Pythagorean triple such that $a^2 + b^2 = c^2$. Then at least one of $a$, $b$, $a + b$ or $a - b$ is divisible by $7$.
Smaller Elements of Pythagorean Triple not both Odd
Let $\left({x, y, z}\right)$ be a Pythagorean triple, i.e. integers such that $x^2 + y^2 = z^2$. Then $x$ and $y$ cannot both be odd.
Integer as Sum of Seven Positive Cubes
Every sufficiently large integer can be expressed as the sum of no more than $7$ positive cubes.
Maximum of Seven Colors Needed for Proper Vertex Coloring on Torus
Let $G$ be a graph embedded on the surface of a torus. $G$ can be assigned a proper vertex $k$-coloring such that $k \le 7$.
Every Element is Directed and Every Two Elements are Included in Third Element implies Union is Directed
Let $P = \left({S, \preceq}\right)$ be an ordered set. Let $A$ be a set of subsets of $S$. Let :$\forall X \in A: X$ is directed. Let :$\forall X, Y \in A: \exists Z \in A: X \cup Y \subseteq Z$ Then $\bigcup A$ is directed.
Every Element is Lower implies Union is Lower
Let $\struct {S, \preceq}$ be an ordered set. Let $A$ be a set of subsets of $S$. Let :$\forall X \in A: X$ is a lower set. Then $\bigcup A$ is a lower set.
Rectangle Divided into Incomparable Subrectangles
Let $R$ be a rectangle. Let $R$ be divided into $n$ smaller rectangles which are pairwise incomparable. Then $n \ge 7$. The smallest rectangle with integer sides that can be so divided into rectangles with integer sides is $13 \times 22$. :600px
If Element Does Not Belong to Ideal then There Exists Prime Ideal Including Ideal and Excluding Element
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a distributive lattice. Let $I$ be an ideal in $L$. Let $x$ be an element of $S$. Suppose $x \notin I$ Then there exists a prime ideal $P$ in $L$: $I \subseteq P$ and $x \notin P$
Definite Integral to Infinity of Exponential of -x by Logarithm of x
Let $\ln t$ denote the natural logarithm function for real $t > 0$. Let $e^{-t}$ denote the real exponential. Then: :$\displaystyle \int_{0^+}^{ \mathop \to +\infty} \ln t \,  e^{-t} \, \mathrm d t = - \gamma$ where the  is an improper integral, and $\gamma$ is the Euler-Mascheroni Constant.
Bottom not in Proper Filter
Let $L = \struct {S, \preceq}$ be a bounded below preordered set. Let $F$ be a filter on $L$. Then $F$ is proper filter  $\bot \notin F$ where $\bot$ denotes the smallest element of $S$.
Rectangle Divided into Differently Shaped Equal Area Subrectangles
Let $R$ be a rectangle. Let $R$ be divided into $n$ smaller rectangles which are of equal area but with different lengths of sides. Then $n \ge 7$.
If Ideal and Filter are Disjoint then There Exists Prime Filter Including Filter and Disjoint from Ideal
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a distributive lattice. Let $I$ be an ideal in $L$. Let $F$ be a filter on $L$ such that :$F \cap I = \O$ Then there exists a prime filter $P$ in $L$: $F \subseteq P$ and $P \cap I = \O$
Determinant of Elementary Row Matrix
Let $\mathbf E$ be an elementary row matrix. The determinant of $\mathbf E$ is as follows:
Dual Distributive Lattice is Distributive
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Then :$L$ is a distributive lattice  :$L^{-1}$ is a distributive lattice where $L^{-1} = \left({S, \succeq}\right)$ denotes the dual of $L$.
Exchange of Rows as Sequence of Other Elementary Row Operations
Let $\mathbf A$ be an $m \times n$ matrix. Let $i, j \in \closedint 1 m: i \ne j$ Let $r_k$ denote the $k$th row of $\mathbf A$ for $1 \le k \le m$: :$r_k = \begin {pmatrix} a_{k 1} & a_{k 2} & \cdots & a_{k n} \end {pmatrix}$ Let $e$ be the elementary row operation acting on $\mathbf A$ as: {{begin-axiom}} {{axiom | n = \text {ERO} 3         | t = Interchange rows $i$ and $j$         | m = r_i \leftrightarrow r_j }} {{end-axiom}} Then $e$ can be expressed as a finite sequence of exactly $4$ instances of the other two elementary row operations. {{begin-axiom}} {{axiom | n = \text {ERO} 1         | t = For some $\lambda \in K_{\ne 0}$, multiply row $i$ by $\lambda$         | m = r_i \to \lambda r_i }} {{axiom | n = \text {ERO} 2         | t = For some $\lambda \in K$, add $\lambda$ times row $j$ to row $i$         | m = r_i \to r_i + \lambda r_j }} {{end-axiom}}
Seven Different Frieze Groups
There are $7$ different symmetry groups for a frieze.
Way Below iff Second Operand Preceding Supremum of Prime Ideal implies First Operand is Element of Ideal
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a distributive complete lattice. Let $x, y \in S$. Then $x \ll y$ : :for every prime ideal $P$ in $L$: $y \preceq \sup P \implies x \in P$
Lower Closure is Prime Ideal for Prime Element
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Let $p \in S$ be a prime element. Then $p^\preceq$ is a prime ideal.
Prime is Pseudoprime (Order Theory)
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be an up-complete lattice. Let $p \in S$ be a prime element. Then $p$ is pseudoprime.
Smallest Prime Number whose Period is of Maximum Length
$7$ is the smallest prime number the period of whose reciprocal, when expressed in decimal notation, is maximum: :$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$
Characterization of Pseudoprime Element by Finite Infima
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a continuous lattice. Let $p \in S$ be a pseudoprime element. Let $A$ be a non-empty finite subset of $S$ such that :$\inf A \ll p$ where $\ll$ denotes the way below relation. Then $\exists a \in A: a \preceq p$
Elementary Row Operations as Matrix Multiplications/Corollary
Let $\mathbf X$ and $\mathbf Y$ be two $m \times n$ matrices that differ by exactly one elementary row operation. Then there exists an elementary row matrix of order $m$ such that: :$\mathbf {E X} = \mathbf Y$
Cyclotomic Polynomial of Prime Index
Let $p$ be a prime number. The '''$p$th cyclotomic polynomial''' is: :$\map {\Phi_p} x = x^{p - 1} + x^{p - 2} + \cdots + x + 1$
Product of Cyclotomic Polynomials
Let $n > 0$ be a (strictly) positive integer. Then: :$\displaystyle \prod_{d \mathop \backslash n} \Phi_d \left({x}\right) = x^n-1$ where: :$\Phi_d \left({x}\right)$ denotes the $d$th cyclotomic polynomial :the product runs over all divisors of $n$.
First Cyclotomic Polynomial
The '''first cyclotomic polynomial''' is: :$\map {\Phi_1} x = x - 1$
Cyclotomic Polynomial has Integer Coefficients
Let $n \in \Z_{>0}$ be a positive integer. Then the $n$th cyclotomic polynomial $\map {\Phi_n} x$ has integer coefficients.
Formal Derivative of Polynomials Satisfies Leibniz's Rule
Let $R$ be a commutative ring with unity. Let $R \left[{X}\right]$ be the polynomial ring over $R$. Let $f, g \in R \left[{X}\right]$ be polynomials. Let $f'$ and $g'$ denote their formal derivatives. Then: :$\left({f g}\right)' = f g' + f' g$
Double Root of Polynomial is Root of Derivative
Let $R$ be a commutative ring with unity. Let $f \in R \left[{X}\right]$ be a polynomial. Let $a \in R$ be a root of $f$ with multiplicity at least $2$. Let $f'$ denote the formal derivative of $f$. Then $a$ is a root of $f'$.
Fano Plane is Unique Projective Plane of Order 2
The '''Fano plane''' is the unique finite projective plane of order $2$.
7 Prime Knots with 7 Crossings
There exist exactly $7$ prime knots which have exactly $7$ crossings.
Prime Divisors of Cyclotomic Polynomials
Let $n \ge 1$ be a positive integer. Let $\map {\Phi_n} x$ denote the $n$th cyclotomic polynomial. Let $a \in \Z$ be an integer such that $\map {\Phi_n} a \ne 0$. Let $p$ be a prime divisor of $\map {\Phi_n} a$. Then $p \equiv 1 \pmod n$ or $p \divides n$.
Sum of Complex Indices of Real Number
Let $r \in \R_{> 0}$ be a (strictly) positive real number. Let $\psi, \tau \in \C$ be complex numbers. Let $r^\lambda$ be defined as the the principal branch of a positive real number raised to a complex number. Then: :$r^{\psi \mathop + \tau} = r^\psi \times r^\tau$
Universal Property of Free Modules
Let $R$ be a ring. Let $M$ be a free $R$-module with basis $\{e_i\mid i\in I\}$. Let $N$ be an $R$-module. Let $\{n_i\mid i\in I\}$ be a family of elements of $N$. Then there exists a unique $R$-module homomorphism that maps $e_i$ to $n_i$ for all $i\in I$.
Odd Squares 7 Less than Nearest Power of 2
There exist exactly $3$ odd squares which are $7$ less than the nearest power of $2$: :$5^2 = 25 = 2^5 - 7$ :$11^2 = 121 = 2^7 - 7$ :$181^2 = 32 \, 761 = 2^{15} - 7$ {{OEIS|A038198}} Note that this sequence includes $1$ and $3$, being all the squares which are $7$ less than a power of $2$. However, for $1$ and $3$, those powers ($8$ and $16$ respectively) are not the nearest power of $2$ ($1$ and $4$ respectively).
Universal Property of Direct Product of Modules
Let $R$ be a ring. Let $N$ be an $R$-module. Let $\left({M_i}\right)_{i \mathop \in I}$ be a family of $R$-modules. Let $M = \displaystyle \prod_{i \mathop \in I} M_i$ be their direct product. Let $\left({\psi_i}\right)_{i \mathop \in I}$ be a family of $R$-module morphisms $N \to M_i$. Then there exists a unique morphism: :$\Psi: N \to M$ such that: : $\forall i: \psi_i = \pi_i \circ \Psi$ where $\pi_i: M \to M_i$ is the $i$th canonical projection.
Universal Property of Direct Sum of Modules
Let $R$ be a ring. Let $N$ be an $R$-module. Let $\left({M_i}\right)_{i \mathop \in I}$ be a family of $R$-modules. Let $M = \displaystyle \bigoplus_{i \mathop \in I} M_i$ be their direct sum. Let $\left({\psi_i}\right)_{i \mathop \in I}$ be a family of $R$-module morphisms $M_i \to N$. Then there exists a unique morphism: :$\Psi: M \to N$ such that: : $\forall i: \psi_i = \Psi \circ \iota_i$ where $\iota_i: M_i \to M$ is the $i$th canonical injection.
Morphism from Ring with Unity to Module
Let $R$ be a ring with unity. Let $M$ be an $R$-module. Then for every $m \in M$ there exists a unique $R$-module morphism: :$\psi: R \to M$ that sends $1$ to $m$.
Universal Property of Free Module on Set
Let $R$ be a ring with unity. Let $R^{\left({I}\right)}$ be the free $R$-module on $I$. Let $M$ be an $R$-module. Let $\left\langle{m_i}\right\rangle_{i \mathop \in I}$ be a family of elements of $M$. Then there exists a unique $R$-module morphism: :$\Psi: R^{\left({I}\right)}\to M$ that sends the $i$th canonical basis element to $m_i$, for all $i\in I$. Moreover: :$\displaystyle \Psi((r_i)_{i \mathop \in I}) = \sum_{i \mathop \in I} r_i m_i$
Multiplicative Auxiliary Relation iff Images are Filtered
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below lattice. Let $\mathcal R$ be an auxiliary relation on $S$. Then $\mathcal R$ is multiplicative  :for all $x \in S$: $\mathcal R\left({x}\right)$ is filtered where $\mathcal R\left({x}\right)$ denotes the $\mathcal R$-image of $x$.
Characterisation of Spanning Set through Free Module Indexed by Set
Let $M$ be a unitary $R$-module. Let $S = \left\langle{m_i}\right\rangle_{i \mathop \in I}$ be a family of elements of $M$. Let $\Psi: R^{\left({I}\right)} \to M$ be the morphism given by Universal Property of Free Module Indexed by Set. Then $S$ is a spanning set of $M$  $\Psi$ is surjective.
Characterisation of Linearly Independent Set through Free Module Indexed by Set
Let $M$ be a unitary $R$-module. Let $S = \left\langle{m_i}\right\rangle_{i \mathop \in I}$ be a family of elements of $M$. Let $\Psi : R^{\left({I}\right)} \to M$ be the module homomorphism given by Universal Property of Free Module on Set. Then the following are equivalent: :$S$ linearly independent :$\Psi$ is injective.
Free Module is Isomorphic to Free Module on Set
Let $M$ be a unitary $R$-module. Let $\mathcal B = \left\langle{b_i}\right\rangle_{i \mathop \in I}$ be a family of elements of $M$. Let $\Psi: R^{\left({I}\right)} \to M$ be the morphism given by Universal Property of Free Module on Set. Then the following are equivalent: :$\mathcal B$ is a basis of $M$ :$\Psi$ is an isomorphism
Direct Product of Modules is Module
Let $R$ be a ring. Let $\left\{\left\langle {M_i,+_i,\circ_i} \right\rangle\right\}_{i \in I}$ be a family of $R$-modules. Let $\left\langle{M, +, \circ}\right\rangle$ be their direct product. Then $\left\langle{M, +, \circ}\right\rangle$ is a module.
Direct Product of Unitary Modules is Unitary Module
Let $R$ be a ring with unity. Let $\left\{ {M_i}\right\}_{i \in I}$ be a family of unitary $R$-modules. Let $\left({M, +, \circ}\right)$ be their direct product. Then $\left({M, +, \circ}\right)$ is a unitary $R$-module.
Auxiliary Relation Image of Element is Upper Set
Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $R$ be an auxiliary relation on $S$. Let $x \in S$. Then $R\left({x}\right)$ is an upper set where $R\left({x}\right)$ denotes the image of $x$ under $R$.
Multiplicative Auxiliary Relation iff Congruent
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below lattice. Let $\mathcal R$ be an auxiliary relation on $S$. Then $\mathcal R$ is multiplicative : :$\forall a, b, x, y \in S: \left({a, x}\right), \left({b, y}\right) \in \mathcal R \implies \left({a \wedge b, x \wedge y}\right) \in \mathcal R$ That is iff $\mathcal R$ is a congruence relation for $\wedge$.
Cube which is One Less than a Square
$8$ is the only cube number which is $1$ less than a square: :$2^3 + 1 = 3^2$
Prime Powers Differing by One
$8$ and $9$ are the only powers of prime numbers which differ by exactly $1$: :$2^3 + 1 = 3^2$
Cubic Fibonacci Numbers
The only Fibonacci numbers which are also cubes are: :$F_1 = 1 = 1^3$ :$F_6 = 8 = 2^3 = 3 + 5$
Monoid Ring of Commutative Monoid over Commutative Ring is Commutative
Let $R$ be a commutative ring. Let $G$ be a commutative monoid. Let $R[G]$ be the monoid ring of $G$ over $R$. Then $R[G]$ is commutative.
3 Non-Parallel Planes divide Space into 8
Let $3$ planes which are pairwise non-parallel be constructed in ordinary $3$-dimensional space. Then that space is divided into $8$ parts by those planes.
8 Mutually Non-Attacking Queens on Chessboard
On a standard chessboard, it is possible to arrange a maximum of $8$ queens so that no queen is attacking any other queen. There are $12$ such arrangements, up to rotation and reflection.
Divisibility by 8
An integer $N$ expressed in decimal notation is divisible by $8$  the $3$ least significant digits of $N$ form a $3$-digit integer divisible by $8$. 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 $8$  : :$100 a_2 + 10 a_1 + a_0$ is divisible by $8$.
No Perfect Magic Cube of Order Less than 5 Exists
Apart from the trivial order $1$ case, no perfect magic cube exists whose order is $4$ or less.
Modulus of Positive Real Number to Complex Power is Positive Real Number to Power of Real Part
Let $z \in \C$ be a complex number. Let $t > 0$ be wholly real. Let $t^z$ be $t$ to the power of $z$ defined on its principal branch. Then: :$\cmod {t^z} = t^{\map \Re z}$
Characterization of Pseudoprime Element when Way Below Relation is Multiplicative
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 pseudoprime element  :$\forall a, b \in S: a \wedge b \ll p \implies a \preceq p \lor b \preceq p$