Fibonacci Prime has Prime Index except for 3

Let $F_n$ denote the $n$th Fibonacci number. Let $F_n$ be a prime number. Then, apart from $F_4 = 3$, $n$ is a prime number.


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 {{LHS}} is a complex Riemann integral : the integral on the {{RHS}} 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 {{LHS}} 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 {{iff}} $\bot \notin F$ 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 {{iff}} :$L^{-1}$ 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 $x \ll y$ {{iff}}: :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 {{iff}} :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 $M$ {{iff}} $\Psi$ 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 {{iff}}: :$\forall a, b, x, y \in S: \left({a, x}\right), \left({b, y}\right) \in \mathcal R \implies \left({a \wedge b, x \wedge y}\right) \in \mathcal R$ 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 $8$ {{iff}} 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 $8$  {{iff}}: :$100 a_2 + 10 a_1 + a_0$ 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 {{iff}} :$\forall a, b \in S: a \wedge b \ll p \implies a \preceq p \lor b \preceq p$


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 $L$ {{iff}} :$x$ 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 $\mathbf A \in \MM_{m \times n}, \; \mathbf B \in \MM_{n \times p}$ {{iff}} $p = m = n = 1$.


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 {{iff}}: :$L$ 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 {{iff}} :$\ll$ 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 :$\displaystyle \map f n = \prod_{d \mathop \divides n} \map g d$ {{iff}}: :$\displaystyle \map g n = \prod_{d \mathop \divides n} \map f d^{\mu \paren {\frac n d} }$


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 {{iff}} $\left({K\left({L}\right), \precsim}\right)$ 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 $f$ {{iff}}: :$\forall x \in \R: \map f {x \bmod L} = \map f x$ 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 {{iff}} $x$ 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 $9$.


123456789 x 9 + 10 = 1111111111

{{begin-eqn}} {{eqn | l = 1 \times 9 + 2       | r = 11 }} {{eqn | l = 12 \times 9 + 3       | r = 111 }} {{eqn | l = 123 \times 9 + 4       | r = 1111 }} {{eqn | l = 1234 \times 9 + 5       | r = 11111 }} {{eqn | l = 12345 \times 9 + 6       | r = 111111 }} {{eqn | l = 123456 \times 9 + 7       | r = 1111111 }} {{eqn | l = 1234567 \times 9 + 8       | r = 11111111 }} {{eqn | l = 12345678 \times 9 + 9       | r = 111111111 }} {{eqn | l = 123456789 \times 9 + 10       | r = 1111111111 }} {{end-eqn}} The above pattern is an instance of the identity: :$\displaystyle \left({b - 1}\right) \sum_{j \mathop = 1}^n j b^{n - j} + n + 1 = \sum_{j \mathop = 0}^n b^j$ where $b = 10$ and $n$ goes from $1$ to $9$.


Algebraic Numbers form Field

The set of real numbers $\R$ forms a field under addition and multiplication: $\left({\R, +, \times}\right)$. {{wtd|correct the statement}}


Linearly Independent over the Rational Numbers iff Linearly Independent over the Integers

Let $z_1, z_2, \ldots, z_n$ be complex numbers. Then: :$z_1, z_2, \ldots, z_n$ are linearly independent over the rational numbers $\Q$ {{iff}}: :$z_1, z_2, \ldots, z_n$ are linearly independent over the integers $\Z$.


Sums of Squares of Diagonals of Order 3 Magic Square

Consider the order 3 magic square: {{:Magic Square/Examples/Order 3}} The sums of the squares of the diagonals, when expressed as $3$-digit decimal numbers, are equal to the sums of the squares of those same diagonals of that same order 3 magic square when reversed. {{improve|Find a way to describe the "diagonals" accurately, as what is being demonstrated here does not match the description.}}


Lambert W of Non-Zero Algebraic Number is Transcendental

Let $W$ denote the (general) Lambert W function. Let $a$ be a non-zero algebraic number. Then, $W \left({a}\right)$ is transcendental.


Way Below in Lattice of Power Set

Let $X$ be a set. Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \preceq}\right)$ be a lattice of power set of $X$ where $\mathord\preceq = \mathord\subseteq \cap \left({\mathcal P\left({X}\right) \times \mathcal P\left({X}\right)}\right)$ Let $x, y \in \mathcal P\left({X}\right)$. Then $x \ll y$ {{iff}} :for every a set $Y$ of subsets of $X$ such that $y \subseteq \bigcup Y$ ::then there exists a finite subset $Z$ of $Y$: $x \subseteq \bigcup Z$ where $\ll$ denotes the way below relation.


Divisibility by 10

An integer $N$ expressed in decimal notation is divisible by $10$ {{iff}} the least significant digit of $N$ is $0$. 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 $10$  {{iff}}: :$a_0 = 0$


Difference of Two Squares cannot equal 2 modulo 4

Let $n \in \Z_{>0}$ be of the form $4 k + 2$ for some $k \in \Z$. Then $n$ cannot be expressed in the form: :$n = a^2 - b^2$ for $a, b \in \Z$.


Compact Closure is Set of Finite Subsets in Lattice of Power Set

Let $X$ be a set. Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \preceq}\right)$ be the lattice of power set of $X$ where $\mathord\preceq = \mathord\subseteq \cap \mathcal P\left({X}\right) \times \mathcal P\left({X}\right)$ Let $x \in \mathcal P\left({X}\right)$. Then $x^{\mathrm{compact} } = \mathit{Fin}\left({x}\right)$ where $\mathit{Fin}\left({x}\right)$ denotes the set of all finite subsets of $x$.


10 is Only Triangular Number that is Sum of Consecutive Odd Squares

$10$ is the only triangular number which is the sum of two consecutive odd squares: :$10 = 1^2 + 3^2$


10 Consecutive Integers contain Coprime Integer

Let $n \in \Z$ be an integer. Let $S := \set {n, n + 1, n + 2, \ldots, n + 9}$ be the set of $10$ consecutive integers starting from $n$. Then at least one element of $S$ is coprime to every other element of $S$.


Two Fifths as Pandigital Fraction

There are $3$ ways $\dfrac 2 5$ can be expressed as a pandigital fraction: :$\dfrac 2 5 = \dfrac {6894} {17235}$ :$\dfrac 2 5 = \dfrac {8694} {21735}$ :$\dfrac 2 5 = \dfrac {9486} {23715}$


Two Sevenths as Pandigital Fraction

There are $6$ ways $\dfrac 2 7$ can be expressed as a pandigital fraction: :$\dfrac 2 7 = \dfrac {3654} {12789}$ :$\dfrac 2 7 = \dfrac {3674} {12859}$ :$\dfrac 2 7 = \dfrac {5342} {18697}$ :$\dfrac 2 7 = \dfrac {7418} {25963}$ :$\dfrac 2 7 = \dfrac {9786} {34251}$ :$\dfrac 2 7 = \dfrac {9862} {34517}$


Two Ninths as Pandigital Fraction

There are $2$ ways $\dfrac 2 9$ can be expressed as a pandigital fraction: :$\dfrac 2 9 = \dfrac {3924} {17658}$ :$\dfrac 2 9 = \dfrac {7596} {34182}$


Closed Form for Tetrahedral Numbers

The closed-form expression for the $n$th tetrahedral number is: :$H_n = \dfrac {n \paren {n + 1} \paren {n + 2} } 6$


Factorial as Product of Two Factorials

Apart from the general pattern, following directly from the definition of the factorial: :$\paren {n!}! = n! \paren {n! - 1}!$ the only known factorial which is the product of two factorials is: :$10! = 6! \, 7!$


Two Thirds as Pandigital Fraction

$\dfrac 2 3$ cannot be expressed as a pandigital fraction.


Three Quarters as Pandigital Fraction

$\dfrac 3 4$ cannot be expressed as a pandigital fraction.


Three Fifths as Pandigital Fraction

$\dfrac 3 5$ cannot be expressed as a pandigital fraction.


Three Sevenths as Pandigital Fraction

$\dfrac 3 7$ cannot be expressed as a pandigital fraction.


Three Eighths as Pandigital Fraction

$\dfrac 3 8$ cannot be expressed as a pandigital fraction.


Four Sevenths as Pandigital Fraction

$\dfrac 4 7$ cannot be expressed as a pandigital fraction.


Five Sevenths as Pandigital Fraction

$\dfrac 5 7$ cannot be expressed as a pandigital fraction.


Six Sevenths as Pandigital Fraction

$\dfrac 6 7$ cannot be expressed as a pandigital fraction.


Five Sixths as Pandigital Fraction

$\dfrac 5 6$ cannot be expressed as a pandigital fraction.


Five Eighths as Pandigital Fraction

$\dfrac 5 8$ cannot be expressed as a pandigital fraction.


Seven Eighths as Pandigital Fraction

$\dfrac 7 8$ cannot be expressed as a pandigital fraction.


Four Ninths as Pandigital Fraction

$\dfrac 4 9$ cannot be expressed as a pandigital fraction.


Five Ninths as Pandigital Fraction

$\dfrac 5 9$ cannot be expressed as a pandigital fraction.


Seven Ninths as Pandigital Fraction

$\dfrac 7 9$ cannot be expressed as a pandigital fraction.


Eight Ninths as Pandigital Fraction

$\dfrac 8 9$ cannot be expressed as a pandigital fraction.


Generating Function for Triangular Numbers

Let $T_n$ denote the $n$th triangular number. Then the generating function for $\sequence {T_n}$ is given as: :$\displaystyle \map G z = \frac z {\paren {1 - z}^3}$


Lattice of Power Set is Algebraic

Let $X$ be a set. Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \preceq}\right)$ be the lattice of power set of $X$ where $\mathord\preceq = \mathord\subseteq \cap \left({\mathcal P\left({X}\right) \times \mathcal P\left({X}\right)}\right)$ Then $L$ is algebraic.


Difference of Squares of Sum and Difference

:$\forall a, b \in \R: \left({a + b}\right)^2 - \left({a - b}\right)^2 = 4 a b$


Is there a Limit to the Multiplicative Persistence of a Number?

It has been conjectured that there may be an upper limit to the multiplicative persistence of a natural number.


Non-Empty Compact Closure is Directed

Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice. Let $x \in S$ such that :$x^{\mathrm{compact} }$ is a non-empty set, where $x^{\mathrm{compact} }$ denotes the compact closure of $x$. Then :$x^{\mathrm{compact} }$ is directed.


Set of Cardinality not Greater than Cardinality of Finite Set is Finite

Let $X, Y$ be sets such that :$\left\vert X \right\vert \le \left\vert Y \right\vert$ and :$Y$ is finite, where $\left\vert X \right\vert$ denotes the cardinality of $X$. Then $X$ is finite.


Finite iff Cardinality Less than Aleph Zero

Let $X$ be a set. Then $X$ is finite {{iff}} $\card X < \aleph_0$ where: :$\card X$ denotes the cardinality of $X$ :$\aleph_0 = \card \N$ by Aleph Zero equals Cardinality of Naturals.


P-adic Valuation of Difference of Powers with Coprime Exponent

Let $x, y \in \Z$ be distinct integers. Let $n \ge 1$ be a natural number. Let $p$ be a prime number. Let: :$p \divides x - y$ and: :$p \nmid x y n$. Then  :$\map {\nu_p} {x^n - y^n} = \map {\nu_p} {x - y}$


One of 4 Consecutive Numbers Greater than 11 is Divisible by Prime Greater than 11

Let $n \in \Z$ such that $n > 11$. Then at least one of the set: :$\set {n, n + 1, n + 2, n + 3}$ is divisible by a prime number greater than $11$.


Ratio of Consecutive Lucas Numbers

For $n \in \N$, let $L_n$ be the $n$th Lucas number. Then: :$\displaystyle \lim_{n \mathop \to \infty} \frac {L_{n + 1} } {L_n} = \phi$ where $\phi = \dfrac {1 + \sqrt 5} 2$ is the golden mean.


Ordered Set of All Mappings is Ordered Set

Let $L = \struct {S, \preceq}$ be an ordered set. Let $X$ be a set. Then $L^X$ is also an ordered set.


Closed Form for Lucas Numbers

The Lucas numbers have a closed-form solution: :$L_n = \phi^n + \paren {1 - \phi}^n = \paren {\dfrac {1 + \sqrt 5} 2}^n + \paren {\dfrac {1 - \sqrt 5} 2}^n$ where $\phi$ is the golden mean. Putting $\hat \phi = 1 - \phi = -\dfrac 1 \phi$ this can be written: :$L_n = \phi^n + \hat \phi^n$


Fibonacci Number 2n equals Fibonacci Number n by Lucas Number n

Let $F_n$ denote the $n$th Fibonacci number. Let $L_n$ denote the $n$th Lucas number. Then: :$F_{2 n} = F_n L_n$


Fibonacci Number 3n in terms of Fibonacci Number n and Lucas Number 2n

Let $F_n$ denote the $n$th Fibonacci number. Let $L_n$ denote the $n$th Lucas number. Then: :$F_{3 n} = F_n \paren {L_{2 n} + \paren {-1}^n}$


Relation between Square of Fibonacci Number and Square of Lucas Number

Let $F_n$ denote the $n$th Fibonacci number. Let $L_n$ denote the $n$th Lucas number. Then: :$5 {F_n}^2 + 4 \paren {-1}^n = {L_n}^2$


12 Pentominoes

There exist $12$ distinct free pentominoes: :600px


18 Fixed Pentominoes

There exist $18$ distinct fixed pentominoes: :600px


Differential of Differentiable Functional is Unique

The differential of a differentiable functional is unique.


Ordered Subset of Ordered Set is Ordered Set

Let $L = \struct {S, \preceq}$ be an ordered set. Let $\struct {S', \preceq'}$ be an ordered subset of $L$. Then $\struct {S', \preceq'}$ is an ordered set.


Finite Subsets form Ideal

Let $X$ be a set. Let $\mathit{Fin}\left({X}\right)$ be the set of all finite subsets of $X$. Then $\mathit{Fin}\left({X}\right)$ is ideal in $\left({\mathcal P\left({X}\right), \subseteq}\right)$ where $\mathcal P\left({X}\right)$ denotes the power set of $X$.


Product of Proper Divisors

Let $n$ be an integer such that $n \ge 1$. Let $\map P n$ denote the product of the proper divisors of $n$. Then: :$\map P n = n^{\map \tau n / 2 - 1}$ where $\map \tau n$ denotes the $\tau$ function of $n$.


Product of Divisors

Let $n$ be an integer such that $n \ge 1$. Let $\map D n$ denote the product of the divisors of $n$. Then: :$\map D n = n^{\map \tau n / 2}$ where $\map \tau n$ denotes the divisor counting function of $n$.


Square of Reversal of Small-Digit Number

Let $n$ be an integer whose decimal representation consists of sufficiently small digits. Then the reversal of the square of $n$ is the square of the reversal of $n$.


12 times Sigma of 12 equals 14 times Sigma of 14

$x = 12$ and $y = 14$ are solutions to the indeterminate equation: :$x \ \sigma \left({x}\right) = y \ \sigma \left({y}\right)$ where $\sigma$ denotes the $\sigma$ function.


12 Knights to Attack or Occupy All Squares on Chessboard

On a standard chessboard, it is a maximum of $12$ knights are needed to ensure all squares are either occipied or under attack.


12 Identical Spheres can touch One Other Sphere

A total of $12$ identical spheres can touch one other such sphere. Each of the outer spheres touch the sphere in the center and $3$ other spheres.


Number of Multidimensional Spheres that can touch One Other Sphere

Let $S_n$ be a spheres of $n$ dimensions with a given radius $r$. Let $\map T n$ denote the number of instances of $S_n$ that can touch one other such instance of $S_n$. The sequence of $\map T n$ begins as follows: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="right" style = "padding: 2px 10px" | $\map T n$  |- | align="right" style = "padding: 2px 10px" | $0$ | align="right" style = "padding: 2px 10px" | $0$ |- | align="right" style = "padding: 2px 10px" | $1$ | align="right" style = "padding: 2px 10px" | $2$ |- | align="right" style = "padding: 2px 10px" | $2$ | align="right" style = "padding: 2px 10px" | $6$ |- | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $12$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $24$ |- | align="right" style = "padding: 2px 10px" | $5$ | align="right" style = "padding: 2px 10px" | $40$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="right" style = "padding: 2px 10px" | $72$ |- | align="right" style = "padding: 2px 10px" | $7$ | align="right" style = "padding: 2px 10px" | $126$ |- | align="right" style = "padding: 2px 10px" | $8$ | align="right" style = "padding: 2px 10px" | $240$ |- | align="right" style = "padding: 2px 10px" | $9$ | align="right" style = "padding: 2px 10px" | $272$ |} {{OEIS|A001116}} </onlyinclude>


Ordering on Closure Operators iff Images are Including

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Let $f, g: S \to S$ be closure operators on $L$. Then: :$f \preceq g$ {{iff}} $g \sqbrk S \subseteq f \sqbrk S$ where :$\preceq$ denotes the ordering on mappings :$f \sqbrk S$ denotes the image of $f$.


Multiple of Perfect Number is Abundant

Let $n$ be a perfect number. Let $m$ be a positive integer such that $m > 1$. Then $m n$ is abundant.


Multiple of Abundant Number is Abundant

Let $n$ be an abundant number. Let $m$ be a positive integer such that $m > 1$. Then $m n$ is abundant.


Divisor of Perfect Number is Deficient

Let $n$ be a perfect number. Let $n = r s$ where $r$ and $s$ are positive integers such that $r > 1$ and $s > 1$. Then $r$ and $s$ are both deficient.


Divisor of Deficient Number is Deficient

Let $n$ be a perfect number. Let $n = k d$ where $r$ is a positive integer. Then $k$ is deficient.


Abundancy Index of Product is greater than Abundancy Index of Proper Factors

Let $n \in \Z_{>0}$ be a composite number such that $n = r s$, where $r, s \in \Z_{>1}$. Then: :$\dfrac {\map \sigma n} n > \dfrac {\map \sigma r} r$ and consequently also: :$\dfrac {\map \sigma n} n > \dfrac {\map \sigma s} s$ where $\map \sigma n$ denotes the $\sigma$ function of $n$. That is, the abundancy index of a composite number is strictly greater than the abundancy index of its divisors.


Condition for Differentiable Functional to have Extremum

Let $S$ be a set of mappings. Let $y, h \in S: \R \to \R$ be real functions.  Let $J \sqbrk y: S \to \R$ be a differentiable functional. Then a necessary condition for the differentiable functional $J \sqbrk {y; h}$ to have an extremum for $y = \hat y$ is: :$\bigvalueat {\delta J \sqbrk {y; h} } {y \mathop = \hat y} = 0$


Ordered Set of Closure Systems is Ordered Set

Let $L = \left({S, \preceq}\right)$ be an ordered set. Then $\operatorname{ClSystems}\left({L}\right)$ is an ordered set, where $\operatorname{ClSystems}\left({L}\right)$ denotes the ordered set of closure systems.


Necessary Condition for Integral Functional to have Extremum for given function

Let $S$ be a set of real mappings such that: :$S = \set {\map y x: \paren {y: S_1 \subseteq \R \to S_2 \subseteq \R}, \paren {\map y x \in C^1 \closedint a b}, \paren {\map y a = A, \map y b = B} }$ Let $J \sqbrk y: S \to S_3 \subseteq \R$ be a functional of the form: :$\displaystyle \int_a^b \map F {x, y, y'} \rd x$  Then a necessary condition for $J \sqbrk y$ to have an extremum (strong or weak) for a given function $\map y x$ is that $\map y x$ satisfy Euler's equation: :$F_y - \dfrac \d {\d x} F_{y'} = 0$


Image of Closure Operator Inherits Infima

Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $f$ be a closure operator on $L$. Then $R = \left({f\left[{S}\right], \precsim}\right)$ inherits infima, where :$\mathord\precsim = \mathord\preceq \cap \left({f\left[{S}\right] \times f\left[{S}\right]}\right)$ :$f\left[{S}\right]$ denotes the image of $f$.


Necessary Condition for Integral Functional to have Extremum for given function/Lemma

Let $\map \alpha x, \map \beta x$ be real functions. Let $\map \alpha x, \map \beta x$ be continuous on $\closedint a b$. Let: :$\forall \map h x \in C^1: \displaystyle \int_a^b \paren {\map \alpha x \map h x + \map \beta x \map {h'} x} \d x = 0$ subject to the boundary conditions: :$\map h a = \map h b = 0$ Then $\map \beta x$ is differentiable. Furthermore: :$\forall x \in \closedint a b: \map {\beta'} x = \map \alpha x$


If Definite Integral of a(x)h(x) vanishes for any C^0 h(x) then C^0 a(x) vanishes

Let $\map \alpha x$ be a continuous real function on the closed real interval $\closedint a b$. Let $\displaystyle \int_a^b \map \alpha x \map h x \rd x = 0$ for every real function $\map h x \in C^0 \closedint a b$ such that $\map h a = 0$ and $\map h b = 0$. {{explain|the notation $C^0 \closedint a b$}} Then $\map \alpha x = 0$ for all $x \in \closedint a b$.


Closure Operator does not Change Infimum of Subset of Image

Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $c: S \to S$ be a closure operator on $L$. Let $X$ be a subset of $c\left[{S}\right]$ such that :$X$ admits an infimum, where $c\left[{S}\right]$ denotes the image of $c$. Then $\inf X = c\left({\inf X}\right)$


Infimum in Ordered Subset

Let $L = \struct {S, \preceq}$ be an ordered set. Let $R = \struct {T, \preceq'}$ be an ordered subset of $L$. Let $X \subseteq T$ such that :$X$ admits an infimum in $L$. Then $\inf_L X \in T$ {{iff}} :$X$ admits an infimum in $R$ and $\inf_R X = \inf_L X$


Operator Generated by Closure System is Closure Operator

Let $L = \left({X, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $S = \left({T, \precsim}\right)$ be a closure system of $L$. Then $\operatorname{operator}\left({S}\right)$ is closure operator, where $\operatorname{operator}\left({S}\right)$ denotes the operator generated by $S$.


Regular Dodecahedron is Dual of Regular Icosahedron

The regular dodecahedron is the dual of the regular icosahedron.


Image of Operator Generated by Closure System is Set of Closure System

Let $L = \left({X, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $S = \left({T, \precsim}\right)$ be a closure system of $L$. Then $\operatorname{operator}\left({S}\right)\left[{X}\right] = T$ where $\operatorname{operator}\left({S}\right)$ denotes the operator generated by $S$.


Regular Icosahedron as Pentagonal Antiprism with Pyramidal Endcaps

The regular icosahedron can be considered as a regular pentagonal antiprism with two regular pentagonal pyramid as end caps. {{stub|Need to construct an icosahedron.}}


Construction of Rhombic Dodecahedron

The rhombic dodecahedron can be constructed as follows: Take a cube $K$ embedded in $3$-dimensional space. Place $6$ more cubes, each congruent with $K$, so that one face of each coincides with a different face of $K$. Join the vertices of $K$ to the centers of the adjacent cubes to describe square pyramids whose apices are the centers of the adjacent cubes and whose bases are the faces of $K$. The polyhedron formed by the $6$ square pyramids so formed, together with $K$, is a rhombic dodecahedron.


Recurring Parts of Multiples of One Thirteenth

The multiples of $\dfrac 1 {13}$ from $\dfrac 1 {13}$ to $\dfrac {12} {13}$ can be divided into two sets of equal size: :one where the digits of the recurring part consists of a cyclic permutation of $076923$ :one where the digits of the recurring part consists of a cyclic permutation of $153846$. :300px


Twelve Factorial plus One is divisible by 13 Squared

:$12! + 1$ is divisible by $13^2$.


Torus can be cut into 13 Pieces with 3 Plane Cuts

A torus can be cut into as many as $13$ separate pieces by $3$ plane cuts.


Operator Generated by Image of Closure Operator is Closure Operator

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Let $c: S \to S$ be a closure operator on $L$. Then $\map {\operatorname {operator} } {\struct {c \sqbrk S, \precsim} } = c$ where :$\mathord \precsim = \mathord \preceq \cap \paren {c \sqbrk S \times c \sqbrk S}$ :$\map {\operatorname {operator} } {\struct {c \sqbrk S, \precsim} }$ denotes the operator generated by $\struct {c \sqbrk S, \precsim}$


Pell's Equation/Examples/13

:$x^2 - 13 y^2 = 1$ has the smallest positive integral solution: :$x = 649$ :$y = 180$


Pell's Equation/Examples/29

:$x^2 - 29 y^2 = 1$ has the smallest positive integral solution: :$x = 9801$ :$y = 1820$


Vanishing First Variational Derivative implies Euler's Equation for Vanishing Variation

Let $\map y x$ be a real function such that $\map y a = A$ and $\map y b = B$. Let $J \sqbrk y$ be a functional of the form: :$\displaystyle J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$ Then: :$\dfrac {\delta J} {\delta y} = 0 \implies F_y - \dfrac \d {\d x} F_{y'} = 0$


Supremum in Ordered Subset

Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $R = \left({T, \preceq'}\right)$ be an ordered subset of $L$. Let $X \subseteq T$ such that :$X$ admits an supremum in $L$. Then $\sup_L X \in T$ {{iff}} :$X$ admits an supremum in $R$ and $\sup_R X = \sup_L X$


Partial Quotients of Continued Fraction Expansion of Irrational Square Root

Let $n \in \Z$ such that $n$ is not a square. Let the continued fraction expansion of $\sqrt n$ be expressed as: :$\left[{a_0, a_1, a_2, \ldots}\right]$ Then the partial quotients of this continued fraction expansion can be calculated as: :$a_r = \left\lfloor{\dfrac{\left\lfloor{\sqrt n}\right\rfloor + P_r} {Q_r} }\right\rfloor$ where: :$P_r = \begin{cases} 0 & : r = 0 \\ a_{r - 1} Q_{r - 1} - P_{r - 1} & : r > 0 \\ \end{cases}$ :$Q_r = \begin{cases} 1 & : r = 0 \\ \dfrac {n - {P_r}^2} {Q_{r - 1} } & : r > 0 \\ \end{cases}$


Operator Generated by Closure System Preserves Directed Suprema iff Closure System Inherits Directed Suprema

Let $L = \left({X, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $S = \left({Y, \precsim}\right)$ be a closure system on $L$. Then $\operatorname{operator}\left({S}\right)$ preserves directed suprema {{iff}} $S$ inherits directed suprema. where $\operatorname{operator}\left({S}\right)$ denotes the operator generated by $S$.


Closure Operator Preserves Directed Suprema iff Image of Closure Operator Inherits Directed Suprema

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $h:S \to S$ be a closure operator on $L$. Then $h$ preserves directed suprema {{iff}} $\left({h \left[{S}\right], \precsim}\right)$ inherits directed suprema. where :$h \left[{S}\right]$ denotes the image of $h$, :$\mathord\precsim = \mathord\preceq \cap \left({h \left[{S}\right] \times h \left[{S}\right]}\right)$


Thirteen Archimedean Polyhedra

There exist exactly $13$ distinct Archimedean polyhedra: :Truncated tetrahedron :Cuboctahedron :Truncated Cube :Truncated octahedron :Rhombicuboctahedron :Truncated cuboctahedron :Snub cube :Icosidodecahedron :Truncated dodecahedron :Truncated icosahedron :Rhombicosidodecahedron :Truncated icosidodecahedron :Snub dodecahedron


Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions

Let $\mathbf y$ be an $n$-dimensional real vector. Let $J \sqbrk {\mathbf y}$ be a functional of the form: $\displaystyle J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ Let: :$\mathbf y \in C^1 \closedint a b$ where $C^1 \closedint a b$ denotes that $\mathbf y$ is continuously differentiable in $\closedint a b$ Let $\mathbf y$ satisfy boundary conditions: :$\map {\mathbf y} a = \mathbf A$  :$\map {\mathbf y} b = \mathbf B$ where $\mathbf A$, $\mathbf B$ are real vectors. Then a necessary condition for $J \sqbrk {\mathbf y}$ to have an extremum (strong or weak) for a given $\mathbf y$ is that they satisfy Euler's equations: $F_{\mathbf y} - \dfrac \d {\d x} F_{\mathbf y'} = 0$


Regular Prism is Semiregular Polyhedron

A regular prism is a semiregular polyhedron.


Regular Antiprism is Semiregular Polyhedron

A regular antiprism is a semiregular polyhedron.


Finer Supremum Precedes Supremum

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $X, Y$ be subsets of $S$ such that :$X$ is finer than $Y$. Then $\sup X \preceq \sup Y$ where $\sup X$ denotes the supremum of $X$.


Thirteen Catalan Polyhedra

There exist exactly $13$ distinct Catalan polyhedra: :Triakis tetrahedron :Triakis octahedron :Disdyakis dodecahedron :Tetrakis hexahedron :Triakis icosahedron :Disdyakis triacontahedron :Pentakis dodecahedron :Rhombic dodecahedron :Rhombic triacontahedron :Deltoidal icositetrahedron :Deltoidal hexecontahedron :Pentagonal icositetrahedron :Pentagonal hexecontahedron


Necessary and Sufficient Condition for Integral Parametric Functional to be Independent of Parametric Representation

Let $x: \R \to \R$ and $y: \R \to \R$ be real functions. Let $J \sqbrk {x, y}$ be a functional of the form $\displaystyle J \sqbrk {x, y} = \int_{t_0}^{t_1} \map \Phi {t, x, y, \dot x, \dot y} \rd t$ where $\dot y$ denotes the derivative of $y$ {{WRT|Differentiation}} $t$: :$\dot y = \dfrac {\d y} {\d t}$ Then $J \sqbrk {x, y}$ depends only on the curve in the $x y$-plane defined by the parametric equations: :$x = \map x t$, $y = \map y t$  and not on the choice of the parametric representation of the curve {{iff}} the integrand $\Phi$ does not involve $t$ explicitly  and is a positive-homogeneous function of degree $1$ in $\dot x$ and $\dot y$.


Intersection of Upper Set with Directed Set is Directed Set

Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $A, B$ be subsets of $S$ such that :$A \cap B \ne \varnothing$ and :$A$ is an upper set, :$B$ is a directed set. Then $A \cap B$ is a directed set.


Convex Set is Contractible

Let $V$ be a topological vector space over $\R$ or $\C$. Let $A\subset V$ be a convex subset. Then $A$ is contractible.


Open Subgroup is Closed

Let $G$ be a topological group. Let $H\leq G$ be an open subgroup. Then $H$ is closed.


Topological Group is Hausdorff iff Identity is Closed

Let $G$ be a topological group. Let $e$ be its identity element. Then $G$ is Hausdorff {{iff}} $\left\{ {e}\right\}$ is closed in $G$.


Jensen's Inequality (Complex Analysis)

Let $D \subset \C$ be an open set with $0 \in D$. Let $R > 0$ be such that $\map B {0, R} \subset D$. Let $f: D \to \C$ be analytic with $\map f 0 \ne 0$. Let $\cmod {\map f z} \le M$ for $\cmod z \le R$. Let $0 < r <R$. Then the number of zeroes of $f$, counted with multiplicity, for $\cmod z \le r$, is at most: :$\dfrac {\map \log {M / \cmod {\map f 0} } } {\map \log {R / r} }$


Complement of Subset with Property (S) is Closed under Directed Suprema

Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $X$ be a subset of $S$ with property (S). Then $\complement_S\left({X}\right)$ is closed under directed suprema.


Lifting The Exponent Lemma for Sums

Let $x, y \in \Z$ be integers with $x + y \ne 0$. Let $n \ge 1$ be an odd natural number. Let $p$ be an odd prime. Let: :$p \divides x + y$ and: :$p \nmid x y$ where $\divides$ and $\nmid$ denote divisibility and non-divisibility respectively. Then: :$\map {\nu_p} {x^n + y^n} = \map {\nu_p} {x + y} + \map {\nu_p} n$ where $\nu_p$ denotes $p$-adic valuation. </onlyinclude>


Lifting The Exponent Lemma for p=2

Let $x, y \in \Z$ be distinct odd integers. Let $n \ge 1$ be a natural number. Let: :$4 \divides x - y$ where $\divides$ denotes divisibility. Then :$\map {\nu_2} {x^n - y^n} = \map {\nu_2} {x - y} + \map {\nu_2} n$ where $\nu_2$ denotes $2$-adic valuation.


Classification of Convex Polyhedra whose Faces are Regular Polygons

The convex polyhedra whose faces are all regular polygons are as follows: :The $5$ Platonic solids :The regular prisms and regular antiprisms, countably infinite in number :The $13$ Archimedean polyhedra :The $92$ Johnson polyhedra.


Chinese Remainder Theorem/General Result 2

Let $n_1, n_2, \ldots, n_k$ be positive integers. Let $b_1, \ldots, b_k$ be integers such that: : $\forall i \ne j: \gcd \left({n_i, n_j}\right) \mathrel \backslash b_i - b_j$ where $\backslash$ denotes divisibility. Then the system of linear congruences: : $x \equiv b_1 \pmod {n_1}$ : $x \equiv b_2 \pmod {n_2}$ : $\ldots$ : $x \equiv b_k \pmod {n_k}$ has a simultaneous solution which is unique modulo $\lcm \left({n_1, \ldots, n_k}\right)$.


Necessary Condition for Integral Functional to have Extremum for given Function/Dependent on Nth Derivative of Function

Let $\map F {x, y, z_1, \ldots, z_n}$ be a function in differentiability class $C^2$ {{WRT}} all its variables. Let $y = \map y x \in C^n\openint a b$ such that: :$\map y a = A_0, \map {y'} a = A_1, \ldots, \map {y^{\paren {n - 1} } } a = A_{n - 1}$ and: :$\map y b = B_0, \map {y'} b = B_1, \ldots, \map {y^{\paren {n - 1} } } b = B_{n - 1}$ Let $J \sqbrk y$ be a functional of the form: :$\displaystyle J \sqbrk y = \int_a^b \map F {x, y, y', \ldots, y^{\paren n} } \rd x$ Then a necessary condition for $J \sqbrk y$ to have an extremum (strong or weak) for a given function $\map y x$ is that $\map y x$ satisfy Euler's equation: :$F_y - \dfrac \d {\d x} F_{y'} + \dfrac {\d^2} {\d x^2} F_{y''}  -\cdots + \paren {-1}^n \dfrac {\d^n} {\d x^n} F_{y^{\paren n} } = 0$


Equivalence of Definitions of Absolute Convergence of Product of Complex Numbers

Let $\left\langle{a_n}\right\rangle$ be a sequence of complex numbers. Let $\log$ denote the complex logarithm. {{TFAE|def = Absolute Convergence of Product}}


Square of Quadratic Gauss Sum

Let $p$ be an odd prime. Let $a$ be an integer coprime to $p$. Let $g \left({a, p}\right)$ denote the quadratic Gauss sum of $a$ and $p$. Then: :${g \left({a, p}\right)}^2 = \left({\dfrac {-1} p}\right) \cdot p$


Absolutely Convergent Product is Convergent

Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field. Let $\mathbb K$ be complete. Let the infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be absolutely convergent. Then it is convergent.


Simplest Variational Problem with Subsidiary Conditions

Let $J \sqbrk y$ and $K \sqbrk y$ be (real) functionals, such that :$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$ :$\ds K \sqbrk y = \int_a^b \map G {x, y, y'} \rd x = l$ where $l$ is a constant. Let $y = \map y x$ be an extremum of $F \sqbrk y$, and satisfy boundary conditions: :$\map y a = A$ :$\map y b = B$ Then, if $y = \map y x$ is not an extremal of $K \sqbrk y$, there exists a constant $\lambda$ such that $y = \map y x$ is an extremal of the functional: :$\ds \int_a^b \paren {F + \lambda G} \rd x$ or, in other words, $y = \map y x$ satisfies: :$F_y - \dfrac {\d} {\d x} F_{y'} + \lambda \paren {G_y - \dfrac {\d} {\d x} G_{y'} } = 0$


Complement of Closed under Directed Suprema Subset is Inaccessible by Directed Suprema

Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $X$ be a closed under directed suprema subset of $S$. Then $\complement_S\left({X}\right)$ is inaccessible by directed suprema.


Complement of Upper Set is Lower Set

Let $\left({S, \preceq}\right)$ be an ordered set. Let $L$ be an upper set. Then $S \setminus L$ is a lower set.


Closed Set iff Lower and Closed under Directed Suprema in Scott Topological Ordered Set

Let $T = \left({S, \preceq, \tau}\right)$ be a relational structure with Scott topology where $\left({S, \preceq}\right)$ is an up-complete ordered set. Let $A$ be a subset of $S$. Then $A$ is closed {{iff}} $A$ is lower and closed under directed suprema.


Complement of Inaccessible by Directed Suprema Subset is Closed under Directed Suprema

Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $X$ be an inaccessible by directed suprema subset of $S$. Then $\complement_S\left({X}\right)$ is closed under directed suprema.


Convergence of Complex Sequence in Polar Form

Let $z \ne 0$ be a complex number with modulus $r$ and argument $\theta$. Let $\sequence {z_n}$ be a sequence of nonzero complex numbers. Let $r_n$ be the modulus of $z_n$ and $\theta_n$ be an argument of $z_n$. Then $z_n$ converges to $z$ {{iff}} the following hold: :$(1): \quad r_n$ converges to $r$ :$(2): \quad$ There exists a sequence $\sequence {k_n}$ of integers such that $\theta_n + 2 k_n \pi$ converges to $\theta$.


Complex Modulus of Sum of Complex Numbers

Let $z_1, z_2 \in \C$ be complex numbers. Let $\theta_1$ and $\theta_2$ be arguments of $z_1$ and $z_2$, respectively. Then: :$\cmod {z_1 + z_2}^2 = \cmod {z_1}^2 + \cmod {z_2}^2 + 2 \cmod {z_1} \cmod {z_2} \, \map \cos {\theta_1 - \theta_2}$


Complex Modulus of Difference of Complex Numbers

Let $z_1, z_2 \in \C$ be complex numbers. Let $\theta_1$ and $\theta_2$ be arguments of $z_1$ and $z_2$, respectively. Then: :$\cmod {z_1 - z_2}^2 = \cmod {z_1}^2 + \cmod {z_2}^2 - 2 \cmod {z_1} \cmod {z_2} \, \map \cos {\theta_1 - \theta_2}$


Uniform Absolute Convergence of Infinite Product of Complex Functions

Let $X$ be a compact metric space. Let $\left\langle{f_n}\right\rangle$ be a sequence of continuous functions $X \to \C$. Let $\displaystyle \sum_{n \mathop = 1}^\infty f_n$ converge uniformly absolutely on $X$. Then: :$f \left({x}\right) = \displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n \left({x}\right)}\right)$ converges uniformly absolutely on $X$ :$f$ is continuous :there exists $n_0 \in \N$ such that $\displaystyle \prod_{n \mathop = n_0}^\infty \left({1 + f_n \left({x}\right)}\right)$ has no zeroes.


Simplest Variational Problem with Subsidiary Conditions for Curve on Surface

Let $J \sqbrk {y, z}$ be a (real) functional of the form: :$\ds J \sqbrk y = \int_a^b \map F {x, y, z, y', z'} \rd x$ Let there exist admissible curves $y, z$ lying on the surface: :$\map g {x, y, z} = 0$ which satisfy boundary conditions: :$\map y a = A_1, \map y b = B_1$ :$\map z a = A_2, \map z b = B_2$ Let $J \sqbrk {y, z}$ have an extremum for the curve $y = \map y x, z = \map z x$. Let $g_y$ and $g_z$ not simultaneously vanish at any point of the surface $g = 0$. Then there exists a function $\map \lambda x$ such that the curve $y = \map y x, z = \map z x$ is an extremal of the functional: :$\ds \int_a^b \paren {F + \map \lambda x g} \rd x$ In other words, $y = \map y x$ satisfies the differential equations: {{begin-eqn}} {{eqn | l = F_y + \lambda g_y - \frac \d {\d x} F_{y'}       | r = 0 }} {{eqn | l = F_z + \lambda g_z - \frac \d {\d x} F_{z'}       | r = 0       | c =  }} {{end-eqn}}


Empty Intersection iff Subset of Relative Complement

Let $S$ be a set. Let $A, B$ be subset of $S$. Then $A \cap B = \varnothing \iff A \subseteq \complement_S\left({B}\right)$


Subset in Subsets

Let $S, B$ be sets. Let $A$ be subset of $S$. Then: :$A \subseteq B \iff \forall x \in S: x \in A \implies x \in B$


Empty Set as Subset

Let $S$ be a set. Let $A$ be a subset of $S$. Then: :$A = \O \iff \forall x \in S: x \notin A$


Pythagorean Triangles whose Areas are Repdigit Numbers

The following Pythagorean triangles have areas consisting of repdigit numbers:


Exponential of Series Equals Infinite Product

Let $\left\langle{z_n}\right\rangle$ be a sequence of complex numbers. Suppose $\displaystyle \sum_{n \mathop = 1}^\infty z_n$ converges to $z \in \C$. Then $\displaystyle \prod_{n \mathop = 1}^\infty \exp \left({z_n}\right)$ converges to $\exp z$.


Absolute Value of Infinite Product

Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\sequence{a_n}$ be a sequence in $\mathbb K$.


Convergence of Series of Complex Numbers by Real and Imaginary Part

Let $\sequence {z_n}$ be a sequence of complex numbers. Then: : the series $\displaystyle \sum_{n \mathop = 1}^\infty z_n$ converges to $Z \in \C$ {{iff}}: : the series: ::$\displaystyle \sum_{n \mathop = 1}^\infty \Re \paren {z_n}$ :and: ::$\displaystyle \sum_{n \mathop = 1}^\infty \Im \paren {z_n}$ :converge to $\Re \paren Z$ and $\Im \paren Z$ respectively.


Lower Closure of Element is Closed under Directed Suprema

Let $L = \struct {S, \preceq}$ be an up-complete ordered set. Let $x \in S$. Then $x^\preceq$ is closed under directed suprema, where $x^\preceq$ denotes the lower closure of $x$.


Pythagorean Triangle whose Hypotenuse and Leg differ by 1

Let $P$ be a Pythagorean triangle whose sides correspond to the Pythagorean triple $T$. Then: : the hypotenuse of $P$ is $1$ greater than one of its legs {{iff}}: : the generator for $T$ is of the form $G = \tuple {n, n + 1}$ where $n \in \Z_{> 0}$ is a (strictly) positive integer.


General Variation of Integral Functional/Dependent on N Functions

Let $J$ be a (real) functional of the form $\ds J \sqbrk {\ldots y_i \ldots} = \int_{x_0}^{x_1} \map F {x, \ldots y_i \ldots, \ldots y_i' \ldots} \rd x, i = \openint 1 n$ Then: $\ds \delta J = \int_{x_0}^{x_1} \sum_{i \mathop = 1}^n \paren {F_{y_i} - \frac \d {\d x} F_{y_i'} } \map {h_i} x + \intlimits {\sum_{i \mathop = 1}^n F_{y_i'} \delta y_i} {x \mathop = x_0} {x \mathop = x_1} + \intlimits {\paren {F - \sum_{i \mathop = 1}^n y_i'F_{y_i'} } \delta x} {x \mathop = x_0} {x \mathop = x_1}$


Kusmin-Landau Inequality

Let $I$ be the half-open interval $\hointl a b$. Let $f: I \to R$ be continuously differentiable. Let $f'$ be monotonic. Let $\norm {f'} \ge \lambda$ on $I$ for some $\lambda \in \R_{>0}$, where $\norm {\, \cdot \,}$ denotes the distance to nearest integer. Then: :$\displaystyle \sum_{n \mathop \in I} e^{2 \pi i \map f n} = \map O {\frac 1 \lambda}$ where the big-O estimate does not depend on $f$.


Closure of Singleton is Lower Closure of Element in Scott Topological Lattice

Let $T = \struct {S, \preceq, \tau}$ be a up-complete topological lattice with Scott topology. Let $x \in S$. Then: :$\set x^- = x^\preceq$ where :$\set x^-$ denotes the topological closure of $\set x$ :$x^\preceq$ denotes the lower closure of $x$.


Pythagorean Triangle cannot be Isosceles

Let $P$ be a Pythagorean triangle. Then $P$ is not isosceles.


Brahmagupta-Fibonacci Identity/Corollary

:$\left({a^2 + b^2}\right) \left({c^2 + d^2}\right) = \left({a c - b d}\right)^2 + \left({a d + b c}\right)^2$


Scott Topological Lattice is T0 Space

Let $T = \left({S, \preceq, \tau}\right)$ be a complete topological lattice with Scott topology. Then $T$ is a $T_0$ space.


Lower Closures are Equal implies Elements are Equal

Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $x, y \in S$ such that :$x^\preceq = y^\preceq$ where $x^\preceq$ denotes the lower closure of $x$. Then $x = y$


General Variation of Integral Functional/Dependent on N Functions/Canonical Variables

Let $\delta J$ be a general variation of integral functional dependent on n functions. Suppose a following coordinate transformation is done: :$\set {x, \ldots y_i, \ldots, \ldots, y_i', \ldots, F} \to \set {x, \ldots, y_i, \ldots, \ldots p', \ldots, H}, i = \tuple {1, \ldots, n}$ Then, in canonical variables: :$\displaystyle \delta J = \int_{x_0}^{x_1} \sum_{i \mathop = 1}^n \paren {F_{y_i} - \dfrac {\d {p_i} } {\d x} } \map {h_i} x \rd x + \intlimits {\sum_{i \mathop = 1}^n p_i \delta y_i - H \delta x} {x \mathop = x_0} {x \mathop = x_1}$ where: {{begin-eqn}} {{eqn | l = \bigvalueat {\delta x} {x \mathop = x_j}       | r = \delta x_j }} {{eqn | l = \bigvalueat {\delta y_i} {x \mathop = x_j}       | r = \delta_i^j }} {{eqn | l = j       | r = \tuple {0, 1} }} {{end-eqn}} {{explain|the meaning of $j {{=}} \tuple {0, 1}$ in this context, and hence the meaning of $x_j$ and $\delta x_j$, and also $\delta_i^j$, all of which are obscure}}


Square of Pythagorean Prime is Hypotenuse of Pythagorean Triangle

Let $p$ be a Pythagorean prime. Then $p^2$ is the hypotenuse of a Pythagorean triangle.


Square of Hypotenuse of Pythagorean Triangle is Difference of two Cubes/Refutation

Let $h$ be the hypotenuse of a Pythagorean triangle. Then it is not necessarily the case that: :$h^2 = a^3 - b^3$ for some $a, b \in \Z_{>0}$.


Pythagorean Triangle from Sum of Reciprocals of Consecutive Same Parity Integers

Let $a, b \in \Z_{>0}$ be (strictly) positive integers such that they are consecutively of the same parity. Let $\dfrac p q = \dfrac 1 a + \dfrac 1 b$. Then $p$ and $q$ are the legs of a Pythagorean triangle.


Rational Number plus Irrational Number is Irrational

Rational number plus irrational number is irrational. That is, let $x \in \Q$, $y \in \R \setminus \Q$ and $x + y = z$. Then $z \in \R \setminus \Q$.


Lower Closure of Element is Topologically Closed in Scott Topological Ordered Set

Let $T = \left({S, \preceq, \tau}\right)$ be a relational structure with Scott topology where $\left({S, \preceq}\right)$ is an up-complete ordered set. Let $x \in S$. Then $x^\preceq$ is topologically closed, where $x^\preceq$ denotes the lower closure of $x$.


Complement of Lower Closure of Element is Open in Scott Topological Ordered Set

Let $T = \struct {S, \preceq, \tau}$ be a relational structure with Scott topology where $\struct {S, \preceq}$ is an up-complete ordered set. Let $x \in S$. Then $\relcomp S {x^\preceq}$ is topologically open, where :$x^\preceq$ denotes the lower closure of $x$, :$\relcomp S {x^\preceq}$ denotes the relative complement of $x^\preceq$.


Open iff Upper and with Property (S) in Scott Topological Lattice

Let $T = \left({S, \preceq, \tau}\right)$ be an up-complete topological lattice. Let $A$ be a subset of $S$. Then $A$ is open {{iff}} $A$ is upper and with property (S).


Necessary Condition for Integral Functional to have Extremum for given Function/Non-differentiable at Intermediate Point

Let $y, F$ be real functions. Let $y$ be continuously differentiable for $x \in \hointr a c \cap  \hointl c b$ and satisfy: :$\map y a = A$ :$\map y b = B$ Let $J\sqbrk y$ be a functional of the form :$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$ Then the functional $J$ has a weak extremum if $y$ satisfies the following system of equations: {{begin-eqn}} {{eqn | l = F_y - \dfrac \d {\d x} F_{y'}       | r = 0 }} {{eqn | l = \bigvalueat {F_{y'} } {x \mathop = c \mathop - 0}       | r = \bigvalueat {F_{y'} } {x \mathop = c \mathop + 0} }} {{eqn | l = \bigvalueat {\paren {F - y' F_{y'} } } {x \mathop = c \mathop - 0}       | r = \bigvalueat {\paren {F - y' F_{y'} } } {x \mathop = c \mathop + 0} }} {{end-eqn}} where, by the use of limit from the left and limit from the right, the following abbreviations are denoted as follows: :$\ds \bigvalueat {\map y x} {x \mathop = c \mathop + 0} = \lim_{x \mathop \to c^+} \map y x$ :$\ds \bigvalueat {\map y x} {x \to x \mathop = c \mathop - 0} = \lim_{x \mathop \to c^-} \map y x$ The last two equations are known as the '''Weierstrass-Erdmann corner conditions'''. {{refactor|move this definition to a separate page once a general principle can be distilled}}


Euler's Equation for Vanishing Variation in Canonical Variables

{{refactor|There are a number of pages linking here with the presentation of the link set as "momenta". This is going to need a definition of its own, but it is not clear what that is from looking at this page.}} Consider the following system of differential equations: :$(1): \quad \begin {cases} F_{y_i} - \dfrac \d {\d x} F_{y_i'} = 0 \\ \dfrac {\d {y_i} } {\d x} = y_i'\end{cases}$ where $i \in \set {1, \ldots, n}$. Let the coordinates $\tuple {x, \family {y_i}_{1 \mathop \le i \mathop \le n}, \family {y_i'}_{1 \mathop \le i \mathop \le n}, F}$ be transformed to canonical variables: :$\tuple {x, \family {y_i}_{1 \mathop \le i \mathop \le n}, \family {p_i}_{1 \mathop \le i \mathop \le n}, H}$ Then the system $(1)$ is transformed into: :$\begin {cases} \dfrac {\d y_i} {\d x} = \dfrac {\partial H} {\partial p_i} \\ \dfrac {\d p_i} {\d x} = -\dfrac {\partial H} {\partial y_i} \end {cases}$


Relational Structure with Topology of Subsets with Property (S) is Topological Space

Let $T = \left({S, \preceq, \tau}\right)$ be a relational structure with topology where :$\left({S, \preceq}\right)$ is an up-complete ordered set, :$\tau$ is the set of all subsets of $S$ with property (S). Then $\left({S, \tau}\right)$ is topological space.


Subgroup of Real Numbers is Discrete or Dense

Let $G$ be a subgroup of the additive group of real numbers. Then one of the following holds: :$G$ is dense in $\R$. :$G$ is discrete and there exists $a \in \R$ such that $G = a \Z$, that is, $G$ is cyclic.


Discrete Subgroup of Real Numbers is Closed

Let $G$ be a subgroup of the additive group of real numbers. Let $G$ be discrete. Then $G$ is closed.


Consecutive Integers with Same Sigma

Let $\sigma: \Z_{>0} \to \Z_{>0}$ be the $\sigma$ function, defined on the strictly positive integers. The equation: :$\map \sigma n = \map \sigma {n + 1}$ is satisfied by integers in the sequence: :$14, 206, 957, 1334, 1364, 1634, 2685, 2974, 4364, 14841, 18873, \ldots$ {{OEIS|A002961}}


Coarser Between Generator Set and Filter is Generator Set of Filter

Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $F$ be a filter on $L$. Let $G$ be a generator set of $F$. Let $A$ be a subset of $S$ such that :$G$ is coarser than $A$ and $A$ is coarser than $F$. Then $A$ is generator set of $F$.


Borsuk-Ulam Theorem

Let $n$ be a positive integer. Let $f: \mathbb S^n \to \R^n$ be a continuous mapping from an $n$-sphere to $\R^n$. Then there exists $x \in \mathbb S^n$ such that $\map f x = \map f {-x}$.


Gershgorin Circle Theorem

Let $n$ be a positive integer. Let $A = \sqbrk {a_{i j} }$ be a complex square matrix of order $n$. Let $\lambda$ be an eigenvalue of $A$. Then there exists $i \in \set {1, 2, \ldots, n}$ such that: :$\lambda \in \map {\mathbb D} {a_{i i}, R_i}$ where: :$\displaystyle R_i = \sum_{j \mathop \ne i} \cmod {a_{ i j} }$ :$\map {\mathbb D} {a, R}$ denotes the complex disk of center $a$ and radius $R$.


Finite Infima Set of Coarser Subset is Coarser than Finite Infima Set

Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $A, B$ be subsets of $S$ such that :$A$ is coarser than $B$. Then $\operatorname{fininfs} \left({A}\right)$ is coarser than $\operatorname{fininfs} \left({B}\right)$ where $\operatorname{fininfs} \left({B}\right)$ denotes the finite infima set of $B$.


Approximations to Equilateral Triangles by Heronian Triangles

The sequence of best approximations to an equilateral triangle by a Heronian triangle begins: :The $\paren {3, 4, 5}$ triangle, with area $6$ :The $\paren {13, 14, 15}$ triangle, with area $84$, where $14 = 4^2 - 2$ :The $\paren {193, 194, 195}$ triangle, where $194 = 14^2 - 2$ :The $\paren {37 \, 633, 37 \, 634, 37 \, 635}$ triangle, where $37 \, 634 = 194^2 - 2$ and so on. {{OEIS|A003010}}


Legendre Transform is Involution

The Legendre transform is an Involution.


Upper Closure of Coarser Subset is Subset of Upper Closure

Let $L = \left({S, \preceq}\right)$ be a preordered set. Let $A, B$ be subsets of $S$ such that :$A$ is coarser than $B$. Then $A^\succeq \subseteq B^\succeq$


Set Coarser than Upper Set is Subset

Let $\left({S, \preceq}\right)$ be a preordered set. Let $A, B$ be subsets of $S$ such that :$A$ is coarser than $B$ and :$B$ is an upper set. Then $A \subseteq B$


Sum of 4 Unit Fractions that equals 1

There are $14$ ways to represent $1$ as the sum of exactly $4$ unit fractions.


Sum of 5 Unit Fractions that equals 1

There are $147$ ways to represent $1$ as the sum of exactly $5$ unit fractions.


Sum of 6 Unit Fractions that equals 1

There are $3462$ ways to represent $1$ as the sum of exactly $6$ unit fractions.


Conditions for Function to be Maximum of its Legendre Transform Two-variable Equivalent

Let $x, p \in \R$. Let $\map f x$ be a strictly convex real function. Let $f^*$ be a Legendre transformed $f$. Let $\map g {x, p} = - \map {f^*} p + x p$ Then: :$\displaystyle \map f x = \max_p \paren {-\map {f^*} p + x p}$ where $\displaystyle \max_p$ maximises the function with respect to a variable $p$.


Set is Coarser than Image of Mapping of Infima

Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $f, g:\N \to S$ be mappings such that :$\forall n \in \N: g\left({n}\right) = \inf \left\{ {f\left({m}\right): m \in \N \land m \le n}\right\}$ Then $f\left[{\N}\right]$ is coarser than $g\left[{\N}\right]$ where $f\left[{\N}\right]$ denotes the image of mapping $f$.


Solutions of Ramanujan-Nagell Equation

Solutions to the Ramanujan-Nagell equation: :$x^2 + 7 = 2^n$ exist for only $5$ values of $n$: :$3, 4, 5, 7, 15$ {{OEIS|A060728}} The corresponding values of $x$ are: :$1, 3, 5, 11, 181$ {{OEIS|A038198}}


Five Ramanujan-Nagell Numbers

There exist exactly $5$ Ramanujan-Nagell numbers: positive integers of the form $2^m - 1$ which are triangular: :$0, 1, 3, 15, 4095$ {{OEIS|A076046}}


Conditions for Functional to be Extremum of Two-variable Functional over Canonical Variable p

Let $y = \map y x$ and $\map F {x, y, y'}$ be real functions. Let $\dfrac {\partial^2 F} {\partial {y'}^2} \ne 0$. Let $\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$ Let $\ds J \sqbrk {y, p} = \int_a^b \paren {-\map H {x, y, p} + p y'} \rd x$, where $H$ is the Hamiltonian of $J \sqbrk y$. Then $\ds J \sqbrk y = \bigvalueat {J \sqbrk {y, p} } {\frac {\delta J \sqbrk{y, p} } {\delta p} \mathop = 0}$


Image of Mapping of Infima is Generator Set of Filter

Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $f, g:\N \to S$ be mappings such that :$\forall n \in \N: g\left({n}\right) = \inf \left\{ {f\left({m}\right): m \in \N \land m \le n}\right\}$ Let $F$ be a filter such that :$f\left[{\N}\right]$ is generator set of $F$, where $f\left[{\N}\right]$ denotes the image of $f$. Then $g\left[{\N}\right]$ is generator set of $F$.


Subset of Set is Coarser than Set

Let $\left({S, \preceq}\right)$ be a preordered set. Let $A, B$ be subset of $S$ such that :$A \subseteq B$ Then $A$ is coarser than $B$.


Definition:Ramanujan-Nagell Number

A '''Ramanujan-Nagell number''' is a positive integer of the form $2^m - 1$ which is also triangular.


Existence of Product of Three Distinct Primes between n and 2n

Let $n \in \Z$ be an integer such that $n > 15$. Then between $n$ and $2 n$ there exists at least one integer which is the product of $3$ distinct prime numbers.


Second Column and Diagonal of Pascal's Triangle consist of Triangular Numbers

The $2$nd column and $2$nd diagonal of Pascal's triangle consists of the set of triangular numbers.