First Order ODE/exp x sine y dx + exp x cos y dy = y sine x y dx + x sine x y dy

The first order ordinary differential equation: :$(1): \quad e^x \sin y \rd x + e^x \cos y \rd y = y \sin x y \rd x + x \sin x y \rd y$ is an exact differential equation with solution: :$e^x \sin y + \cos x y = C$


Linear Second Order ODE/(1 + x^2) y'' + x y' = 0

The second order ODE: :$\paren {1 + x^2} y'' + x y' = 0$ has the general solution: :$y = C_1 \map \ln {x + \sqrt {x^2 + 1} } + C_2$


Circumscribing Circle about Regular Pentagon

About any given regular pentagon it is possible to circumscribe a circle. {{:Euclid:Proposition/IV/14}}


Pólya-Vinogradov Inequality

Let $p$ be a positive odd prime. Then: :$\forall m, n \in \N: \displaystyle \size {\sum_{k \mathop = m}^{m + n} \paren {\frac k p} } < \sqrt p \, \ln p$ where $\paren {\dfrac k p}$ is the Legendre symbol.


Multiplication of Real Numbers is Left Distributive over Subtraction

{{:Euclid:Proposition/V/5}} That is, for any numbers $a, b$ and for any integer $m$: : $m a - m b = m \paren {a - b}$


Linear Second Order ODE/y'' + y = 0/y(0) = 2, y'(0) = 3

The second order ODE: :$(1): \quad y'' + y = 0$ with initial conditions: :$\map y 0 = 2$ :$\map {y'} 0 = 3$ has the particular solution: :$y = 3 \sin x + 2 \cos x$


Legendre's Differential Equation/(1 - x^2) y'' - 2 x y' + 2 y = 0

The special case of Legendre's differential equation: :$(1): \quad \paren {1 - x^2} y'' - 2 x y' + 2 y = 0$ has the general solution: :$y = C_1 x + C_2 \paren {\dfrac x 2 \, \map \ln {\dfrac {1 + x} {1 - x} } - 1}$


Second Order ODE/y'' - f(x) y' + (f(x) - 1) y = 0

The second order ODE: :$(1): \quad y'' - \map f x y' + \paren {\map f x - 1} y = 0$ has the general solution: :$\displaystyle y = C_1 e^x + C_2 e^x \int e^{-2 x + \int \map f x \rd x} \rd x$


Linear Second Order ODE/2 x^2 y'' + 10 x y' + 8 y = 0

The second order ODE: :$(1): \quad 2 x^2 y'' + 10 x y' + 8 y = 0$ has the general solution: :$y = C_1 x^{-2} + C_2 x^{-2} \ln x$


Divisors obey Distributive Law

{{:Euclid:Proposition/VII/5}} In modern algebraic language: :$\displaystyle a = \frac 1 n b, c = \frac 1 n d \implies a + c = \frac 1 n \left({b + d}\right)$


Real Numbers under Addition Modulo 1 form Group

Let $S = \left\{{x \in \R: 0 \le x < 1}\right\}$. Let $\circ: S \times S \to S$ be the operation defined as: : $x \circ y = x + y - \left \lfloor {x + y} \right \rfloor$ That is, $\circ$ is defined as addition modulo $1$. Then $\left({S, \circ}\right)$ is a group.


Linear Second Order ODE/y'' + y = 2 cosine x

The second order ODE: :$(1): \quad y'' + y = 2 \cos x$ has the general solution: :$y = C_1 \sin x + C_2 \cos x + x \cos x$


Linear Second Order ODE/y'' + y = cosecant x

The second order ODE: :$(1): \quad y'' + y = \csc x$ has the general solution: :$y = C_1 \sin x + C_2 \cos x - x \cos x + \sin x \map \ln {\sin x}$


Quaternions Defined by Ordered Pairs

Consider the quaternions $\Bbb H$ as numbers in the form: : $a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ where: : $a, b, c, d$ are real numbers; : $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ are entities related to each other in the following way: {{begin-eqn}}      {{eqn | l = \mathbf i \mathbf j = - \mathbf j \mathbf i       | r = \mathbf k       | c =  }} {{eqn | l = \mathbf j \mathbf k = - \mathbf k \mathbf j       | r = \mathbf i       | c =  }} {{eqn | l = \mathbf k \mathbf i = - \mathbf i \mathbf k       | r = \mathbf j       | c =  }} {{eqn | l = \mathbf i^2 = \mathbf j^2 = \mathbf k^2 = \mathbf i \mathbf j \mathbf k       | r = - \mathbf 1       | c =  }} {{end-eqn}} Now consider the quaternions $\Bbb H$ defined as ordered pairs $\left({x, y}\right)$ where $x, y \in \C$ are complex numbers, on which the operation of multiplication is defined as follows: Let $w = a_1 + b_1 i, x = c_1 + d_1 i, y = a_2 + b_2 i, z = c_2 + d_2 i$ be complex numbers. Then $\left({w, x}\right) \left({y, z}\right)$ is defined as: :$\left({w, x}\right) \left({y, z}\right) := \left({w y - z \overline x, \overline w z + x y}\right)$ where $\overline w$ and $\overline x$ are the complex conjugates of $w$ and $x$ respectively. These two definitions are equivalent.


Linear Second Order ODE/y'' - 2 y' - 3 y = 64 x exp -x

The second order ODE: :$(1): \quad y'' - 2 y' - 3 y = 64 x e^{-x}$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-x} - e^{-x} \paren {8 x^2 + 4 x + 1}$


Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = x (x + 1)^2

The second order ODE: :$(1): \quad \paren {x^2 + x} y'' + \paren {2 - x^2} y' - \paren {2 + x} y = x \paren {x + 1}^2$ has the general solution: :$y = C_1 e^x + \dfrac {C_2} x - x - 1 - \dfrac {x^2} 3$


Position of Cart attached to Wall by Spring under Damping/Critically Damped/x = x0 at t = 0

Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the horizontal position of $C$ at time $t$ can be expressed as: :$x = x_0 e^{-a t} \left({1 + a t}\right)$


Ring of Square Matrices over Real Numbers

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {\map {\mathcal M_\R} n, +, \times}$ denote the ring of square matrices of order $n$ over $\R$. Then $\struct {\map {\mathcal M_\R} n, +, \times}$ is a ring with unity, but is not a commutative ring.


Resonance Frequency is less than Natural Frequency

Consider a physical system $S$ whose behaviour is defined by the second order ODE: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = K \cos \omega x$ where: :$K \in \R: k > 0$ :$a, b \in \R_{>0}: b < a$ Then the resonance frequency of $S$ is smaller than the natural frequency of the associated second order ODE: :$(2): \quad \dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = 0$


Circle is Bisected by Diameter

A circle is bisected by a diameter.


Power to Characteristic Power of Field is Monomorphism

Let $F$ be a field whose characteristic is $p$ where $p \ne 0$. Let $n \in \Z_{\ge 0}$ be any positive integer. Let $\phi_n: F \to F$ be the mapping on $F$ defined as: :$\forall x \in F: \map {\phi_n} x = x^{p^n}$ Then $\phi_n$ is a (field) monomorphism.


Logarithmic Spiral is Equiangular

The logarithmic spiral is '''equiangular''', in the following sense: Let $P = \left\langle{r, \theta}\right\rangle$ be a point on a logarithmic spiral $S$ expressed in polar coordinates as: :$r = a e^{b \theta}$ Then the angle $\psi$ that the tangent makes to the radius vector of $S$ is constant.


Area Enclosed by First Turn of Archimedean Spiral

Let $S$ be the Archimedean spiral defined by the equation: :$r = a \theta$ The area $\mathcal A$ enclosed by the first turn of $S$ and the polar axis is given by: :$\mathcal A = \dfrac {4 \pi^3 a^2} 3$ :500px


Trisecting the Angle/Archimedean Spiral

Let $\alpha$ be an angle which is to be trisected. This can be achieved by means of an Archimedean spiral.


Upper Adjoint Preserves All Infima

Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets. Let $g: S \to T$ be an upper adjoint of Galois connection. Then $g$ preserves all infima.


Trisecting the Angle/Parabola

Let $\alpha$ be an angle which is to be trisected. This can be achieved by means of a parabola.


Cardinality of Set of Induced Equivalence Classes of Injection

Let $f: S \to T$ be a mapping. Let $\mathcal R_f \subseteq S \times S$ be the relation induced by $f$: :$\tuple {s_1, s_2} \in \mathcal R_f \iff \map f {s_1} = \map f {s_2}$ Let $f$ be an injection. Then there are $\card S$ different $\mathcal R_f$-classes.


Multiplicative Group of Positive Rationals is Non-Cyclic

Let $\struct {\Q_{>0}, \times}$ be the multiplicative group of positive rational numbers. Then $\struct {\Q_{>0}, \times}$ is not a cyclic group.


Top equals to Relative Pseudocomplement in Brouwerian Lattice

Let $\struct {S, \vee, \wedge, \preceq}$ be a Brouwerian lattice with greatest element $\top$. Let $a, b \in S$. Then :$\top = a \to b$ {{iff}} $a \preceq b$


Approximate Value of Nth Prime Number

The $n$th prime number is approximately $n \ln n$.


Fortissimo Space is not Compact

Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $T$ is not a compact space.


Fortissimo Space is not Sequentially Compact

Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $T$ is not sequentially compact.


Rational Number is Algebraic of Degree 1

Let $r \in \Q$ be a rational number. Then $r$ is an algebraic number of degree $1$.


Double Pointed Fortissimo Space is Lindelöf

Let $T = \struct {S, \tau}$ be a Fortissimo space. Let $T \times D$ be the double pointed topology on $T$. Then $T \times D$ is a Lindelöf space.


Double Pointed Fortissimo Space is Weakly Countably Compact

Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Let $T \times D$ be the double pointed topology on $T$. Then $T \times D$ is weakly countably compact.


Double Pointed Fortissimo Space is not Pseudocompact

Let $T = \struct {S, \tau}$ be a Fortissimo space. Let $T \times D$ be the double pointed topology on $T$. Then $T \times D$ is not pseudocompact.


Modified Fort Space is not Locally Connected

Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is not locally connected.


Clopen Sets in Modified Fort Space

Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Let $A \subseteq S$ be both closed and open in $T$. If $a \in A$, then $b \in A$ as well. That is, any clopen set of $T$ must contain '''both''' or '''neither''' of $a$ and $b$.


Maximum Rate of Change of Y Coordinate of Cycloid

Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian plane at a constant speed such that the center moves with a velocity $\mathbf v_0$ in the direction of increasing $x$. Consider a point $P$ on the circumference of this circle. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The maximum rate of change of $y$ is $\mathbf v_0$, which happens when $\theta = \dfrac \pi 2 + 2 n \pi$ where $n \in \Z$.


Vertices in Locally Finite Graph

Let $G$ be a locally finite graph. Then if $G$ is infinite, it contains an infinite number of vertices.


Relation between Equations for Hypocycloid and Epicycloid

Consider the hypocycloid defined by the equations: :$x = \paren {a - b} \cos \theta + b \map \cos {\paren {\dfrac {a - b} b} \theta}$ :$y = \paren {a - b} \sin \theta - b \map \sin {\paren {\dfrac {a - b} b} \theta}$ By replacing $b$ with $-b$, this converts to the equations which define an epicycloid: :$x = \paren {a + b} \cos \theta - b \map \cos {\paren {\dfrac {a + b} b} \theta}$ :$y = \paren {a + b} \sin \theta - b \map \sin {\paren {\dfrac {a + b} b} \theta}$


Derivative of Hyperbolic Sine Function

:$\map {\dfrac \d {\d x} } {\sinh u} = \cosh u \dfrac {\d u} {\d x}$


Radius of Curvature in Cartesian Form

Let $C$ be a curve defined by a real function which is twice differentiable. Let $C$ be embedded in a cartesian plane. The '''radius of curvature''' $\rho$ of $C$ at a point $P = \tuple {x, y}$ is given by: :$\rho = \dfrac {\paren {1 + y'^2}^{3/2} } {\size {y''} }$ where: :$y' = \dfrac {\d y} {\d x}$ is the derivative of $y$ {{WRT|Differentiation}} $x$ at $P$ :$y'' = \dfrac {\d^2 y} {\d x^2}$ is the second derivative of $y$ {{WRT|Differentiation}} $x$ at $P$.


Derivative of Hyperbolic Tangent Function

:$\map {\dfrac \d {\d x} } {\tanh u} = \sech^2 u \dfrac {\d u} {\d x}$


Quaternion Multplication is not Commutative

The operation of multplication on the quaternions $H$ is not commutative.


Vectors in Three Dimensional Space with Cross Product forms Lie Algebra

Let $S$ be the set of vectors in $3$ dimensional Euclidean space. Let $\times$ denote the vector cross product on $S$. Then $\struct {S, \times}$ is a Lie algebra.


Tangent Line to Convex Graph

Let $f$ be a real function that is continuous on some closed interval $\left[{a \,.\,.\, b}\right]$ and differentiable and convex on the open interval $\left({a \,.\,.\, b}\right)$.  Then all the tangent lines to $f$ are below the graph of $f$. {{explain|"below"}}


Characterization of Lower Semicontinuity

Let $f: S \to \overline \R$ be an extended real valued function. Let $S$ be endowed with a topology $\tau$. The following are equivalent: :$(1): \quad$ $f$ is lower semicontinuous (LSC) on $S$. :$(2): \quad$ The epigraph $\map {\operatorname{epi}} f$ of $f$ is a closed set in $S \times \R$ with the product topology. :$(3): \quad$ All lower level sets of $f$ are closed in $S$.


Power is Well-Defined/Rational

Let $x \in \R_{> 0}$ be a (strictly) positive real number. Let $q$ be a rational number. Then $x^q$ is  well-defined.


Integral of Arcsine Function

:$\displaystyle \int \arcsin x \rd x = x \arcsin x + \sqrt {1 - x^2} + C$ for $x \in \closedint {-1} 1$.


Ordinal is Subset of Ordinal Class

Suppose $A$ is an ordinal. Then: :$A \subseteq \On$ where $\On$ represents the class of all ordinals.


Power Function on Base Greater than One is Strictly Increasing/Integer

Let $a \in \R$ be a real number such that $a > 1$. Let $f: \Z \to \R$ be the real-valued function defined as: :$\map f k = a^k$ where $a^k$ denotes $a$ to the power of $k$. Then $f$ is strictly decreasing.


Subset Products of Normal Subgroup with Normal Subgroup of Subgroup

Let $G$ be a group. Let: :$(1): \quad H$ be a subgroup of $G$ :$(2): \quad K$ be a normal subgroup of $H$ :$(3): \quad N$ be a normal subgroup of $G$ Then: :$N K \lhd N H$ where: : $N K$ and $N H$ denote subset product : $\lhd$ denotes the relation of being a normal subgroup.


Intersection with Normal Subgroup is Normal/Examples/Subset Product of Normal Subgroup with Intersection

Let $\struct G$ be a group whose identity is $e$. Let $H_1, H_2$ be subgroups of $G$. Let: : $N_1 \lhd H_1$ : $N_2 \lhd H_2$ where $\lhd$ denotes the relation of being a normal subgroup. Then: :$N_1 \paren {H_1 \cap N_2} \lhd N_1 \paren {H_1 \cap H_2}$


Quotient Group of Direct Products

Let $G$ and $G'$ be groups. Let: :$H \lhd G$ :$H' \lhd G'$ where $\lhd$ denotes the relation of being a normal subgroup. Then: :$\paren {G \times G'} / \paren {H \times H'}$ is isomorphic to $\paren {G / H} \times \paren {G' / H'}$ where: :$G \times G'$ denotes the group direct product of $H$ and $H'$ :$G / H$ denotes the quotient group of $G$ by $H$.


Set of Non-Negative Real Numbers is not Well-Ordered by Usual Ordering

The set of non-negative real numbers $\R_{\ge 0}$ is not well-ordered under the usual ordering $\le$.


Characteristic Subgroup is Transitive

Let $G$ be a group. Let $H$ be a characteristic subgroup of $G$. Let $K$ be a characteristic subgroup of $H$. Then $K$ is a characteristic subgroup of $G$.


Integral Resulting in Arcsecant

:$\displaystyle \int \frac 1 {x \sqrt{x^2 - a^2} }\ \mathrm dx = \begin{cases} \dfrac 1 {\left\vert{a}\right\vert} \operatorname {arcsec} \dfrac x {\left\vert{a}\right\vert} + C & : x > \left\vert{a}\right\vert \\ -\dfrac 1 {\left\vert{a}\right\vert} \operatorname {arcsec} \dfrac x {\left\vert{a}\right\vert} + C & : x < -\left\vert{a}\right\vert \end{cases}$ where $a$ is a constant.


Upper Bound of Natural Logarithm/Corollary

:$\forall s \in \R_{>0}: \ln x \le \dfrac {x^s} s$


Finite Abelian Group is Solvable

Let $G$ be a finite abelian group. Then $G$ is solvable.


Powers Drown Logarithms/Corollary

:$\displaystyle \lim_{y \mathop \to 0_+} y^r \ln y = 0$


Logarithm of Logarithm in terms of Natural Logarithms

Let $b, x \in \R_{>0}$ be (strictly) positive real numbers. Then: :$\map {\log_b} {\log_b x} = \dfrac {\map \ln {\ln x} - \map \ln {\ln b} } {\ln b}$ where $\ln x$ denotes the natural logarithm of $x$.


Intersection is Subset of Union of Intersections with Complements

Let $R, S, T$ be sets. Then: :$S \cap T \subseteq \paren {R \cap S} \cup \paren {\overline R \cap T}$ where $\overline R$ denotes the complement of $R$.


Adjoining Commutes with Inverting

Let $H$ be a Hilbert space. Let $A \in B \left({H, K}\right)$ be a bounded linear operator. Let $A^{-1} \in B \left({K, H}\right)$ be an inverse for $A$. Then the adjoint of $A$, $A^*$, is invertible. Furthermore, $\left({A^*}\right)^{-1} = \left({A^{-1}}\right)^*$.


Sum of Elements in Inverse of Combinatorial Matrix

Let $C_n$ be the combinatorial matrix of order $n$ given by: :$C_n = \begin{bmatrix} x + y & y & \cdots & y \\ y & x + y & \cdots & y \\ \vdots & \vdots & \ddots & \vdots \\ y & y & \cdots & x + y \end{bmatrix}$ Let $C_n^{-1}$ be its inverse, from Inverse of Combinatorial Matrix: :$b_{i j} = \dfrac {-y + \delta_{i j} \left({x + n y}\right)} {x \left({x + n y}\right)}$ where $\delta_{i j}$ is the Kronecker delta. The sum of all the elements of $C_n^{-1}$ is: :$\displaystyle \sum_{1 \mathop \le i, \ j \mathop \le n} b_{i j} = \dfrac n {x + n y}$


Elements of Inverse of Hilbert Matrix are Integers

Let $H_n$ be the Hilbert matrix of order $n$: :$\begin{bmatrix} a_{i j} \end{bmatrix} = \begin{bmatrix} \dfrac 1 {i + j - 1} \end{bmatrix}$ Consider its inverse $H_n^{-1}$. All the elements of $H_n^{-1}$ are integers.


Auxiliary Relation is Transitive

Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $\mathcal R$ be relation on $S$ satisfying conditions $(i)$ and $(ii)$ of auxiliary relation. Then :$\mathcal R$ is a transitive relation.


Ordered Set of Auxiliary Relations is Complete Lattice

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $\operatorname {Aux} \left({L}\right)$ be the set of all auxiliary relations on $S$. Let $P = \left({\operatorname {Aux} \left({L}\right), \precsim}\right)$ be an ordered set where $\precsim \mathop = \subseteq \restriction_{\operatorname {Aux} \left({L}\right) \times \operatorname {Aux} \left({L}\right)}$ Then :$P$ is a complete lattice.


Zorn's Lemma Implies Axiom of Choice

If Zorn's Lemma is true, then so must the Axiom of Choice be.


Number minus Modulo is Integer Multiple

Let $x, y \in \R$ be real numbers. Let $x \bmod y$ denote the modulo operation: :$x \bmod y := \begin{cases} x - y \left \lfloor {\dfrac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$ where $\left \lfloor {\dfrac x y}\right \rfloor$ denotes the floor of $\dfrac x y$. Let $y < 0$. Then: :$x - \left({x \bmod y}\right)$ is an integer multiple of $y$.


Modulo Operation/Examples/5 mod 3

:$5 \bmod 3 = 2$


Modulo Multiplication is Well-Defined/Warning

Let $z \in \R$ be a real number. Let: :$a \equiv b \pmod z$ and: :$x \equiv y \pmod z$ where $a, b, x, y \in \R$. Then it does '''not''' necessarily hold that: : $a x \equiv b y \pmod z$


Constant to Power of Number of Distinct Prime Divisors is Multiplicative Function

Let $c \in \R$ be a constant. Let $f: \N \to \R$ denotes the mapping defined as: :$\forall n \in \N: f \left({n}\right) = c^k$ where $k$ is number of distinct primes that divide $n$. Then $f$ is multiplicative.


Clopen Sets in Finite Complement Topology

Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then the only clopen sets of $T$ are $S$ and $\O$.


Number of Digits in Factorial

Let $n!$ denote the factorial of $n$. The number of digits in $n!$ is approximately: :$1 + \left\lfloor{\dfrac 1 2 \left({\log_{10} 2 + \log_{10} \pi}\right) + \dfrac 1 2 \log_{10} n + n \left({\log_{10} n - \log_{10} e}\right)}\right\rfloor$ when $n!$ is shown in decimal notation. This evaluates to: :$1 + \left\lfloor{\left({n + \dfrac 1 2}\right) \log_{10} n - 0.43429 \ 4481 \, n + 0.39908 \ 9934}\right\rfloor$


Transformation of Unit Matrix into Inverse

Let $\mathbf A$ be a square matrix of order $n$ of the matrix space $\map {\MM_\R} n$. Let $\mathbf I$ be the unit matrix of order $n$. Suppose there exists a sequence of elementary row operations that reduces $\mathbf A$ to $\mathbf I$. Then $\mathbf A$ is invertible. Futhermore, the same sequence, when performed on $\mathbf I$, results in the inverse of $\mathbf A$.


Sum over k of r-kt Choose k by r over r-kt by s-(n-k)t Choose n-k by s over s-(n-k)t

For $n \in \Z_{\ge 0}$: :$\displaystyle \sum_k A_k \left({r, t}\right) A_{n - k} \left({s, t}\right) = A_n \left({r + s, t}\right)$ where $A_n \left({x, t}\right)$ is the polynomial of degree $n$ defined as: :$A_n \left({x, t}\right) = \dbinom {x - n t} n \dfrac x {x - n t}$ where $x \ne n t$.


Integer Multiples Greater than Positive Integer Closed under Addition

Let $n \Z$ be the set of integer multiples of $n$. Let $p \in \Z: p \ge 0$ be a positive integer. Let $S \subseteq n \Z$  be defined as: :$S := \set {x \in n \Z: x > p}$ that is, the set of integer multiples of $n$ greater than $p$. Then the algebraic structure $\struct {S, +}$ is closed under addition.


Signed Stirling Number of the First Kind of n+1 with 0

:$\map s {n + 1, 0} = 0$


Subset of Natural Numbers under Max Operation is Monoid

Let $S \subseteq \N$ be a subset of the natural numbers $\N$. Let $\left({S, \max}\right)$ denote the algebraic structure formed from $S$ and the max operation. Then $\left({S, \max}\right)$ is a monoid. Its identity element is the smallest element of $S$.


Stirling Number of the Second Kind of n+1 with 2

:$\displaystyle {n + 1 \brace 2} = 2^n - 1$


Preceding is Approximating Relation

Let $\left({S, \preceq}\right)$ be an ordered set. Then $\preceq$ is an approximating relation on $S$.


Substitution Instance of WFF is WFF

Let $\mathbf A$ be a WFF of predicate logic. Let $\tau$ be a term of predicate logic. Let $x \in \mathrm{VAR}$ be a variable. Let $\mathbf A \left({x \gets \tau}\right)$ be the substitution instance of $\mathbf A$ substituting $\tau$ for $x$. Then $\mathbf A \left({x \gets \tau}\right)$ is a WFF.


Sequence Converges to Within Half Limit/Real Numbers

Let $\sequence {x_n}$ be a sequence in $\R$. Let $\sequence {x_n}$ be convergent to the limit $l$. That is, let $\displaystyle \lim_{n \mathop \to \infty} x_n = l$. Suppose $l > 0$. Then: : $\exists N: \forall n > N: x_n > \dfrac l 2$ Similarly, suppose $l < 0$. Then: : $\exists N: \forall n > N: x_n < \dfrac l 2$


Binomial Coefficient/Examples/Number of Bridge Hands

The total number $N$ of possible different hands for a game of [https://en.wikipedia.org/wiki/Contract_bridge bridge] is: :$N = \dfrac {52!} {13! \, 39!} = 635 \ 013 \ 559 \ 600$


Symmetric Difference is Subset of Union of Symmetric Differences

Let $R, S, T$ be sets. Then: :$R * S \subseteq \left({R * T}\right) \cup \left({S * T}\right)$ where $R * S$ denotes the symmetric difference between $R$ and $S$.


Continuous iff Meet-Continuous and There Exists Smallest Auxiliary Approximating Relation

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Then: :$L$ is continuous {{iff}} :$L$ is meet-continuous and there exists the smallest auxiliary approximating relation on $S$ That is: :$L$ is continuous {{iff}} :$L$ is meet-continuous and there exists an auxiliary approximating relation $\mathcal R$ on $S$ ::for every auxiliary approximating relation $\mathcal Q$ on $S$: $\mathcal R \subseteq \mathcal Q$


Sum over k of r+tk choose k by s-tk choose n-k

Let $n \in \Z_{\ge 0}$ be a non-negative integer. Then: :$\displaystyle \sum_k \dbinom {r + t k} k \dbinom {s - t k} {n - k} = \sum_{k \mathop \ge 0} \dbinom {r + s - k} {n - k} t^k$ where $\dbinom {r + t k} k$ etc. denotes a binomial coefficient.


Homogeneous System has Zero Vector as Solution

Every homogeneous system of linear equations has the zero vector as a solution.


Summation over Lower Index of Unsigned Stirling Numbers of the First Kind

Let $n \in \Z_{\ge 0}$ be a positive integer. Then: :$\displaystyle \sum_k \left[{n \atop k}\right] = n!$ where: :$\displaystyle \left[{n \atop k}\right]$ denotes an unsigned Stirling number of the first kind :$n!$ denotes the factorial of $n$.


Co-Countable Measure is Probability Measure

Let $X$ be an uncountable set. Let $\mathcal A$ be the $\sigma$-algebra of countable sets on $X$. Then the co-countable measure $\mu$ on $X$ is a probability measure.


Supremum of Ideals is Upper Adjoint

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below continuous join semilattice. Let $\mathit{Ids}\left({L}\right)$ be the set of all ideals in $L$. Let $P = \left({\mathit{Ids}\left({L}\right), \precsim}\right)$ be an ordered set where $\mathord \precsim = \subseteq\restriction_{\mathit{Ids}\left({L}\right)\times \mathit{Ids}\left({L}\right)}$ Let $f: \mathit{Ids}\left({L}\right) \to S$ be a mapping such that :$\forall I \in \mathit{Ids}\left({L}\right): f\left({I}\right) = \sup I$ Then $f$ is an upper adjoint of Galois connection.


Summation of Summation over Divisors of Function of Two Variables

Let $c, d, n \in \Z$. Then: :$\displaystyle \sum_{d \mathop \divides n} \sum_{c \mathop \divides d} \map f {c, d} = \sum_{c \mathop \divides n} \sum_{d \mathop \divides \paren {n / c} } \map f {c, c d}$ where $c \divides d$ denotes that $c$ is a divisor of $d$.


Summation over k of Ceiling of k over 2

:$\displaystyle \sum_{k \mathop = 1}^n \left \lceil{\dfrac k 2}\right \rceil = \left \lceil{\dfrac {n \left({n + 2}\right)} 4}\right \rceil$


Open Rectangles Closed under Intersection

Let $\left(({\mathbf a \,.\,.\, \mathbf b}\right))$ and $\left(({\mathbf c \,.\,.\, \mathbf d}\right))$ be open $n$-rectangles. Then $\left(({\mathbf a \,.\,.\, \mathbf b}\right)) \cap \left(({\mathbf c \,.\,.\, \mathbf d}\right))$ is also an open $n$-rectangle.


Euclidean Borel Sigma-Algebra Closed under Scalar Multiplication

Let $\mathcal B \left({\R^n}\right)$ be the Borel $\sigma$-algebra on $\R^n$. Let $B \in \mathcal B$, and let $t \in \R_{>0}$. Then also $t \cdot B := \left\{{t \mathbf b: \mathbf b \in B}\right\} \in \mathcal B$.


Lebesgue Measure of Scalar Multiple

Let $\lambda^n$ be the $n$-dimensional Lebesgue measure on $\R^n$ equipped with the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$. Let $B \in \mathcal B$, and let $t \in \R_{>0}$. Then $\lambda^n \left({t \cdot B}\right) = t^n \lambda^n \left({B}\right)$, where $t \cdot B$ is the set $\left\{{t \mathbf b: \mathbf b \in B}\right\}$.


Membership of Set of Strictly Positive Integers is Replicative Function

Let $f: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map f x = \sqbrk {x \in \Z_{> 0} }$ where $\sqbrk \cdots$ is Iverson's convention. Then $f$ is a replicative function.


Membership of Equivalence Class of m mod pi is Replicative Function

Let $f: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map f x = \sqbrk {\exists r \in \Q, \exists m \in \Z: x = r \pi + m}$ where $\sqbrk {\cdots}$ is Iverson's convention. Then $f$ is a replicative function.


Sum of Replicative Functions is Replicative

Let $f: \R \to \R$ and $g: \R \to \R$ be real functions. Let $f$ and $g$ both be replicative functions. Then the pointwise sum of $f$ and $g$ is also a replicative function.


Meet Irreducible iff Finite Infimum equals Element

Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $x \in S$. Then :$x$ is meet irreducible {{iff}} :for every non-empty finite subset $A$ of $S$: $x = \inf A \implies x \in A$


Sum over j of Function of Floor of mj over n

Let $f$ be a real function. Then: :$\displaystyle \sum_{0 \mathop \le j \mathop < n} \map f {\floor {\dfrac {m j} n} } = \sum_{0 \mathop \le r \mathop < m} \ceiling {\dfrac {r n} m} \paren {\map f {r - 1} - \map f r} + n \map f {m - 1}$


Isomorphism of External Direct Products/General Result

Let: : $(1): \quad \displaystyle \left({S, \circ}\right) = \prod_{k \mathop = 1}^n S_k = \left({S_1, \circ_1}\right) \times \left({S_2, \circ_2}\right) \times \cdots \times \left({S_n, \circ_n}\right)$ : $(2): \quad \displaystyle \left({T, \ast}\right) = \prod_{k \mathop = 1}^n T_k = \left({T_1, \ast_1}\right) \times \left({T_2, \ast_2}\right) \times \cdots \times \left({T_n, \ast_n}\right)$ be external direct products of algebraic structures. Let $\phi_k: \left({S_k, \circ_k}\right) \to \left({T_k, \ast_k}\right)$ be an isomorphism for each $k \in \left[{1 \,.\,.\, n}\right]$. Then: :$\phi: \left({s_1, \ldots, s_n}\right) \to \left({\phi_1 \left({s_1}\right), \ldots, \phi_n \left({s_n}\right)}\right)$ is an isomorphism from $\left({S, \circ}\right)$ to $\left({T, \ast}\right)$.


Cancellable Infinite Semigroup is not necessarily Group

Let $\struct {S, \circ}$ be a semigroup whose underlying set is infinite. Let $\struct {S, \circ}$ be such that all elements of $S$ are cancellable. Then it is not necessarily the case that $\struct {S, \circ}$ is a group.


Latin Square is not necessarily Cayley Table of Group

While it is true that the Cayley table of a (finite) group is in the form of a Latin square it is not necessarily the case that a Latin square is the Cayley table of a group.


Projection is Epimorphism/General Result

Let $\left({S, \circ}\right)$ be the external direct product of the algebraic structures $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_k, \circ_k}\right), \ldots, \left({S_n, \circ_n}\right)$. Then: :for each $j \in \left[{1 \,.\,.\, n}\right]$, $\operatorname{pr}_j$ is an epimorphism from $\left({S, \circ}\right)$ to $\left({S_j, \circ_j}\right)$ where $\operatorname{pr}_j: \left({S, \circ}\right) \to \left({S_j, \circ_j}\right)$ is the $j$th projection from $\left({S, \circ}\right)$ to $\left({S_j, \circ_j}\right)$.


External Direct Product of Projection with Canonical Injection/General Result

Let $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \dotsc, \struct {S_j, \circ_j}, \dotsc, \struct {S_n, \circ_n}$ be algebraic structures with identities $e_1, e_2, \dotsc, e_j, \dotsc, e_n$ respectively. Let $\displaystyle \struct {S, \circ} = \prod_{i \mathop = 1}^n \struct {S_i, \circ_i}$ be the external direct product of $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \dotsc, \struct {S_j, \circ_j}, \dotsc, \struct {S_n, \circ_n}$. Let $\pr_j: \struct {S, \circ} \to \struct {S_j, \circ_j}$ be the $j$th projection from $\struct {S, \circ}$ to $\struct {S_j, \circ_j}$. Let $\inj_j: \struct {S_j, \circ_j} \to \struct {S, \circ}$ be the canonical injection from $\struct {S_j, \circ_j}$ to $\struct {S, \circ}$. Then: :$\pr_j \circ \inj_j = I_{S_j}$ where $I_{S_j}$ is the identity mapping from $S_j$ to $S_j$.


Pre-Image Sigma-Algebra on Codomain is Sigma-Algebra

Let $X, X'$ be sets, and let $f: X \to X'$ be a mapping. Let $\Sigma$ be a $\sigma$-algebra on $X$. Denote with $\Sigma'$ the pre-image $\sigma$-algebra on the domain of $f$. Then $\Sigma'$ is a $\sigma$-algebra on $X'$.


Multiplicative Group of Reduced Residues Modulo 8 is Klein Four-Group

Let $K_4$ denote the Klein $4$-group. Let $R_4$ be the multiplicative group of reduced residues Modulo $8$. Then $K_4$ and $R_4$ are isomorphic algebraic structures.


Properties of Relation Not Preserved by Restriction

If a relation is: * serial, * non-reflexive, * non-symmetric, * non-transitive or * non-connected it is impossible to state without further information whether or not any restriction of that relation has the same properties.


Totally Ordered Set is Well-Ordered iff Subsets Contain Infima

Let $\left({S, \preccurlyeq}\right)$ be a totally ordered set. Then $\left({S, \preccurlyeq}\right)$ is a well-ordered set iff every non-empty subset of $T \subseteq S$ has an infimum such that $\inf \left({T}\right) \in T$.


Cantor Set has Zero Lebesgue Measure

Let $\mathcal C$ be the Cantor set. Let $\lambda$ be the Lebesgue measure on the Borel $\sigma$-algebra $\mathcal B \left({\R}\right)$ on $\R$. Then $\mathcal C$ is $\mathcal B \left({\R}\right)$-measurable, and $\lambda \left({\mathcal C}\right) = 0$. That is, $\mathcal C$ is a $\lambda$-null set.


Factorization Lemma/Real-Valued Function

Then a mapping $g: X \to \R$ is $\map \sigma f \, / \, \map {\mathcal B} \R$-measurable {{iff}}: :There exists a $\Sigma \, / \, \map {\mathcal B} \R$-measurable mapping $\tilde g: Y \to \R$ such that $g = \tilde g \circ f$ where: :$\map \sigma f$ denotes the $\sigma$-algebra generated by $f$ :$\map {\mathcal B} \R$ denotes the Borel $\sigma$-algebra on $\R$


Non-Zero-Sum Game as Zero-Sum Game

Let $G$ be a non-zero-sum game for $n$ players. Then $G$ can be modelled as a zero-sum game for $n + 1$ players.


Two-Person Zero-Sum Game is Non-Cooperative

A two-person zero-sum game necessarily has to be non-cooperative.


Simple Graph with Finite Vertex Set is Finite

Let $G$ be a simple graph. Suppose that the vertex set of $G$ is finite. Then $G$ is a finite graph. That is to say, its edge set is also finite.


Tarski's Geometry is Complete/Corollary

Tarski's geometry does not contain minimal arithmetic.


Symmetric Closure of Symmetric Relation

Let $\mathcal R$ be a relation on a set $S$. Let $\mathcal R^\leftrightarrow$ be the symmetric closure of $\mathcal R$. Then $\mathcal R = \mathcal R^\leftrightarrow$.


Eluding Game has no Saddle Point

The eluding game has no saddle point.


Prime Element iff Element Greater is Top

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a Boolean lattice. Let $p \in S$ such that :$p \ne \top$ Then :$p$ is prime element {{iff}} :$\forall x \in S: \left({ p \prec x \implies x = \top }\right)$


Relation Isomorphism Preserves Antisymmetry

Let $\left({S, \mathcal R_1}\right)$ and $\left({T, \mathcal R_2}\right)$ be relational structures. Let $\left({S, \mathcal R_1}\right)$ and $\left({T, \mathcal R_2}\right)$ be (relationally) isomorphic. Then $\mathcal R_1$ is an antisymmetric relation {{iff}} $\mathcal R_2$ is also an antisymmetric relation.


There are exactly four anomalously cancelling vulgar fractions having two-digit numerator and denominator when expressed in base $10$ notation: {{:Anomalous Cancellation on 2-Digit Numbers/Examples}}


Intersection of Strict Lower Closures in Toset

Let $\left({S, \preceq}\right)$ be a totally ordered set. Let $a,b \in S$. Then: :$a^\prec \cap b^\prec = \left({\min \left({a, b}\right)}\right)^\prec$ where: : $a^\prec$ denotes strict lower closure of $a$ : $\min$ denotes the min operation.


Intermediate Value Theorem (Topology)

Let $X$ be a connected topological space. Let $\struct {Y, \preceq, \tau}$ be a totally ordered set equipped with the order topology. Let $f: X \to Y$ be a continuous mapping. Let $a$ and $b$ are two points of $a, b \in X$ such that: :$\map f a \prec \map f b$ Let: :$r \in Y: \map f a \prec r \prec \map f b$ Then there exists a point $c$ of $X$ such that: :$\map f c = r$


Equivalence of Definitions of Upper Wythoff Sequence

The following definitions of the upper Wythoff sequence are equivalent:


Positive Integer is Sum of Consecutive Positive Integers iff not Power of 2

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then $n$ can be expressed as the sum of $2$ or more consecutive (strictly) positive integers {{iff}} $n$ is not a power of $2$.


Baire-Osgood Theorem

Let $X$ be a Baire space. Let $Y$ be a metrizable topological space Let $f: X \to Y$ be a mapping which is the pointwise limit of a sequence $\left \langle{f_n}\right\rangle$ in $C \left({X, Y}\right)$. {{explain|$C \left({X, Y}\right)$}} Let $D \left({f}\right)$ be the set of points where $f$ is discontinuous. Then $D \left({f}\right)$ is a meager subset of $X$.


Ramanujan's Infinite Nested Roots

:$3 = \sqrt {1 + 2 \sqrt {1 + 3 \sqrt { 1 + \cdots} } }$


Product of Three Consecutive Integers is never Perfect Power

Let $n \in \Z_{> 1}$ be a (strictly) positive integer. Then: :$\paren {n - 1} n \paren {n + 1}$ cannot be expressed in the form $a^k$ for $a, k \in \Z$ where $k \ge 2$. That is, the product of $3$ consecutive (strictly) positive integers can never be a perfect power.


Maximal Element need not be Greatest Element

Let $\struct {S, \preccurlyeq}$ be an ordered set. Let $M \in $ be a maximal element of $S$. Then $M$ is not necessarily the greatest element of $S$.


Maximal Ideal of Division Ring

Let $\left({D, +, \circ}\right)$ be a Division Ring whose zero is $0$. :Let $\left({J, +, \circ}\right)$ be a maximal ideal of $D$. Then $J = \left\{{0}\right\}$.


Identity Mapping is Ordered Ring Automorphism

Let $\struct {S, +, \circ, \preceq}$ be an ordered ring. Then the identity mapping $I_S: S \to S$ is an ordered ring automorphism.


Number of Binary Digits in Power of 10/Example/1000

When expressed in binary notation, the number of digits in $1000$ is $10$.


Characteristic of Integral Domain is Zero or Prime

Let $\struct {D, +, \circ}$ be an integral domain. Let $\operatorname{Char} \left({D}\right)$ be the characteristic of $D$. Then $\operatorname{Char} \left({D}\right)$ is either $0$ or a prime number.


Number is Sum of Five Cubes

Let $n \in \Z$ be an integer. Then $n$ can be expressed as the sum of $5$ cubes (either positive or negative) in an infinite number of ways.


Exponential is of Exponential Order Real Part of Index

Let $\map f t = e^{\psi t}$ be the complex exponential function, where $t \in \R, \psi \in \C$. Let $a = \map \Re \psi$. Then $e^{\psi t}$ is of exponential order $a$.


Scalar Product with Multiple of Unity

:$\paren {n \cdot 1_R} \circ x = n \cdot x$ that is: :$\paren {\map {\paren {+_R}^n} {1_R} } \circ x = \map {\paren {+_G}^n} x$


Pythagorean Triangle from Fibonacci Numbers

Take $4$ consecutive Fibonacci numbers: :$F_n, F_{n + 1}, F_{n + 2}, F_{n + 3}$ Let: :$a := F_n F_{n + 3}$ :$b := 2 F_{n + 1} F_{n + 2}$ :$c := F_{2 n + 3}$ Then: :$a^2 + b^2 = c^2$ and: :$\dfrac {a b} 2 = F_n \times F_{n + 1} \times F_{n + 2} \times F_{n + 3}$ That is, if the legs of a right triangle are the product of the outer terms and twice the inner terms, then: :the hypotenuse is the Fibonacci number whose index is half the sum of the indices of the four given Fibonacci numbers. :the area is the product of the four given Fibonacci numbers.


Conditions for Homogeneity/Plane

The plane $P = \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 = \gamma$ is homogeneous iff $\gamma = 0$.


Field Homomorphism Preserves Subfields

Let $\struct {F_1, +_1, \circ_1}$ and $\struct {F_2, +_2, \circ_2}$ be fields. Let $\phi: F_1 \to F_2$ be a field homomorphism such that $\phi$ is not the trivial homomorphism. If $K$ is a subfield of $F_1$, then $\phi \sqbrk K$ is a subfield of $F_2$.


Fermat Quotient of 2 wrt p is Square iff p is 3 or 7

Let $p$ be a prime number. The Fermat quotient of $2$ with respect to $p$: :$\map {q_p} 2 = \dfrac {2^{p - 1} - 1} p$ is a square {{iff}} $p = 3$ or $p = 7$.


Integer Addition is Well-Defined

Let $\struct {\N, +}$ be the semigroup of natural numbers under addition. Let $\struct {\N \times \N, \oplus}$ be the (external) direct product of $\struct {\N, +}$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$. Let $\boxtimes$ be the cross-relation defined on $\N \times \N$ by: :$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$ Let $\eqclass {x, y} {}$ denote the equivalence class of $\tuple {x, y}$ under $\boxtimes$. The operation $\oplus$ on these equivalence classes is well-defined, in the sense that: {{begin-eqn}} {{eqn | l = \eqclass {a_1, b_1} {}       | r = \eqclass {a_2, b_2} {}       | c =  }} {{eqn | l = \eqclass {c_1, d_1} {}       | r = \eqclass {c_2, d_2} {}       | c =  }} {{eqn | ll= \leadsto       | l = \eqclass {a_1, b_1} {} \oplus \eqclass {c_1, d_1} {}       | r = \eqclass {a_2, b_2} {} \oplus \eqclass {c_2, d_2} {}       | c =  }} {{end-eqn}}


Preimage of Serial Relation is Domain

Let $\mathcal R$ be a serial relation on $S$. Then the preimage of $\mathcal R$ is $S$ (the domain of $\mathcal R$).


Closed Form for Octagonal Numbers

The closed-form expression for the $n$th octagonal number is: :$O_n = n \left({3 n - 2}\right)$


Number of Distinct Deltahedra is Unlimited

There are an unlimited number of distinct deltahedra.


Set Intersection Preserves Subsets/Corollary/Proof 1

Let $A, B, S$ be sets. Then: :$A \subseteq B \implies A \cap S \subseteq B \cap S$


Characteristic of Increasing Mapping from Toset to Order Complete Toset

Let $\struct {S, \preceq}$ and $\struct {T, \preccurlyeq}$ be tosets. Let $T$ be order complete. Let $H \subseteq S$ be a subset of $S$. Let $f: H \to T$ be an increasing mapping from $H$ to $T$. Then: :$f$ has an extension to $S$ which is increasing {{iff}}: :for all $A \subseteq H$: if $A$ is bounded in $S$, then $f \sqbrk A$ is bounded in $T$ where $f \sqbrk A$ denotes the image set of $A$ under $f$.


A.E. Equal Positive Measurable Functions have Equal Integrals

Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f, g: X \to \overline \R_{\ge 0}$ be positive $\mu$-measurable functions. Suppose that $f = g$ almost everywhere. Then: :$\displaystyle \int f \rd \mu = \int g \rd \mu$


Characterization of Euler's Number by Inequality

Let $a$ be a (strictly) positive real number. Then: :$a = e \iff \forall x \in \R: a^x \ge x + 1$ where $e$ denotes Euler's number.


Limit of x to the x

Let $f: \R \to \R$ be defined on $\left [{0 \,.\,.\, \to} \right)$ with $f \left({x}\right) = x^x$. Then: :$\displaystyle \lim_{x \to 0^+} x^x = 1$ Equivalently, from the definition of power: :$\displaystyle \lim_{x \to 0^+} \exp \left({x \ln x}\right) = 1$


Schanuel's Conjecture Implies Transcendence of Log Pi

Let Schanuel's Conjecture be true. Then the logarithm of $\pi$ (pi): :$\ln \pi$ is transcendental.


Cauchy-Bunyakovsky-Schwarz Inequality/Lebesgue 2-Space

Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f, g: X \to \R$ be $\mu$-square integrable functions, that is $f, g \in \map {\LL^2} \mu$, Lebesgue $2$-space. Then: :$\displaystyle \int \size {f g} \rd \mu \le \norm f_2^2 \cdot \norm g_2^2$ where $\norm {\, \cdot \,}_2$ is the $2$-norm.


Preimage of Intersection under Relation/Family of Sets

Let $S$ and $T$ be sets. Let $\family {T_i}_{i \mathop \in I}$ be a family of subsets of $T$. Let $\RR \subseteq S \times T$ be a relation. Then: :$\ds \RR^{-1} \sqbrk {\bigcap_{i \mathop \in I} T_i} \subseteq \bigcap_{i \mathop \in I} \RR^{-1} \sqbrk {T_i}$ where $\ds \bigcap_{i \mathop \in I} T_i$ denotes the intersection of $\family {T_i}_{i \mathop \in I}$.


Image of Intersection under Injection/Family of Sets

Let $S$ and $T$ be sets. Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$. Let $f: S \to T$ be a mapping. Then: :$\displaystyle f \sqbrk {\bigcap_{i \mathop \in I} S_i} = \bigcap_{i \mathop \in I} f \sqbrk {S_i}$ {{iff}} $f$ is an injection.


Lines through Center Square of Order 3 Magic Square are in Arithmetic Sequence

Consider the order 3 magic square: {{:Magic Square/Examples/Order 3}} Each of the lines through the center cell contain $3$ integers in arithmetic sequence.


Omega Constant is Transcendental

The omega constant is transcendental.


Divisibility by Power of 10

Let $r \in \Z_{\ge 1}$ be a strictly positive integer. An integer $N$ expressed in decimal notation is divisible by $10^r$ {{iff}} the last $r$ digits of $N$ are all $0$. 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 $10^r$  {{iff}}: :$a_0 + a_1 10 + a_2 10^2 + \cdots + a_r 10^r = 0$


Correspondence Theorem (Set Theory)

Let $S$ be a set. Let $\RR \subseteq S \times S$ be an equivalence relation on $S$. Let $\mathscr A$ be the set of partitions of $S$ associated with equivalence relations $\RR'$ on $S$ such that: :$\tuple {x, y} \in \RR \iff \tuple {x, y} \in \RR'$ Then there exists a bijection $\phi$ from $\mathscr A$ onto the set of partitions of the quotient set $S / \RR$.


Factorial as Product of Three Factorials

This general pattern can be used to find a factorial which is the product of three factorials: :$\left({\left({n!}\right)!}\right)! = n! \left({n! - 1}\right)! \left({\left({n!}\right)! - 1}\right)!$ while there are instances of factorials which do not fit that pattern.


Right Quasigroup if (1-3) Parastrophe of Magma is Magma

Let $\struct {S, \circ}$ be a magma. Let the $(1-3)$ parastrophe of $\struct {S, \circ}$ be a magma. Then $\struct {S, \circ}$ is a right quasigroup.


Left Quasigroup if (2-3) Parastrophe of Magma is Magma

Let $\struct {S, \circ}$ be a magma. Let the $\paren {2 - 3}$ parastrophe of $\struct {S, \circ}$ be a magma. Then $\struct {S, \circ}$ is a left quasigroup.


Product of Cardinals is Associative

Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be cardinals. Then: : $\mathbf a \left({\mathbf b \mathbf c}\right) = \left({\mathbf a \mathbf b}\right) \mathbf c$ where $\mathbf a \mathbf b$ denotes the product of $\mathbf a$ and $\mathbf b$.


Fundamental Theorem of Line Integrals

Let $\mathcal C$ be a smooth curve given by the vector function $\mathbf r \left({t}\right)$ for $a \le t \le b$. Let $f$ be a differentiable function of two or three variables whose gradient vector $\nabla f$ is continuous on $\mathcal C$. Then: :$\displaystyle \int_\mathcal C \nabla f \cdot d \mathbf r = f \left({\mathbf r \left({b}\right)}\right) - f \left({\mathbf r \left({a}\right)}\right)$


Abelian Group Induces Commutative B-Algebra

Let $\left({G, \circ}\right)$ be an abelian group whose identity element is $e$. Let $*$ be the binary operation on $G$ defined as: :$\forall a, b \in G: a * b = a \circ b^{-1}$ where $b^{-1}$ is the inverse element of $b$ under the operation $\circ$. Then the algebraic structure $\left({G, *}\right)$ is a commutative $B$-algebra. That is: :$\forall a, b \in G: a * \left({0 * b}\right) = b * \left({0 * a}\right)$


Sum of Cardinals is Associative

Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be cardinals. Then: : $\mathbf a + \left({\mathbf b + \mathbf c}\right) = \left({\mathbf a + \mathbf b}\right) + \mathbf c$ where $\mathbf a + \mathbf b$ denotes the sum of $\mathbf a$ and $\mathbf b$.


Lucas Number 2n in terms of Square of Lucas Number n

Let $L_n$ denote the $n$th Lucas number. Then: :$L_{2 n} = {L_n}^2 + 2 \left({-1}\right)^n$


Natural Numbers as Cardinals

The natural numbers $\N = \set {0, 1, 2, 3, \ldots}$ can be defined as the set of cardinals.


Young's Inequality for Convolutions

Let $p, q, r \in \R_{\ge 1}$ satisfy: :$1 + \dfrac 1 r = \dfrac 1 p + \dfrac 1 q$ Let $L^p \left({\R^n}\right)$, $L^q \left({\R^n}\right)$, and $L^r \left({\R^n}\right)$ be Lebesgue spaces with seminorms $\left\Vert{\cdot}\right\Vert_p$, $\left\Vert{\cdot}\right\Vert_q$, and $\left\Vert{\cdot}\right\Vert_r$ respectively. Let $f \in L^p \left({\R^n}\right)$ and $g \in L^q \left({\R^n}\right)$. Then the convolution $f * g$ is in $L^r \left({\R^n}\right)$ and the following inequality is satisfied: :$\left\Vert{f * g}\right\Vert_r \le \left\Vert{f}\right\Vert_p \cdot \left\Vert{g}\right\Vert_q$


Conditions for C^1 Smooth Solution of Euler's Equation to have Second Derivative

Let $\map y x:\R \to \R$ be a real function. Let $\map F {x, y, y'}:\R^3 \to \R$ be a real function. Suppose $\map F {x, y, y'}$ has continuous first and second derivatives {{WRT|Differentiation}} all its arguments. Suppose $y$ has a continuous first derivative and satisfies Euler's equation:  :$F_y - \dfrac \d {\d x} F_{y'} = 0$ Suppose: :$\map {F_{y' y'} } {x, \map y x, \map y x'} \ne 0$ Then $\map y x$ has continuous second derivatives.


If Double Integral of a(x, y)h(x, y) vanishes for any C^2 h(x, y) then C^0 a(x, y) vanishes

Let $\alpha \left({x, y}\right)$, $h \left({x, y}\right)$ be functions in $\R$. Let $\alpha \in C^0$ in a closed region $R$ whose boundary is $\Gamma$. Let $h \in C^2$ in $R$ and $h = 0$ on $\Gamma$. Let: : $\displaystyle \int \int_R \alpha \left({x, y}\right) h \left({x, y}\right) \rd x \rd y = 0$ Then $\alpha \left({x, y}\right)$ vanishes everywhere in $R$.


Simple Variable End Point Problem

Let $y$ and $F$ be mappings. {{explain|Define their domain and codomain}} Suppose the endpoints of $y$ lie on two given vertical lines $x = a$ and $x = b$. Suppose $J$ is a functional of the form :$(1): \quad J \sqbrk y = \displaystyle \int_a^b \map F {x, y, y'} \rd x$ and has an extremum for a certain function $\hat y$. Then $y$ satisfies the system of equations :$\begin {cases} F_y - \dfrac \d {\d x} F_{y'} = 0 \\ \bigvalueat {F_{y'} } {x \mathop = a} = 0 \\ \bigvalueat {F_{y'} } {x \mathop = b} = 0 \end {cases}$


Ordered Set of Closure Operators and Dual Ordered Set of Closure Systems are Isomorphic

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Then $\operatorname{Closure}\left({L}\right)$ and $\operatorname{ClSystems}\left({L}\right)^{-1}$ are order isomorphic where :$\operatorname{Closure}\left({L}\right)$ denotes the ordered set of closure operators of $L$, :$\operatorname{ClSystems}\left({L}\right)$ denotes the ordered set of closure systems oj $L$, :$\operatorname{ClSystems}\left({L}\right)^{-1}$ denotes the dual to $\operatorname{ClSystems}\left({L}\right)$.


Euler's Equation for Vanishing Variation is Invariant under Coordinate Transformations

Euler's Equation for Vanishing Variation is invariant under coordinate transformations.


Minimal Infinite Successor Set is Limit Ordinal

Let $\omega$ denote the minimal infinite successor set. Then $\omega$ is a limit ordinal.


Ordinals Isomorphic to the Same Well-Ordered Set

Let $A$ and $B$ be ordinals. Let $\left({\prec, S}\right)$ be a strict well-ordering. Let $\left({\in, A}\right)$ and $\left({\prec, S}\right)$ be order isomorphic. Let $\left({\in, B}\right)$ and $\left({\prec, S}\right)$ be order isomorphic. Then: : $A = B$


Topological Group is T1 iff T2

Let $G$ be a topological group. Then $G$ is a $T_1$ space {{iff}} $G$ is Hausdorff.


Logarithm of Infinite Product of Complex Numbers

Let $\sequence {z_n}$ be a sequence of nonzero complex numbers. {{TFAE}} :$(1): \quad$ The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty z_n$ converges to $z \in \C_{\ne 0}$. :$(2): \quad$ The series $\displaystyle \sum_{n \mathop = 1}^\infty \log z_n$ converges to $\log z + 2 k \pi i$ for some integer $k \in \Z$.


Infinite Ordinal can be expressed Uniquely as Sum of Limit Ordinal plus Finite Ordinal

Let $x$ be an ordinal. Suppose $x$ satisfies $\omega \subseteq x$. Then $x$ has a unique representation as $\paren {y + z}$ where $y$ is a limit ordinal and $z$ is a finite ordinal.


Legs of Pythagorean Triangle used as Generator for another Pythagorean Triangle

Let $a$ and $b$ be the legs of a Pythagorean triangle $P_1$. Let $\tuple {a, b}$ be used as the generator for a new Pythagorean triangle $P_2$. Then the hypotenuse of $P_2$ is the square of the hypotenuse of $P_1$.


Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation

Let $\Phi = \map {\Phi} {x, \family {y_i}_{1 \mathop \le i \mathop \le n}, \family {p_i}_{1 \mathop \le i \mathop \le n} }$ be a real function. Let $H$ be Hamiltonian. Then a necessary and sufficient condition for $\Phi$ to be the first integral of Euler's Equations is :$\dfrac {\partial \Phi} {\partial x} + \sqbrk{\Phi, H} = 0$


Homotopic Paths Implies Homotopic Composition

Let $T = \left({S, \tau}\right)$ be a topological space. Let $f_1, f_2, g_1, g_2: \left[{0 \,.\,.\, 1}\right] \to S$ be paths in $T$. Let $f_1$ be homotopic to $f_2$ and $g_1$ be homotopic to $g_2$. Then the concatenated paths $f_1 * g_1$ and $f_2 * g_2$ are homotopic.


Product of Two Triangular Numbers to make Square

Let $T_n$ be a triangular number. Then there is an infinite number of $m \in \Z_{>0}$ such that $T_n \times T_m$ is a square number.


Numbers of form 31 x 16^n are sum of 16 Powers of 4

Let $m \in \Z$ be an integer of the form $31 \times 16^n$ for $n \in \Z_{\ge 0}$. Then in order express $m$ as the sum of powers of $4$, you need $16$ of them.


Element of Ordered Set of Topology is Dense iff is Everywhere Dense

Let $T = \struct {S, \tau}$ be a topological space. Let $P = \struct {\tau, \preceq}$ be an ordered set where $\mathord\preceq = \mathord\subseteq \cap \paren {\tau \times \tau}$ Let $A \in \tau$. Then $A$ is a dense element in $P$ {{iff}} $A$ is everywhere dense.


Element Commutes with Product of Commuting Elements/General Theorem

Let $(S,\circ)$ be a semigroup. Let $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ be a sequence of terms of $S$. Let $b \in S$. If $b$ commutes with $a_k$ for each $k \in \left[{1 \,.\,.\, n}\right]$, then $b$ commutes with $a_1 \circ \cdots \circ a_n$.


Set is Small Class

Let $x$ be a set. Then $x$ is a small class.


Group Induced by B-Algebra Induced by Group

Let $\left({S, \circ}\right)$ be a group. Let $\left({S, *}\right)$ be the $B$-algebra described on Group Induces $B$-Algebra. Let $\left({S, \circ'}\right)$ be the group described on $B$-Algebra Induces Group. Then $\left({S, \circ'}\right) = \left({S, \circ}\right)$.


Category of Ordered Sets is Category

Let $\mathbf{OrdSet}$ be the category of ordered sets. Then $\mathbf{OrdSet}$ is a metacategory.


Magic Hexagon of Order 3 is Unique

Apart from the trivial order $1$ magic hexagon, there exists only one magic hexagon: the order $3$ magic hexagon: {{:Definition:Order 3 Magic Hexagon}}


Universal Class is Proper

Let $V$ denote the universal class. Then $V$ is a proper class.


Squares Ending in 5 Occurrences of 2-Digit Pattern

Let $n$ be a square number whose decimal representation ends in the pattern $\mathtt {xyxyxyxyxy}$. Then $\mathtt {xy}$ is one of: :$21, 29, 61, 69, 84$ The smallest examples of such numbers are: {{begin-eqn}} {{eqn | l = 508 \, 853 \, 989^2       | r = \phantom {0 \,} 258 \, 932 \, 38 \mathbf {2 \, 121 \, 212 \, 121} }} {{eqn | l = 162 \, 459 \, 327^2       | r = \phantom {0 \, 0} 26 \, 393 \, 03 \mathbf {2 \, 929 \, 292 \, 929} }} {{eqn | l = 1 \, 318 \, 820 \, 881^2       | r = 1 \, 739 \, 288 \, 51 \mathbf {6 \, 161 \, 616 \, 161} }} {{eqn | l = 541 \, 713 \, 187^2       | r = \phantom {0 \,} 293 \, 453 \, 17 \mathbf {6 \, 969 \, 696 \, 969} }} {{eqn | l = 509 \, 895 \, 478^2       | r = \phantom {0 \,} 259 \, 993 \, 39 \mathbf {8 \, 484 \, 848 \, 484} }} {{end-eqn}}


Sufficient Condition for Twice Differentiable Functional to have Minimum

Let $J$ be a twice differentiable functional. Let $J$ have an extremum for $y=\hat y$. Let the second variation $\delta^2 J \sqbrk {\hat y; h}$ be strongly positive {{WRT}} $h$. Then $J$ acquires the minimum for $y = \hat y$ .


Discrete Category is Order Category

Let $\mathbf{Dis} \left({S}\right)$ be a discrete category. Then $\mathbf{Dis} \left({S}\right)$ is also an order category.


Category of Monoids is Category

Let $\mathbf{Mon}$ be the category of monoids. Then $\mathbf{Mon}$ is a metacategory.


Cayley's Representation Theorem/General Case

Let $\struct {G, \cdot}$ be a group. Then there exists a permutation group $P$ on some set $S$ such that: :$G \cong P$ That is, such that $G$ is isomorphic to $P$.


Permutation of Cosets/Corollary 1

Let $G$ be a group. Let $H \le G$ such that $\index G H = n$ where $n \in \Z$. Then: : $\exists N \lhd G: N \lhd H: n \divides \index G H \divides n!$


Cartesian Product is Small iff Inverse is Small

Let $A$ and $B$ be classes. Then the Cartesian product $A \times B$ is a small class {{iff}} $B \times A$ is small.


Order Isomorphism on Foundational Relation preserves Foundational Structure

Let $A_1$ and $A_2$ be classes. Let $\prec_1$ and $\prec_2$ be relations. Let $\phi: \left({A_1, \prec_1}\right) \to \left({A_2, \prec_2}\right)$ be an order isomorphism. Then $\left({A_1, \prec_1}\right)$ is a foundational structure iff $\left({A_2, \prec_2}\right)$ is also a foundational structure.


Kleene Closure is Free Monoid

Let $S$ be a set. Let $S^*$ be its Kleene closure, and let $i: S \to S^*$ be the insertion of generators. Then $\left({S^*, i}\right)$ is a free monoid over $S$.


Transitive Closure Always Exists (Set Theory)

Let $S$ be a set. Let $G$ be a mapping such that $\map G x = x \cup \bigcup x$. {{explain|Domain and range of $G$ needed}} Let $F$ be defined using the Principle of Recursive Definition: :$\map F 0 = S$ :$\map F {n^+} = \map G {\map F n}$ Let $\displaystyle T = \bigcup_{n \mathop \in \omega} \map F n$. Then: :$T$ is a set and is transitive :$S \subseteq T$ :If $R$ is transitive and $S \subseteq R$, then $T \subseteq R$. That is, given any set $S$, there is an explicit construction for its transitive closure.


Stabilizer of Polynomial

Let $n \in \Z: n > 0$. Let $\map f {x_1, x_2, \ldots, x_n}$ be a polynomial in $n$ variables $x_1, x_2, \ldots, x_n$. Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi, \rho \in S_n$. Let the group action $\pi * f$ be defined as the permutation on the polynomial $f$ by $\pi$. Then the stabilizer of $f$ is the set of permutations on $n$ letters which fix $f$.


Stabilizer of Element of Group Acting on Itself is Trivial

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $*$ be the group action of $\struct {G, \circ}$ on itself by the rule: :$\forall g, h \in G: g * h = g \circ h$ Then the stabilizer of an element $x \in G$ is given by: :$\Stab x = \set e$


Even Perfect Number except 6 is Congruent to 1 Modulo 9

Let $n$ be an even perfect number, but not $6$. Then: :$n \equiv 1 \pmod 9$


Divisibility of Sum of 3 Fourth Powers

Let $n \in \Z_{\ge 0}$ be the sum of three $4$th powers. Then: :$n$ is divisible by $5$ {{iff}} all three addends are also divisible by $5$ :$n$ is divisible by $29$ {{iff}} all three addends are also divisible by $29$.


Prime-Generating Quadratics of form 2 a squared plus p

The quadratic form: :$2 a^2 + p$ yields prime numbers for $a = 0, 1, \ldots, p - 1$ for values of $p$: :$3, 5, 11, 29$


Trivial Group is Initial Object

Let $\mathbf{Grp}$ be the category of groups. Let $1 = \left\{{e}\right\}$ be the trivial group. Then $1$ is an initial object of $\mathbf{Grp}$.


Rank is Ordinal

Let $S$ be a small class The rank of $S$ is an ordinal.


Group Direct Product of Cyclic Groups/Corollary

Let $n_1, n_2, \ldots, n_s$ be a finite sequence of integers, all greater than $1$, such that for any pair of them $n_i$ and $n_j$, $n_1 \perp n_j$. Let $G_i$ be a cyclic group of order $n_i$ for each $i: 1 \le i \le s$. Then $G_1 \times G_2 \times \cdots \times G_s$ is cyclic of order $n_1 n_2 \ldots n_s$.


Compact Closure of Element is Principal Ideal on Compact Subset iff Element is Compact

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below algebraic join semilattice. Let $P = \left({K \left({L} \right), \precsim}\right)$ be an ordered subset of $L$ where $K \left({L} \right)$ denotes the compact subset of $L$. Let $x \in S$. Then $x^{\mathrm{compact} }$ is principal ideal in $P$ {{iff}} $x$ is a compact element.


Arithmetic Sequence of 4 Terms with 3 Distinct Prime Factors

The arithmetic sequence: :$30, 66, 102, 138$ is the smallest of $4$ terms which consists entirely of positive integers each with $3$ distinct prime factors.


Identity Morphism is Terminal Object in Slice Category

Let $\mathbf C$ be a metacategory, and let $C \in \mathbf C_0$ be an object of $\mathbf C$. Let $\operatorname{id}_C: C \to C$ be the identity morphism for $C$. Then $\operatorname{id}_C$ is a terminal object in the slice category $\mathbf C \mathop / C$.


Identity Morphism is Initial Object in Coslice Category

Let $\mathbf C$ be a metacategory, and let $C \in \mathbf C_0$ be an object of $\mathbf C$. Let $\operatorname{id}_C: C \to C$ be the identity morphism for $C$. Then $\operatorname{id}_C$ is an initial object in the coslice category $C \mathop / \mathbf C$.


Ordinal Subset is Well-Ordered

Let $S$ be a class. Let every element of $S$ be an ordinal. Then $\struct {S, \in}$ is a strict well-ordering.


Real Symmetric Matrix is Hermitian

Every real symmetric matrix is Hermitian.


Sufficient Conditions for Weak Extremum

Let $J$ be a functional such that: :$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$ :$\map y a = A$ :$\map y b = B$ Let $y = \map y x$ be an extremum. Let the strengthened Legendre's Condition hold. Let the strengthened Jacobi's Necessary Condition hold. {{explain|specific links to those strengthened versions}} Then the functional $J$ has a weak minimum for $y = \map y x$.


Equivalence Class of Fixed Element/Corollary

:$i \notin \Fix \sigma$ {{iff}} $\eqclass i {\RR_\sigma}$ contains more than one element


Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence

There are an infinite number of rows of Pascal's triangle which contain $3$ integers in arithmetic sequence.


Rows in Pascal's Triangle containing Numbers in Geometric Sequence

There exist no rows of Pascal's triangle which contain $3$ integers in geometric sequence.


Rows in Pascal's Triangle containing Numbers in Harmonic Sequence

There exist no rows of Pascal's triangle which contain $3$ integers in harmonic sequence.


Element of Leibniz Harmonic Triangle is Sum of Numbers Below

The elements in the Leibniz harmonic triangle are the sum of the elements immediately below them. {{refactor|Rework this as another definition of LHT, establishing that column and diagonal $0$ are defined as the reciprocals.}}


Characterization of Prime Element in Inclusion Ordered Set of Topology

Let $T = \struct {S, \tau}$ be a topological space. Let $L = \struct {\tau, \preceq}$  be an inclusion ordered set of $\tau$. Let $Z \in \tau$. Then $Z$ is prime element in $L$ {{iff}}: :$\forall X, Y \in \tau: X \cap Y \subseteq Z \implies X \subseteq Z \lor Y \subseteq Z$


Permutation is Cyclic iff At Most One Non-Trivial Orbit

Let $S$ be a set. Let $\rho: S \to S$ be a permutation on $S$. Then: :$\rho$ is a cyclic permutation {{iff}}: :$S$ has no more than one orbit under $\rho$ with more than one element.


Class of Cardinals Contains Minimal Infinite Successor Set

Let $\mathcal N$ denote the class of all cardinal numbers. Then: :$\omega \subseteq \mathcal N$ Where $\omega$ denotes the minimal infinite successor set.


Nonlimit Ordinal Cofinal to One

Let $x$ be a nonlimit non-empty ordinal. Let $\operatorname{cof}$ denote the cofinal relation.  Let $1$ denote the ordinal one. Then: :$\operatorname{cof} \left({x, 1}\right)$


GCD of Polynomials does not depend on Base Field

Let $E / F$ be a field extension. Let $P, Q \in F \sqbrk X$ be polynomials. Let: :$\gcd \set {P, Q} = R$ in $F \sqbrk X$ :$\gcd \set {P, Q} = S$ in $E \sqbrk X$. Then $R = S$. In particular, $S \in F \sqbrk X$.


Sequences of 4 Consecutive Integers with Falling Sigma

The following ordered quadruple of consecutive integers have sigma values which are strictly decreasing: :$44, 45, 46, 47$ :$104, 105, 106, 107$


Pairs of Consecutive Integers with 6 Divisors

The following sequence of integers are those $n$  which fulfil the equation: :$\tau \left({n}\right) = \tau \left({n + 1}\right) = 6$ where $\tau \left({n}\right)$ denotes the $\tau$ function. That is, they are the first of pairs of consecutive integers which each have $6$ divisors: :$44, 75, 98, 116, 147, 171, 242, 243, 244, 332, \ldots$ {{OEIS|A049103}}


Intersection of Subsemigroups/General Result

Let $\mathbb S$ be a set of subsemigroups of $\left({S, \circ}\right)$, where $\mathbb S \ne \varnothing$. Then the intersection $\bigcap \mathbb S$ of the members of $\mathbb S$ is itself a subsemigroup of $\left({S, \circ}\right)$. Also, $\bigcap \mathbb S$ is the largest subsemigroup of $\left({S, \circ}\right)$ contained in each member of $\mathbb S$.


Abelian Group of Order Twice Odd has Exactly One Order 2 Element

Let $G$ be an abelian group whose identity element is $e$. Let the order of $G$ be $2 n$ such that $n$ is odd. Then there exists exactly one $g \in G$ with $g \ne e$ such that $g = g^{-1}$.


Numbers which Multiplied by 2 are the Reverse of when Added to 2

{{begin-eqn}} {{eqn | l = 47 + 2       | r = 49 }} {{eqn | l = 47 \times 2       | r = 94 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 497 + 2       | r = 499 }} {{eqn | l = 497 \times 2       | r = 994 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 4997 + 2       | r = 4999 }} {{eqn | l = 4997 \times 2       | r = 9994 }} {{end-eqn}} ... and so on:


Third Isomorphism Theorem/Groups/Corollary 1

Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $q: G \to \dfrac G N$ be the quotient epimorphism from $G$ to the quotient group $\dfrac G N$. Let $K$ be the kernel of $q$. Then: :$\dfrac G N \cong \dfrac {G / K} {N / K}$


Morphisms-Only Metacategory Induces Metacategory

Let $\mathbf C$ be a morphisms-only metacategory. Then $\mathbf C$ induces a metacategory $\mathbf C'$, as follows (phrased to fit with Characterization of Metacategory via Equations): :Define $\mathbf C'_1$ to be the collection $\mathbf C_1$ of morphisms of $\mathbf C$. :Define $\mathbf C'_0$ to be the image of the operation $\operatorname{dom}$ on $\mathbf C_1$. :Define $\operatorname{id}$ to be the identity on $\mathbf C'_0$, and take $\operatorname{dom}$ and $\operatorname{cod}$ as in $\mathbf C$. :Define $\circ$ to be as in $\mathbf C$; i.e., $g \circ f$ is the unique element of $\mathbf C_1$ with $R_\circ \left({g, f, g \circ f}\right)$.


Smallest Pair of Quasiamicable Numbers

The smallest pair of quasiamicable numbers is $48$ and $75$.


Equivalence of Definitions of Quasiamicable Numbers

Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers. {{TFAE|def = Quasiamicable Numbers}}


Duality Principle (Category Theory)/Formal Duality

=== Morphisms-Only Category Theory === Let $\Sigma$ be a statement in the language of category theory. Suppose $\Sigma$ is provable from the axioms for morphisms-only category theory $\mathrm{MOCT}$: :$\mathrm{MOCT} \vdash \Sigma$ Then the dual statement $\Sigma^*$ is also provable from these axioms, i.e.: :$\mathrm{MOCT} \vdash \Sigma^*$ === Object Category Theory === Let $\mathrm{CT}$ be the collection of seven axioms on Characterization of Metacategory via Equations. Suppose a statement $\Sigma$ about metacategories follows from the axioms $\mathrm{CT}$. Then so does its dual statement $\Sigma^*$.


Logarithm of Divergent Product of Real Numbers/Zero

The following are equivalent: * The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$. * The series $\displaystyle \sum_{n \mathop = 1}^\infty \log a_n$ diverges to $-\infty$.


Disjoint Union is Coproduct in Category of Sets

Let $\mathbf{Set}$ be the category of sets. Let $S$ and $T$ be sets. Then their disjoint union $S \sqcup T$ is a coproduct in $\mathbf{Set}$.


Continuous Implies Locally Bounded

Let $X$ be a topological space. Let $M$ be a metric space. Let $f: X \to M$ be continuous. Then $f$ is locally bounded.


Image of Canonical Injection is Normal Subgroup

Let $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ be groups with identity elements $e_1$ and $e_2$ respectively. Let $\struct {G_1 \times G_2, \circ}$ be the group direct product of $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$. Let: :$\inj_1: \struct {G_1, \circ_1} \to \struct {G_1 \times G_2, \circ}$ be the canonical injection from $\struct {G_1, \circ_1}$ to $\struct {G_1 \times G_2, \circ}$ :$\inj_2: \struct {G_2, \circ_2} \to \struct {G_1 \times G_2, \circ}$ be the canonical injection from $\struct {G_2, \circ_2}$ to $\struct {G_1 \times G_2, \circ}$. Then: :$(1): \quad \Img {\inj_1} \lhd \struct {G_1 \times G_2, \circ}$ :$(2): \quad \Img {\inj_2} \lhd \struct {G_1 \times G_2, \circ}$ That is, the images of the canonical injections are normal subgroups of the group direct product of $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$.


Period of Reciprocal of 53 is of Quarter Maximal Length

The decimal expansion of the reciprocal of $53$ has $\dfrac 1 4$ the maximum period, that is: $13$: :$\dfrac 1 {53} = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3$ {{OEIS|A007450}}


Pointwise Addition on Real-Valued Functions is Associative

Let $f, g, h: S \to \R$ be real-valued functions. Let $f + g: S \to \R$ denote the pointwise sum of $f$ and $g$. Then: :$\paren {f + g} + h = f + \paren {g + h}$


Pointwise Addition on Integer-Valued Functions is Associative

Let $f, g, h: S \to \Z$ be integer-valued functions. Let $f + g: S \to \Z$ denote the pointwise sum of $f$ and $g$. Then: :$\paren {f + g} + h = f + \paren {g + h}$


Pointwise Addition on Rational-Valued Functions is Associative

Let $f, g, h: S \to \Q$ be rational-valued functions. Let $f + g: S \to \Q$ denote the pointwise sum of $f$ and $g$. Then: :$\paren {f + g} + h = f + \paren {g + h}$


Dirichlet Convolution Preserves Multiplicativity

Let $f, g: \N \to \C$ be multiplicative arithmetic functions. Then their Dirichlet convolution $f * g$ is again multiplicative.


Closed Form for Pentatope Numbers

The closed-form expression for the $n$th pentatope number is: :$P_n = \dfrac {n \paren {n + 1} \paren {n + 2} \paren {n + 3} } {24}$


Necessary and Sufficient Condition for First Order System to be Field for Second Order System

Let $\mathbf y$, $\mathbf f$, $\boldsymbol \psi$ be N-dimensional vectors. Let $\boldsymbol\psi$ be continuously differentiable. Then $\forall x \in \closedint a b$ the first-order system of differential equations: :$\mathbf y' = \map {\boldsymbol \psi} {x, \mathbf y}$ is a field for the second-order system :$\mathbf y'' = \map {\mathbf f} {x, \mathbf y, \mathbf y'}$ {{iff}} $\boldsymbol \psi$ satisfies : :$\displaystyle \frac {\partial \boldsymbol \psi} {\partial x} + \sum_{i \mathop = 1}^N \frac {\partial \boldsymbol \psi} {\partial y_i} \psi_i = \map {\mathbf f} {x, \mathbf y, \boldsymbol \psi}$ That is, every solution to Hamilton-Jacobi system is a field for the original system.


Tetrahedral Number as Sum of Squares

:$H_n = \displaystyle \sum_{k \mathop = 0}^{n / 2} \paren {n - 2 k}^2$ where $H_n$ denotes the $n$th tetrahedral number.


Smooth Homotopy is an Equivalence Relation

Let $X$ and $Y$ be smooth manifolds. Let $K \subseteq X$ be a (possibly empty) subset of $X$. Let $\mathcal C^\infty \left({X, Y}\right)$ be the set of all smooth mappings from $X$ to $Y$. Define a relation $\sim$ on $\mathcal C \left({X, Y}\right)$ by $f \sim g$ if $f$ and $g$ are smoothly homotopic relative to $K$. Then $\sim$ is an equivalence relation.


Positive Even Integers as Sum of 2 Composite Odd Integers in 2 Ways

Let $n \in \Z_{>0}$ be a positive even integer. Let $n$ be such that it cannot be expressed as the sum of $2$ odd positive composite integers in at least $2$ different ways. Then $n$ belongs to the set: :$\left\{ {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 38, 40, 44, 46, 52, 56, 62, 68}\right\}$ {{OEIS|A284788}}


Continuous Image of Path-Connected Set is Path-Connected

Let $M_1, M_2$ be metric spaces whose metrics are $d_1, d_2$ respectively. Let $f: M_1 \to M_2$ be a continuous mapping. Let $S \subseteq M_1$ be a path-connected subspace of $M_1$. Then $f \sqbrk S$ is a a path-connected subspace of $M_2$.


Quotient Mapping is Coequalizer

Let $\mathbf{Set}$ be the category of sets. Let $S$ be a Set, and let $\mathcal R \subseteq S \times S$ be an equivalence relation on $S$. Let $r_1, r_2: \mathcal R \to S$ be the projections corresponding to the inclusion mapping $\mathcal R \hookrightarrow S \times S$. Let $q: S \to S / \mathcal R$ be the quotient mapping induced by $\mathcal R$. Then $q$ is a coequalizer of $r_1$ and $r_2$ in $\mathbf{Set}$.


Reciprocal of 81

The decimal expansion of the reciprocal of $81$ has a particularly interesting pattern: :$\dfrac 1 {81} = 0 \cdotp \dot 01234 \, 567 \dot 9$


Open Real Interval is not Closed Set/Corollary

Let: :$I_a = \openint \gets a$ :$I_b = \openint b \to$ be unbounded open real intervals. Then neither $I_a$ nor $I_b$ are closed sets of $\R$.


Necessary and Sufficient Condition for First Order System to be Mutually Consistent

Let $\mathbf y$, $\boldsymbol \psi$ be N-dimensional vectors. Let $g$ be a twice differentiable mapping. Let :$(1): \quad \map {\boldsymbol \psi} {x, \mathbf y} = \map {\mathbf y'} {x, \mathbf y}$ :$(2): \quad \mathbf p \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } = \map {g_{\mathbf y} } {x, \mathbf y}$ where $\mathbf p$ is a momentum. Then the boundary conditions defined by $(1)$ are mutually consistent {{iff}} the mapping $\map g {x, \mathbf y}$ satisfies the Hamilton-Jacobi equation: :$(3): \quad \dfrac {\partial g} {\partial x} + \map H {x, \mathbf y, \dfrac {\partial g} {\partial \mathbf y} } = 0$


Reciprocal of 89

The decimal expansion of the reciprocal of $89$ contains the Fibonacci sequence: :$\dfrac 1 {89} = 0 \cdotp \dot 01123 \, 59550 \, 56179 \, 77528 \, 08988 \, 76404 \, 49438 \, 20224 \, 719 \dot 1$


Reciprocal as Sum of Fibonacci Numbers by Negative Powers of 10

:$\displaystyle \sum_{k \mathop \ge 0} \dfrac {F_k} {10^{k + 1} } = \dfrac 1 {89}$ where $F_k$ is the $k$th Fibonacci number: :$F_0 = 0, F_1 = 1, F_k = F_{k - 1} + F_{k - 2}$ That is: <pre> 1 / 89 = 0.0        + 0.01        + 0.001        + 0.0002        + 0.00003        + 0.000005        + 0.0000008        + 0.00000013        + 0.000000021        + 0.0000000034        + 0.00000000055        + .............. </pre>


Continuous Function on Compact Space is Uniformly Continuous

Let $\R^n$ be the $n$-dimensional Euclidean space. Let $S \subseteq \R^n$ be a compact subspace of $\R^n$. Let $f: S \to \R$ be a continuous function. Then $f$ is uniformly continuous in $\R$.


91 is Pseudoprime to 35 Bases less than 91

$91$ is a Fermat pseudoprime in $35$ bases less than itself: :$3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 27, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 64, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88, 90$


Reciprocal of 97

The decimal expansion of the reciprocal of $97$ has the maximum period, that is: $96$: :$\dfrac 1 {97} = 0 \cdotp \dot 01030 \, 92783 \, 50515 \, 46391 \, 75257 \, 73195 \, 87628 \, 86597 \, 93814 \, 43298 \, 96907 \, 21649 \, 48453 \, 60824 \, 74226 \, 80412 \, 37113 \, 40206 \, 18556 \, \dot 7$ {{OEIS|A007450}}


Reciprocal of 98

The decimal expansion of the reciprocal of $98$ starts with the powers of $2$: :$\dfrac 1 {98} = 0 \cdotp 0 \dot 1020 \, 40816 \, 32653 \, 06122 \, 44897 \, 95918 \, 36734 \, 69387 \, 75 \dot 5$ {{OEIS|A021102}}


Reciprocal of 103

:$\dfrac 1 {103} = 0 \cdotp \dot 00970 \, 87378 \, 64077 \, 66990 \, 29126 \, 21359 \, 223 \dot 3$


Infinite Set in Compact Space has Omega-Accumulation Point

Let $\struct {X, \tau}$ be a compact topological space. Let $A \subseteq X$ be infinite. Then $A$ has an $\omega$-accumulation point in $X$.


Cube of 11 is Palindromic

::$11^3 = 1331$


Fourth Power of 11 is Palindromic

::$11^4 = 14641$


Local Membership of Equalizer

Let $\mathbf C$ be a metacategory. Let $e: E \to C$ be the equalizer of $f,g : C \to D$. Then a variable element $z: Z \to C$ is a local member of $e$ iff $f \circ z = g \circ z$: :$z \in_C e \iff f \circ z = g \circ z$


Topological Equivalence is Equivalence Relation

Let $A$ be a set. Let $\mathcal D$ be the set of all metrics on $A$. Let $\sim$ be the relation on $\mathcal D$ defined as: :$\forall d_1, d_2 \in \mathcal D: d_1 \sim d_2 \iff d_1$ is topologically equivalent to $d_2$ Then $\sim$ is an equivalence relation.


Prime Gaps of 10

The following pairs of consecutive prime numbers are those whose difference is $10$: :$\tuple {139, 149}, \tuple {181, 191}, \tuple {241, 251}, \tuple {283, 293}, \ldots$ {{OEIS|A031928|order = lower}}


Euclidean Plane is Abstract Geometry

The Euclidean plane $\left({\R^2, L_E}\right)$ is an abstract geometry.


Factorions Base 10

The following positive integers are the only factorions base $10$: :$1, 2, 145, 40 \, 585$


Strong Separation Theorem

Let $C \subset \R^\ell$ be closed and convex. Let $D = \set {\mathbf v} \subset C^c$. Then $C$ and $D$ can be strongly separated.


Pairs of Integers whose Product with Tau Value are Equal

Let $\tau \left({n}\right)$ denote the $\tau$ function: the number of divisors of $n$. The following pairs of integers $T$ have the property that $m \tau \left({m}\right)$ is equal for each $m \in T$: :$\left\{ {18, 27}\right\}$ :$\left\{ {24, 32}\right\}$ :$\left\{ {56, 64}\right\}$


Numbers that cannot be made Prime by changing 1 Digit

The following positive integers cannot be made into prime numbers by changing just one digit: :$200, 202, 204, 205, 206, 208, \ldots$ {{OEIS|A192545}}


Triangular Numbers which are also Pentagonal

The sequence of triangular numbers which are also pentagonal begins: :$1, 210, 40 \, 755, 7 \, 906 \, 276, 1 \, 533 \, 776 \, 805, 297 \, 544 \, 793 \, 910, \ldots$ {{OEIS|A014979}}


For Complete Ritz Sequence Continuous Functional approaches its Minimal Value

Let $J$ be a continuous functional. Let $\sequence {\phi_n}$ be a complete Ritz sequence. {{explain|The concept of $\sequence {\phi_n}$ does not appear to be related in any way to the statement of the theorem.}} Then: :$\displaystyle \lim_{n \mathop \to \infty} \mu_n = \mu$ where $\displaystyle \mu = \inf_y J \sqbrk y$.


17 Wallpaper Groups

There are $17$ wallpaper groups.


Closure of Infinite Union may not equal Union of Closures

Let $T$ be a topological space. Let $I$ be an infinite indexing set. Let $\family {H_i}_{i \mathop \in I}$ be an indexed family of subsets of a set $S$. Let $\displaystyle H = \bigcup_{i \mathop \in I} H_i$ be the union of $\family {H_i}_{i \mathop \in I}$. Then it is not always the case that: :$\displaystyle \bigcup_{i \mathop \in I} \map \cl {H_i} = \map \cl {\bigcup_{i \mathop \in I} H_i}$ where $\map \cl {H_i}$ denotes the closure of $H_i$.


Set of Reciprocals of Positive Integers is Nowhere Dense in Reals

Let $N$ be the set defined as: :$N := \set {\dfrac 1 n: n \in \Z_{>0} }$ where $\Z_{>0}$ is the set of (strictly) positive integers. Let $\R$ denote the real number line with the usual (Euclidean) metric. Then $N$ is nowhere dense in $\R$.


Numbers in Even-Even Amicable Pair are not Divisible by 3

Let $\tuple {m_1, m_2}$ be an amicable pair such that both $m_1$ and $m_2$ are even. Then neither $m_1$ nor $m_2$ is divisible by $3$.


Construction of Regular 257-Gon

It is possible to construct a regular polygon with $257$ sides) using a compass and straightedge construction.


Young's Inequality for Increasing Functions/Equality

Let $a_0$ and $b_0$ be strictly positive real numbers. Let $f: \closedint 0 {a_0} \to \closedint 0 {b_0}$ be a strictly increasing bijection. Let $a$ and $b$ be real numbers such that $0 \le a \le a_0$ and $0 \le b \le b_0$. Then $b = \map f a$ {{iff}}: :$\displaystyle a b = \int_0^a \map f u \rd u + \int_0^b \map {f^{-1} } v \rd v$ where $\displaystyle \int$ denotes the Darboux integral.


Filters of Lattice of Power Set form Bounded Below Ordered Set

Let $X$ be a set. Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be an inclusion lattice of power set of $X$. Let $F = \left({\mathit{Filt}\left({L}\right), \subseteq}\right)$ be an inclusion ordered set, where $\mathit{Filt}\left({L}\right)$ denotes the set of all filters on $L$. Then $F$ is bounded below and $\bot_F = \left\{{X}\right\}$ where $\bot_F$ denotes the smallest element of $F$.


Deterministic Time Hierarchy Theorem

Let $\map f n$ be a time-constructible function. Then there exists a decision problem which: :can be solved in worst-case deterministic time $\map f {2 n + 1}^3$ but: :cannot be solved in worst-case deterministic time $\map f n$. In other words, the complexity class $\map {\mathsf {DTIME} } {\map f n} \subsetneq \map {\mathsf {DTIME} } {\map f {2 n + 1}^3}$.


Points in Product Spaces are Near Open Sets

Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of topological spaces, where $I$ is an arbitrary index set. Let $X = \displaystyle \prod_{i \mathop \in I} X_i$ be the product space of $\family {X_i}_{i \mathop \in I}$. Let $U$ be nonempty open subset of $X$. Let $x$ be a point in $X$. For each point $y$ in $X$, let $\map K y = \set {i \in I : y_i \ne x_i}$. Then there exists a point $u$ in $U$ such that $\map K u$ is finite.


Equivalence of Definitions of Limit of Function in Metric Space

{{TFAE|def = Limit of Function (Metric Space)|view = Limit of Function|context = Metric Space|contextview = Metric Spaces}} Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $c$ be a limit point of $M_1$. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ ''except possibly'' at $c$. Let $L \in M_2$.


Equivalence of Definitions of Field of Quotients

Let $D$ be an integral domain. Let $F$ be a field. {{TFAE|def = Field of Quotients}}


Products of 2-Digit Pairs which Reversed reveal Same Product

The following positive integers can be expressed as the product of $2$ two-digit numbers in $2$ ways such that the factors in one of those pairs is the reversal of each of the factors in the other: :$504, 756, 806, 1008, 1148, 1209, 1472, 1512, 2016, 2208, 2418, 2924, 3024, 4416$ <!-- fascists won't include the damn thing {{OEIS|A289978}}  SIEG HEIL! SIEG HEIL!-->


Poulet Numbers which are also Magic Constant for Magic Square

The sequence of Poulet numbers which are also the magic constant of a magic square begins: :$1105, 2465, \ldots$


Length of Contour is Well-Defined

Let $C_1, \ldots, C_n$ be directed smooth curves. Let $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$. Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$. Suppose that $\sigma_i: \closedint {c_i} {d_i} \to \C$ is a reparameterization of $C_i$ for all $i \in \set {1, \ldots, n}$ Then: :$\displaystyle \sum_{i \mathop = 1}^n \int_{a_i }^{b_i} \size {\map {\gamma_i'} t} \rd t = \sum_{i \mathop = 1}^n \int_{c_i}^{d_i} \size {\map {\sigma_i'} t} \rd t$ and all real integrals in the equation are defined.


Equivalence of Definitions of Change of Basis Matrix

Let $R$ be a ring with unity. Let $G$ be a finite-dimensional unitary $R$-module. Let $A = \sequence {a_n}$ and $B = \sequence {b_n}$ be ordered bases of $G$. {{TFAE|def = Change of Basis Matrix}}


Numbers not Expressible as Sum of Fewer than 19 Fourth Powers

The following positive integer are the only ones which cannot be expressed as the sum of fewer than $19$ fourth powers: :$79, 159, 239, 319, 399, 479, 559$ {{OEIS|A046050}}


Free Module on Set is Free

Let $R$ be a ring with unity. Let $I$ be a set. Let $R^{\paren I}$ be the free $R$-module on $I$. Then $R^{\paren I}$ is a free $R$-module.


Pandigital Square Equation

The following equations, which include each digit from $1$ to $9$ inclusive, are the only ones of their kind: {{begin-eqn}} {{eqn | l = 567^2       | r = 321 \, 489 }} {{eqn | l = 854^2       | r = 729 \, 316 }} {{end-eqn}}


Sequence of 11 Primes by Trebling and Adding 16

The process of multiplication by $3$ and then adding $16$ produces a sequence of $11$ primes when starting from $587$: :$587, 1777, 5347, 16 \, 057, 48 \, 187, 144 \, 577, 433 \, 747, 1 \, 301 \, 257, 3 \, 903 \, 787, 11 \, 711 \, 377, 35 \, 134 \, 147$


Consecutive Sophie Germain Primes cannot be Pair of Twin Primes

Let $p$ and $p + 2$ be twin primes. Then unless $p = 3$ it is not possible for both $p$ and $p + 2$ to be Sophie Germain primes.


Ring Without Unity may have Quotient Ring with Unity

Let $\struct {R, +, \circ}$ be a ring. Let $I$ be an ideal of $R$. Let $\struct {R / I, +, \circ}$ be the associated quotient ring. Then $\struct {R / I, +, \circ}$ may have a unity even if $\struct {R, +, \circ}$ has not.


Stone Space is Topological Space

Let $\struct {B, \preceq, \wedge, \vee}$ be a non-empty Boolean algebra. Let $\struct {U, \tau}$ be the Stone space of $B$. Then $\struct {U, \tau}$ is a topological space.


Sum of 714 and 715

The sum of $714$ and $715$ is a $4$-digit integer which has $6$ anagrams which are prime.


Limits of Real and Imaginary Parts

Let $f: D \to \C$ be a complex function, where $D \subseteq \C$. Let $z_o \in D$ be a complex number. Suppose $f$ is continuous at $z_0$. Then: :$(1): \quad \displaystyle \lim_{z \to z_o} \operatorname{Re} \left({f \left({z}\right) }\right) = \operatorname{Re} \left({ \lim_{z \to z_o} f \left({z}\right) }\right)$ :$(2): \quad \displaystyle \lim_{z \to z_o} \operatorname{Im} \left({f \left({z}\right) }\right) = \operatorname{Im} \left({ \lim_{z \to z_o} f \left({z}\right) }\right)$ Here, $\operatorname{Re} \left({f \left({z}\right) }\right) $ denotes the real part of $f \left({z}\right)$, and $\operatorname{Im} \left({f \left({z}\right) }\right) $ denotes the imaginary part of $f \left({z}\right)$.


Equivalence of Definitions of Synthetic Basis

Let $S$ be a set. {{TFAE|def = Synthetic Basis}}


Continuous iff Mapping at Element is Supremum of Compact Elements

Let $L = \left({S, \preceq_1, \tau_1}\right)$ and $R = \left({T, \preceq_2, \tau_2}\right)$ be complete algebraic topological lattices with Scott topologies. Let $f: S \to T$ be a mapping. Then $f$ is continuous {{iff}} :$\forall x \in S: f \left({x}\right) = \sup \left\{ {f \left({w}\right): w \in S \land w \preceq_1 x \land w}\right.$ is compact$\left.{}\right\}$


Sum of Pandigital Triplet of 3-Digit Primes

The smallest integer which is the sum of a set of $3$ three-digit primes using all $9$ digits from $1$ to $9$ once each is $999$: :$149 + 263 + 587 = 999$


Reciprocal of 1089

:$\dfrac 1 {1089} = 0 \cdotp \dot 00091 \, 82736 \, 45546 \, 37281 \, 9 \dot 1$


Square Numbers which are Sum of Sequence of Odd Cubes

The sequence of square numbers which can be expressed as the sum of a sequence of odd cubes from $1$ begins: :$1, 1225, 1 \, 413 \, 721, 1 \, 631 \, 432 \, 881, \dotsc$ {{OEIS|A046177}} The sequence of square roots of this sequence is: :$1, 35, 1189, 40 \, 391, \dotsc$ {{OEIS|A046176}}


Smallest Triplet of Consecutive Integers Divisible by Cube

The smallest sequence of triplets of consecutive integers each of which is divisible by a cube greater than $1$ is: :$\tuple {1375, 1376, 1377}$


Sequence of Composite Mersenne Numbers

The sequence of Mersenne numbers which are composite begins: :$2047, 8 \, 388 \, 607, 536 \, 870 \, 911, 137 \, 438 \, 953 \, 471, 2 \, 199 \, 023 \, 255 \, 551,\ldots$ {{OEIS|A065341}} The sequence of corresponding indices $p$ such that $2^p - 1$ is composite begins: :$11, 23, 29, 37, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, \ldots$ {{OEIS|A054723}} The sequence of corresponding integers $n$ such that the $n$th prime number $p \left({n}\right)$ is such that $2^{p \left({n}\right)} - 1$ is composite begins: :$5, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, \ldots$ {{OEIS|A135980}}


Numbers Reversed when Multiplying by 4

Numbers of the form $\sqbrk {21 \paren 9 78}_{10}$ are reversed when they are multiplied by $4$: {{begin-eqn}} {{eqn | l = 2178 \times 4       | r = 8712 }} {{eqn | l = 21 \, 978 \times 4       | r = 87 \, 912 }} {{eqn | l = 219 \, 978 \times 4       | r = 879 \, 912 }} {{end-eqn}} and so on.


Relational Structure admits Lower Topology

Let $R = \left({S, \preceq}\right)$ be a relational structure. Then there exists a relational structure with lower topology $T = \left({S, \preceq, \tau}\right)$ such that $T$ is a topological space.


2520 equals Sum of 4 Divisors in 6 Ways

The number $2520$ can be expressed as the sum of $4$ of its divisors in $6$ different ways: {{begin-eqn}} {{eqn | l = 2520       | r = 1260 + 630 + 504 + 126 }} {{eqn | r = 1260 + 630 + 421 + 210 }} {{eqn | r = 1260 + 840 + 360 + 60 }} {{eqn | r = 1260 + 840 + 315 + 105 }} {{eqn | r = 1260 + 840 + 280 + 140 }} {{eqn | r = 1260 + 840 + 252 + 168 }} {{end-eqn}} This is the maximum possible number of ways it is possible to express an integer as the sum of $4$ of its divisors.


Operating on Ordered Group Inequalities

If $x \prec y$ and $z \prec w$, then $x \circ z \prec y \circ w$. If $x \prec y$ and $z \preceq w$, then $x \circ z \prec y \circ w$. If $x \preceq y$ and $z \prec w$, then $x \circ z \prec y \circ w$. If $x \preceq y$ and $z \preceq w$, then $x \circ z \preceq y \circ w$.


Reflexive Closure of Antisymmetric Relation is Antisymmetric

Let $S$ be a set. Let $\mathcal R$ be an antisymmetric relation on $S$. Let $\mathcal R^=$ be the reflexive closure of $\mathcal R$. Then $\mathcal R^=$ is also antisymmetric.


Conditional is not Left Self-Distributive/Formulation 2

While this holds: :$\vdash \paren {\paren {p \implies q} \implies r} \implies \paren {\paren {p \implies r} \implies \paren {q \implies r} }$ its converse does not: :$\not \vdash \paren {\paren {p \implies r} \implies \paren {q \implies r} } \implies \paren {\paren {p \implies q} \implies r}$


Self-Distributive Law for Conditional/Formulation 2

:$\vdash \paren {p \implies \paren {q \implies r} } \iff \paren {\paren {p \implies q} \implies \paren {p \implies r} }$


Rule of Exportation/Formulation 2

:$\vdash \paren {\paren {p \land q} \implies r} \iff \paren {p \implies \paren {q \implies r} }$


Cube of 20 is Sum of Sequence of 4 Consecutive Cubes

:$20^3 = \displaystyle \sum_{k \mathop = 11}^{14} k^3$ That is: :$20^3 = 11^3 + 12^3 + 13^3 + 14^3$


Largest Integer not Sum of Two Abundant Numbers

The largest integer which is not the sum of $2$ abundant numbers is $20 \, 161$.


Smallest Integer using Three Words in English Description

The smallest integer which uses exactly $3$ words in its standard (British) English description is: :$21 \, 000$: '''twenty-one thousand''' counting hyphenations as separate words.


Carmichael Number with 4 Prime Factors

$41 \, 041$ is the smallest Carmichael number with $4$ prime factors: :$41 \, 041 = 7 \times 11 \times 13 \times 41$


Tableau Extension Lemma/General Statement/Proof 1

Let $T$ be a finite propositional tableau. Let its hypothesis set $\mathbf H$ be finite. {{:Tableau Extension Lemma/General Statement}}


Reciprocal of 142,857

:$\dfrac 1 {142 \, 857} = 0 \cdotp \dot 00000 \, \dot 7$


Numbers whose Fourth Root equals Number of Divisors

There are $4$ positive integers whose $4$th root equals the number of its divisors: {{begin-eqn}} {{eqn | l = 1       | r = 1^4       | c =  }} {{eqn | l = 625       | r = 5^4       | c =  }} {{eqn | l = 6561       | r = 9^4       | c =  }} {{eqn | l = 4 \, 100 \, 625       | r = 45^4       | c =  }} {{end-eqn}} {{OEIS|A143026}}


Smallest 5 Consecutive Primes in Arithmetic Sequence

The smallest $5$ consecutive primes in arithmetic sequence are: :$9 \, 843 \, 019 + 30 n$ for $n = 0, 1, 2, 3, 4$. Note that while there are many longer arithmetic sequences of far smaller primes, those primes are not consecutive.


Smallest Triplet of Primitive Pythagorean Triangles with Same Area

The smallest set of $3$ primitive Pythagorean triangles which all have the same area are: :the $4485-5852-7373$ triangle :the $3059-8580-9109$ triangle :the $1380-19 \, 019-19 \, 069$ triangle. That area is $13 \, 123 \, 110$.


Biconditional is Transitive/Formulation 1/Proof 2

:$p \iff q, q \iff r \vdash p \iff r$


Law of Identity/Formulation 2/Proof 2

: $\vdash p \implies p$


Exchange of Order of Indexed Summations/Rectangular Domain

Let $D = \closedint a b \times \closedint c d$ be the cartesian product. Let $f: D \to \mathbb A$ be a mapping Then we have an equality of indexed summations: :$\displaystyle \sum_{i \mathop = a}^b \sum_{j \mathop = c}^d \map f {i, j} = \sum_{j \mathop = c}^d \sum_{i \mathop = a}^b \map f {i, j}$


Sum over Union of Finite Sets

Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S$ and $T$ be finite sets. Let $f: S \cup T \to \mathbb A$ be a mapping. Then we have the equality of summations over finite sets: :$\displaystyle \sum_{u \mathop \in S \mathop \cup T} \map f u = \sum_{s \mathop \in S} \map f s + \sum_{t \mathop \in T} \map f t - \sum_{v \mathop \in S \mathop \cap T} \map f v$


Rule of Idempotence/Conjunction/Formulation 1/Proof

: $p \dashv \vdash p \land p$


Exchange of Order of Summations over Finite Sets/Subset of Cartesian Product

Let $D\subset S \times T$ be a subset. Let $\pi_1 : D \to S$ and $\pi_2 : D \to T$ be the restrictions of the projections of $S\times T$. Then we have an equality of summations over finite sets: :$\displaystyle \sum_{s \mathop \in S} \sum_{t \mathop \in \pi_2 \left({\pi_1^{-1} \left({s}\right)}\right)} f \left({s, t}\right) = \sum_{t \mathop \in T} \sum_{s \mathop \in \pi_1 \left({\pi_2^{-1} \left({t}\right)}\right)} f \left({s, t}\right)$


Canonical Homomorphism to Polynomial Ring is Ring Monomorphism

Let $R$ be a commutative ring with unity. Let $(R[X], \iota, X)$ be a polynomial ring over $R$ in one indeterminate $X$. Then the canonical homomorphism $\iota : R \to R[X]$ is a ring monomorphism.


Equivalence of Definitions of Consistent Proof System

{{TFAE|def = Consistent (Logic)/Proof System/Propositional Logic|view = Consistent Proof System for Propositional Logic}} Let $\LL_0$ be the language of propositional logic. Let $\mathscr P$ be a proof system for $\LL_0$.


Irreducible Component is Closed

Let $T = \left({S, \tau}\right)$ be a topological space. Let $Y$ be an irreducible component of $T$. Then $Y$ is closed in $T$.


Homotopy Characterisation of Simply Connected Sets

Let $T = \struct {S, \tau}$ be a topological space. Let $X$ be a subset of $S$. Then $X$ is simply connected {{iff}} the following conditions hold: :$(1): \quad $ $X$ is path-connected. :$(2): \quad $ All paths in $X$ with the same initial points and final points are freely homotopic.


Largest Pandigital Square including Zero

The largest pandigital square (in the sense where pandigital includes the zero) is $9 \, 814 \, 072 \, 356$: :$9 \, 814 \, 072 \, 356 = 99 \, 066^2$


Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 1/Proof

:$p \lor \paren {q \land r} \dashv \vdash \paren {p \lor q} \land \paren {p \lor r}$


Multiply Perfect Number of Order 6

The number defined as: :$n = 2^{36} \times 3^8 \times 5^5 \times 7^7 \times 11 \times 13^2 \times 19 \times 31^2$ ::$\times \ 43 \times 61 \times 83 \times 223 \times 331 \times 379 \times 601 \times 757 \times 1201$ ::$\times \ 7019 \times 112 \, 303 \times 898 \, 423 \times 616 \, 318 \, 177$ is multiply perfect of order $6$.


Rule of Distribution/Disjunction Distributes over Conjunction/Right Distributive/Formulation 1/Proof

:$\left({q \land r}\right) \lor p \dashv \vdash \left({q \lor p}\right) \land \left({r \lor p}\right)$


Functional Equation for Completed Riemann Zeta Function

Let $\xi : \C \to \C$ be the completed Riemann zeta function. Let $s\in \C$ be a complex number. Then: :$\map \xi s = \map \xi {1 - s}$


Largest Integer whose Digits taken in Pairs all form Distinct Primes

The largest integer which has the property that every pair of its digits taken together is a distinct prime number is $619 \, 737 \, 131 \, 179$.


Smallest 22 Primes in Arithmetic Sequence

The smallest $22$ primes in arithmetic sequence are: :$11 \, 410 \, 337 \, 850 \, 553 + 4 \, 609 \, 098 \, 694 \, 200 n$ for $n = 0, 1, \ldots, 21$.


Smallest Cunningham Chain of the Second Kind of Length 13

The smallest Cunningham chain of the second kind of length $13$ is: :$758 \, 083 \, 947 \, 856 \, 951$, $1 \, 516 \, 167 \, 895 \, 713 \, 901$, $3 \, 032 \, 335 \, 791 \, 427 \, 801$, $6 \, 064 \, 671 \, 582 \, 855 \, 601$, $12 \, 129 \, 343 \, 165 \, 711 \, 201$, $24 \, 258 \, 686 \, 331 \, 422 \, 401$, $48 \, 517 \, 372 \, 662 \, 844 \, 801$, $97 \, 034 \, 745 \, 325 \, 689 \, 601$, $194 \, 069 \, 490 \, 651 \, 379 \, 201$, $388 \, 138 \, 981 \, 302 \, 758 \, 401$, $776 \, 277 \, 962 \, 605 \, 516 \, 801$, $1 \, 552 \, 555 \, 925 \, 211 \, 033 \, 601$, $3 \, 105 \, 111 \, 850 \, 422 \, 067 \, 201$


Repunit Expressed using Power of 10

The repunit number $R_n$ can be expressed as: :$R_n = \dfrac {10^n - 1} 9$


Repunit in Base 9 is Triangular

Let $m$ be a repunit base $9$. Then $m$ is a triangular number.


Largest nth Power which has n Digits

The largest $n$th power which has $n$ digits is $9^{21}$: :$9^{21} = 109 \, 418 \, 989 \, 131 \, 512 \, 359 \, 209$


Left-Truncatable Prime/Examples/357,686,312,646,216,567,629,137

The largest left-truncatable prime is $357 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$.


Smallest Integer Divisible by All Numbers from 1 to 100

The smallest positive integer which is divisible by each of the integers from $1$ to $100$ is: :$69 \, 720 \, 375 \, 229 \, 712 \, 477 \, 164 \, 533 \, 808 \, 935 \, 312 \, 303 \, 556 \, 800$


Integer which is Multiplied by 9 when moving Last Digit to First

Let $N$ be the positive integer: :$N = 10 \, 112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 808 \, 988 \, 764 \, 044 \, 943 \, 820 \, 224 \, 719$ $N$ is the smallest positive integer $N$ such that if you move the last digit to the front, the result is the positive integer $9 N$.


Yoneda Embedding Theorem

Let $C$ be a locally small category. Let $\mathbf{Set}$ be the category of sets. Let $[C^{\operatorname{op}}, \mathbf{Set}]$ be the contravariant functor category. Then the Yoneda embedding $h_- : C \to [C^{\operatorname{op}}, \mathbf{Set}]$ is a fully faithful embedding.


Titanic Prime whose Digits are all Odd

The integer defined as: :$1358 \times 10^{3821} - 1$ is a titanic prime all of whose digits are odd. That is: :$1357 \paren 9_{3821}$ where $\paren a_b$ means $b$ instances of $a$ in a string.


Gigantic Palindromic Prime

The integer defined as: :$10^{11 \, 810} + 1 \, 465 \, 641 \times 10^{5902} + 1$ is a gigantic prime which is also palindromic. That is: :$1(0)_{5901}1465641(0)_{5901}1$ where $\left({a}\right)_b$ means $b$ instances of $a$ in a string.


Existence of Topological Space which satisfies no Separation Axioms

There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied.


Relation Compatible with Group Operation is Reflexive or Antireflexive

Let $\struct {G, \circ}$ be a group. Let $\RR$ be a relation on $G$ that is compatible with $\circ$. Then $\RR$ is reflexive or antireflexive.


Existence of Compact Hausdorff Space which is not T5

There exists at least one example of a compact $T_2$ (Hausdorff) space which is not a $T_5$ space.


Existence of Hausdorff Space which is not Completely Hausdorff

There exists at least one example of a topological space which is a $T_2$ (Hausdorff) space, but is not also a completely Hausdorff space.


Existence of Semiregular Topological Space which is not Urysohn Space

There exists at least one example of a semiregular topological space which is not a Urysohn space.


Existence of Compact Space which is not Sequentially Compact

There exists at least one example of a compact topological space which is not also a sequentially compact space.


Norm of Vector Cross Product

Let $\mathbf a$ and $\mathbf b$ be vectors in the Euclidean space $\R^3$. Let $\times$ denote the vector cross product. Then: :$(1): \quad$ $\left\Vert{ \mathbf a \times \mathbf b }\right\Vert^2 = \left\Vert{\mathbf a}\right\Vert^2 \left\Vert{\mathbf b}\right\Vert^2 - \left({\mathbf a \cdot \mathbf b}\right)^2$ :$(2): \quad$ $\left\Vert{ \mathbf a \times \mathbf b }\right\Vert = \left\Vert{\mathbf a}\right\Vert \left\Vert{\mathbf b}\right\Vert \left\vert{\sin \theta}\right\vert$ where $\theta$ is the angle between $\mathbf a$ and $\mathbf b$, or an arbitrary number if $\mathbf a$ or $\mathbf b$ is the zero vector.


Existence of Separable Space which is not Second-Countable

There exists at least one example of a separable topological space which is not also a second-countable space.


Existence of Paracompact Space which is not Compact

There exists at least one example of a paracompact topological space which is not also a compact topological space.