hcju/proofwikips huggingface dataset:
Definition:Convergent Product/Number Field/Nonzero Sequence
Let $\sequence{a_n}$ be a sequence of nonzero elements of $\mathbb K$. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is '''convergent'''  its sequence of partial products converges to a nonzero limit $a\in\mathbb K\setminus\{0\}$.
Definition:Convergent Product/Number Field/Arbitrary Sequence
Let $\sequence{a_n}$ be a sequence of elements of $\mathbb K$. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is '''convergent''' : * There exists $n_0\in\N$ such that the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some $b\in\mathbb K\setminus\{0\}$. The sequence of partial products of $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is then convergent to some $a\in\mathbb K$.
Definition:Convergent Product/Arbitrary Field/Arbitrary Sequence
Let $\sequence {a_n}$ be a sequence of elements of $\mathbb K$. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is '''convergent''' : : There exists $n_0 \in \N$ such that the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some $b \in \mathbb K \setminus \set 0$. The sequence of partial products of $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is then convergent to some $a \in \mathbb K$.
Definition:Convergent Product/Arbitrary Field/Nonzero Sequence
Let $\sequence {a_n}$ be a sequence of nonzero elements of $\mathbb K$. The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is '''convergent'''  its sequence of partial products converges to a nonzero limit $a \in \mathbb K \setminus \set 0$.
Definition:Group of Permutation Matrices
The '''group of permutation matrices''' of order $n$ is the set of all $n \times n$ permutation matrices with multiplication of matrices. {{Disambiguate|Definition:Order}}
Definition:Change of Basis Matrix/Definition 1
The '''matrix of change of basis from $A$ to $B$''' is the matrix whose columns are the coordinate vectors of the elements of the '''new basis''' $\left \langle {b_n} \right \rangle$ relative to the '''original basis''' $\left \langle {a_n} \right \rangle$.
Definition:Change of Basis Matrix/Definition 2
Let $I_G$ be the identity linear operator on $G$. Let $\left[{I_G; \left \langle {a_n} \right \rangle, \left \langle {b_n} \right \rangle}\right]$ be the matrix of $I_G$ relative to $\left \langle {b_n} \right \rangle$ and $\left \langle {a_n} \right \rangle$. Then $\left[{I_G; \left \langle {a_n} \right \rangle, \left \langle {b_n} \right \rangle}\right]$ is called the '''matrix corresponding to the change of basis from $\left \langle {a_n} \right \rangle$ to $\left \langle {b_n} \right \rangle$'''.
Definition:Basis of Module
Let $R$ be a ring with unity. Let $\left({G, +_G, \circ}\right)_R$ be a unitary $R$-module. === Definition 1 === {{:Definition:Basis of Module/Definition 1}} === Definition 2 === {{:Definition:Basis of Module/Definition 2}}
Definition:Basis of Vector Space/Definition 1
A '''basis of $G$''' is a linearly independent subset of $G$ which is a generator for $G$.
Definition:Basis of Vector Space/Definition 2
A '''basis''' is a maximal linearly independent subset of $G$.
Definition:Basis of Vector Space
Let $K$ be a division ring. Let $\struct {G, +_G, \circ}_R$ be a vector space over $K$. === Definition 1 === {{:Definition:Basis of Vector Space/Definition 1}} === Definition 2 === {{:Definition:Basis of Vector Space/Definition 2}}
Definition:Field of Directions
Consider the following system of differential equations: :$(1): \quad \mathbf y' = \map {\mathbf f} {x, \mathbf y, \mathbf y'}$ where $\mathbf y$ is an $n$-dimensional vector. Let the boundary conditions be prescribed $\forall x \in \closedint a b$: :$\mathbf y' = \map {\boldsymbol \psi} {x, \mathbf y}$ Let these boundary conditions be consistent for all $x_1, x_2 \in \closedint a b$. Then the family of mutually consistent boundary conditions is called a '''field of directions''' for the given system $(1)$. That is, the first-order system is valid in an interval instead of a countable set of points.
Definition:Angle/Unit/Minute
The '''minute (of arc)''' is a measurement of plane angles, symbolized by $'$. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''minute''' }} {{eqn | r = 60       | c = seconds }} {{eqn | r = \dfrac 1 {60}       | c = degree of arc (by definition) }} {{end-eqn}}
Definition:Angle/Unit/Second
The '''second (of arc)''' is a measurement of plane angles, symbolized by $''$. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''second''' }} {{eqn | r = \dfrac 1 {60}       | c = minute of arc (by definition) }} {{eqn | r = \dfrac 1 {60 \times 60} = \dfrac 1 {3600}       | c = degree of arc }} {{end-eqn}}
Definition:Second
=== Second of Time === {{:Definition:Second of Time}} === Second of Arc === {{:Definition:Second of Arc}}
Definition:Minute
=== Minute of Time === {{:Definition:Minute of Time}} === Minute of Arc === {{:Definition:Minute of Arc}}
Definition:Splitting Polynomial
Let $K$ be a field. Let $P \in K \sqbrk X$ be a polynomial. === Definition 1 === The polynomial $P$ '''splits (completely)''' in $K$  it is a product of polynomials of degree $1$. === Definition 2 === The polynomial $P$ '''splits (completely)''' in $K$  every irreducible factor of $P$ has degree $1$.
Definition:Self-Adjoint Boundary Conditions
Consider the functional $J \sqbrk {\mathbf y}$, such that: :$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ Let the momenta of $J$ be: :$\mathbf p = \nabla_{\mathbf y'} \map F {x, \mathbf y, \mathbf y'}$ Let the following boundary conditions hold: :$\map {\mathbf y'} a = \bigvalueat {\map {\boldsymbol \psi} {\mathbf y} } {x \mathop = a}$ If: :$\exists \map g {x, \mathbf y}: \bigvalueat {\map {\mathbf p} {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } } {x \mathop = a} = \bigvalueat {\nabla_{\mathbf y'} \map g {x, \mathbf y} } {x \mathop = a}$ then the boundary conditions are called '''self-adjoint'''.
Definition:Unitary Group
Let $n$ be a positive integer. The '''unitary group''' $\operatorname{U}_n$ is the group of all $n\times n$ unitary matrices. === Unitary Group of Hilbert Space === Let $H$ be a Hilbert space. Its '''unitary group''' is the group consisting of its unitary operators.
Definition:Special Unitary Group
Let $n$ be a positive integer. The '''special unitary group''' $\operatorname{SU}_n$ is the group of all $n\times n$ unitary matrices with determinant $1$: :$\operatorname{SU}_n = \operatorname{U}_n \cap \operatorname{SL}_n(\C)$
Definition:Standard Basis Matrix
Let $R$ be a ring with unity. Let $m, n \ge 1$ be positive integers. Let $i, j \in \set {1, \ldots, m} \times \set {1, \ldots, n}$. The $\tuple {i, j}$th '''standard basis matrix''' is the $m \times n$ matrix which is $0$ everywhere except a $1$ at the $\tuple {i, j}$th indices.
Definition:Coordinate Vector
Let $R$ be a ring with unity. Let $M$ be a free $R$-module of dimension $n$. Let $B = \left \langle {b_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ be an ordered basis of $M$. Let $x\in M$. If $\lambda_1,\ldots,\lambda_n\in R$ are such that $x = \displaystyle\sum_{i=1}^n \lambda_i b_i$, then $(\lambda_1,\ldots,\lambda_n)^\intercal \in R^n$ is the '''coordinate vector''' of $x$ with respect to $B$. This can be denoted: $[x]_B$.
Definition:Highly Composite Number
Let $n \in \Z_{>0}$ be a positive integer. Then $n$ is '''highly composite''' : :$\forall m \in \Z_{>0}, m < n: \map \tau m < \map \tau n$ where $\map \tau n$ is the $\tau$ function of $n$.
Definition:Highly Composite Number/Sequence
The sequence of highly composite numbers begins: :$1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, \ldots$
Definition:Internal Direct Sum of Modules
Let $R$ be a ring. Let $M$ be an $R$-module. Let $(M_i)_{i\in I}$ be a family of submodules. === Definition 1 === {{:Definition:Internal Direct Sum of Modules/Definition 1}} === Definition 2 === {{:Definition:Internal Direct Sum of Modules/Definition 2}} === Definition 3 === {{:Definition:Internal Direct Sum of Modules/Definition 3}}
Definition:Seventh Power
A '''seventh power''' is an integer which can be expressed as the $7$th power of an integer.
Definition:Internal Direct Sum of Modules/Definition 1
$M$ is the '''internal direct sum''' of $(M_i)_{i\in I}$  every $m\in M$ can be written uniquely as a sum $\sum m_i$ with each $m_i\in M_i$.
Definition:Internal Direct Sum of Modules/Definition 2
$M$ is the '''internal direct sum''' of $(M_i)_{i\in I}$ : * $\displaystyle\bigcup_{i\in I}M_i$ generates $M$ * For all $i\in I$, $M_i\cap \displaystyle\sum_{j\neq i} M_j = \{0\}$
Definition:Internal Direct Sum of Modules/Definition 3
Let $\displaystyle \bigoplus_{i \mathop \in I} M_i$ be the external direct sum of $\left\langle{M_i}\right\rangle_{i \mathop \in I}$. $M$ is the '''internal direct sum''' of $\left\langle{M_i}\right\rangle_{i \mathop \in I}$  the mapping given by Universal Property of Direct Sum of Modules is an isomorphism onto $M$.
Definition:Basis of Module/Definition 1
A '''basis of $G$''' is a linearly independent subset of $G$ which is a generator for $G$.
Definition:Basis of Module/Definition 2
Let $\mathcal B = \family {b_i}_{i \mathop \in I}$ be a family of elements of $M$. Let $\Psi: R^{\paren I} \to M$ be the homomorphism given by Universal Property of Free Module on Set. Then $\mathcal B$ is a '''basis'''  $\Psi$ is an isomorphism.
Definition:Relative Matrix of Bilinear Form
Let $R$ be a ring with unity. Let $M$ be a free $R$-module of finite dimension $n>0$. Let $\mathcal B = \left \langle {b_m} \right \rangle$ be an ordered basis of $M$. Let $f : M\times M\to R$ be a bilinear form. The '''matrix of $f$ relative to $\mathcal B$''' is the $n\times n$ matrix $\mathbf M_{f, \mathcal B}$ where: :$\displaystyle \forall \left({i, j}\right) \in \left[{1 \,.\,.\, n}\right] \times \left[{1 \,.\,.\, n}\right] : (\mathbf M_{f, \mathcal B})_{ij} = f \left({b_i, b_j}\right)$
Definition:Mutually Consistent Boundary Conditions/wrt Functional
Let $J$ be a (real) functional, such that: :$\displaystyle J = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ where its Euler's equations are: :$\nabla_{\mathbf y'} F - \dfrac \d {\d x} \nabla_{\mathbf y} F = 0$ Consider the following boundary conditions: :$\bigvalueat {\mathbf y} {x \mathop = x_1} = \bigvalueat {\map {\boldsymbol \psi^{\paren 1} } {\mathbf y} } {x \mathop = x_1}$ :$\bigvalueat {\mathbf y} {x \mathop = x_2} = \bigvalueat {\map {\boldsymbol \psi^{\paren 2} } {\mathbf y} } {x \mathop = x_2}$ If they are consistent  the Euler equations, then these boundary conditions are called '''mutually consistent'''  the functional $J$.
Definition:Endomorphism Ring of Module
Let $R$ be a ring. Let $M$ be an $R$-module. The '''endomorphism ring''' $\operatorname{End}_R(M)$ is the ring of all endomorphisms of $M$ where: : Multiplication is composition of mappings : Addition is point-wise
Definition:Module Endomorphism
Let $R$ be a ring. Let $M$ be an $R$-module. An '''module endomorphism''' of $M$ is a module homomorphism from $M$ to itself.
Definition:Associated Quadratic Form
Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$. Let $b : V\times V \to \mathbb K$ be a bilinear form. The '''quadratic form associated to $b$''' is the quadratic form: :$q : V \to \mathbb K : x \mapsto b(x, x)$
Definition:Associated Bilinear Form
Let $\mathbb K$ be a field of characteristic $\operatorname{char} \mathbb K\neq2$. Let $V$ be a vector space over $\mathbb K$. Let $q : V \to \mathbb K$ be a quadratic form. The '''bilinear form associated to $q$''' is the bilinear form: :$b : V \times V \to \mathbb K : (v,w) \mapsto \frac12\left( q(v+w) - q(v) - q(w) \right)$
Definition:Matrix Congruence
Let $R$ be a commutative ring with unity. Let $n$ be a positive integer. Let $\mathbf A$ and $\mathbf B$ be square matrices of order $n$ over $R$. Then $\mathbf A$ and $\mathbf B$ are '''congruent'''  there exists an invertible matrix $\mathbf P\in R^{n\times n}$ such that $\mathbf B = \mathbf P^\intercal \mathbf A \mathbf P$.
Definition:Symmetric Bilinear Form
Let $R$ be a ring Let $M$ be an $R$-module. Let $b: M \times M \to R$ be a bilinear form. Then $b$ is '''symmetric''' : :$\forall v, w, \in M: b \left({v, w}\right) = b \left({w, v}\right)$
Definition:Represented by Quadratic Form
Let $R$ be an integral domain. Let $M$ be an $R$-module. Let $q : M \to R$ be a quadratic form. Let $a\in R$. Then '''$q$ represents $a$ (over $R$)'''  there exists $x\in M\setminus\{0\}$ with $q(x) = a$.
Definition:Isotropic Quadratic Form
Let $\mathbb K$ be a field of characteristic $\operatorname{char}\mathbb K \neq 2$. Let $V$ be a vector space over $\mathbb K$. Let $q : V\times V \mapsto \mathbb K$ be a quadratic form. Then $q$ is '''isotropic'''  it represents $0$. That is: $q(v) = 0$ for some $v\in V\setminus\{0\}$. === Anisotropic Quadratic Form === A quadratic form that is not isotropic is said to be '''anisotropic'''.
Definition:Alternating Bilinear Form
Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$. Let $b$ be a bilinear form on $V$. Then $b$ is '''alternating''' : : $\forall v \in V: b \left({v, v}\right) = 0$
Definition:Reflexive Bilinear Form
Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$. Let $b$ be a bilinear form on $V$. Then $b$ is '''reflexive''' : : $\forall v, w \in V: b \left({v, w}\right) = 0 \implies b \left({w, v}\right) = 0$
Definition:Apéry's Constant
'''Apéry's constant''' is the value of the infinite sum: :$\map \zeta 3 = \displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$ where $\zeta$ denotes the Riemann zeta function. Its approximate value is given by: :$\map \zeta 3 \approx 1 \cdotp 20205 \, 69031 \, 59594 \, 28539 \, 97381 \, 61511 \, 44999 \, 07649 \, 86292 \ldots$
Definition:Discriminant of Bilinear Form
Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$ of finite dimension $n>0$. Let $b : V\times V \to \mathbb K$ be a bilinear form on $V$. Let $A$ be the matrix of $b$ relative to an ordered basis of $V$. If $b$ is nondegenerate, its '''discriminant''' is the equivalence class of the determinant $\det A$ in the quotient group $\mathbb K^\times / (\mathbb K^\times)^2$. If $b$ is degenerate, its '''discriminant''' is $0$.
Definition:Degenerate Bilinear Form
Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$. Let $b : V\times V \to \mathbb K$ be a bilinear form on $V$. Then $b$ is '''degenerate'''  there exists $v\in V\setminus\{0\}$ such that $b(v,u) = 0$ for all $u\in V$. === Nondegenerate Bilinear Form === {{:Definition:Nondegenerate Bilinear Form}}
Definition:Nondegenerate Bilinear Form
A bilinear form on $V$ which is not degenerate is '''nondegenerate'''.
Definition:Quadratic Space
Let $\mathbb K$ be a field. A '''quadratic space''' over $\mathbb K$ is a pair $\left({V, q}\right)$ where: :$V$ is a vector space over $\mathbb K$ of finite dimension $n>0$ :$q$ is a quadratic form on $V$.
Definition:Orthogonal (Bilinear Form)
Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$. Let $b: V \times V \to \mathbb K$ be a reflexive bilinear form on $V$. Let $v,w\in V$. Then $v$ and $w$ are '''orthogonal (with respect to $b$)'''  $b \left({v, w}\right) = b \left({w, v}\right) = 0$ This is denoted: $v \perp w$. === Orthogonal Subsets === {{:Definition:Orthogonal (Bilinear Form)/Subsets}} === Orthogonal Complement === {{:Definition:Orthogonal (Bilinear Form)/Orthogonal Complement}} === Radical === {{:Definition:Orthogonal (Bilinear Form)/Radical}}
Definition:Orthogonal (Bilinear Form)/Radical
The '''radical''' of $V$ is the orthogonal complement of $V$: :$\operatorname{rad}(V) = V^\perp$
Definition:Orthogonal (Bilinear Form)/Orthogonal Complement
Let $S\subset V$ be a subset. The '''orthogonal complement of $S$ (with respect to $b$)''' is the set of all $v \in V$ which are orthogonal to all $s \in S$. This is denoted: $S^\perp$. If $S = \left\{ {v}\right\}$ is a singleton, we also write $v^\perp$.
Definition:Relative Matrix of Quadratic Form
Let $\mathbb K$ be a field of characteristic $\operatorname{char}\mathbb K \neq2$. Let $V$ be a vector space over $\mathbb K$ of finite dimension $n>0$. Let $\mathcal B$ be an ordered basis of $V$. Let $q$ be a quadratic form on $V$. Its '''matrix  relative to $\mathcal B$''' is the matrix of its associated bilinear form relative to $\mathcal B$, denoted $\mathbf M_{q, \mathcal B}$.
Definition:Symplectic Basis
Let $\mathbb K$ be a field. Let $\struct {V, f}$ be a bilinear space over $\mathbb K$ of finite dimension $2 n > 0$. Let $\BB = \tuple {b_1, c_1, \ldots, b_n, c_n}$ be an ordered basis of $V$. Then $\BB$ is '''symplectic''' : :$\map f {b_i, b_j} = \map f {c_i, c_j} = 0$ for all $i, j$ :$\map f {b_i, c_j} = \delta_{i j}$ for all $i, j$ where $\delta$ denotes Kronecker delta. That is,  the matrix of $f$ relative to $\BB$ has the form: :$\begin{pmatrix}  0 & 1 \\ -1 & 0 \\  & &  0 & 1 \\  & & -1 & 0 \\  & & & & \ddots \\  & & & & &  0 & 1 \\   & & & & & -1 & 0 \end{pmatrix}$
Definition:Bilinear Space
Let $\mathbb K$ be a field. A '''bilinear space''' over $\mathbb K$ is a pair $\left({V, f}\right)$ where: :$V$ be a vector space over $\mathbb K$ of finite dimension $n > 0$ :$f$ is a bilinear form on $V$.
Definition:Orthogonal Basis
=== Orthogonal Basis of Vector Space === {{:Definition:Orthogonal Basis of Vector Space}} === Orthogonal Basis of Inner Product Space === {{:Definition:Orthogonal Basis of Inner Product Space}} === Orthogonal Basis of Bilinear Space === {{:Definition:Orthogonal Basis of Bilinear Space}}
Definition:Orthogonal Basis/Bilinear Space
Let $\mathbb K$ be a field. Let $\struct {V, f}$ be a bilinear space over $\mathbb K$ of finite dimension $n > 0$. Let $\BB = \tuple {b_1, \ldots, b_n}$ be an ordered basis of $V$. Then $\BB$ is '''orthogonal''' : :$\map f {b_i, b_j} = 0$ for $i \ne j$. That is,  the matrix of $f$ relative to $\mathcal B$ is diagonal.
Definition:Internal Orthogonal Sum (Bilinear Space)
Let $\mathbb K$ be a field. Let $\left({V, f}\right)$ be a reflexive bilinear space over $\mathbb K$. Let $U, W \subset V$ be subspaces of $V$. Then $V$ is the '''internal orthogonal (direct) sum''' of $U$ and $W$ : :$V = U \oplus W$, that is, $V$ is the internal direct sum of $U$ and $W$ :$U \perp W$, that is, $U$ and $W$ are orthogonal. This is denoted: $V = U\oplus W$.
Definition:Reflexive Bilinear Space
Let $\mathbb K$ be a field. Let $\left({V, f}\right)$ be a bilinear space over $\mathbb K$. Then $\left({V, f}\right)$ is '''reflexive'''  $f$ is a reflexive bilinear form.
Definition:Orthogonal (Bilinear Form)/Subsets
Let $S, T \subset V$ be subsets. Then $S$ and $T$ are '''orthogonal'''  for all $s\in S$ and $t\in T$, $s$ and $t$ are orthogonal: $s \perp t$.
Definition:Field of Quotients/Definition 2
A '''field of quotients''' of $D$ is a pair $\struct {F, \iota}$ such that: :$(1): \quad F$ is a field :$(2): \quad \iota: D \to F$ is a ring monomorphism  :$(3): \quad$ If $K$ is a field with $\iota \sqbrk D \subset K \subset F$, then $K = F$. That is, the '''field of quotients''' of an integral domain $D$ is the smallest field containing $D$ as a subring.
Definition:Field of Quotients/Definition 3
A '''field of quotients''' of $D$ is a pair $\struct {F, \iota}$ where: :$(1): \quad$ $F$ is a field :$(2): \quad$ $\iota : D \to F$ is a ring monomorphism :$(3): \quad$ it satisfies the following universal property: ::::For every field $E$ and for every ring monomorphism $\varphi: D \to E$, there exists a unique field homomorphism $\bar \varphi: F \to E$ such that $\varphi = \bar \varphi \circ \iota$
Definition:Field of Quotients/Definition 1
A '''field of quotients''' of $D$ is a pair $\struct {F, \iota}$ where: :$(1): \quad$ $F$ is a field :$(2): \quad$ $\iota : D \to F$ is a ring monomorphism :$(3): \quad \forall z \in F: \exists x \in D, y \in D_{\ne 0}: z = \dfrac {\map \iota x} {\map \iota y}$
Definition:Field of Directions/Functional
Let $\mathbf y$ be an $N$-dimensional vector. Let the functional $J$ be such that: :$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x $ Let the following be a family of boundary conditions, presribed $\forall x \in \closedint a b$: :$\mathbf y' = \map {\boldsymbol \psi} {x, \mathbf y}$ Let these boundary conditions be self-adjoint and consistent $\forall x_1, x_2 \in \closedint a b$. Then these boundary conditions are called '''field of directions''' of the functional $J$.
Definition:Pluperfect Digital Invariant
Let $n \in \Z_{>0}$ be a positive integer. $n$ is a '''pluperfect digital invariant'''  when expressed in decimal notation the $n$th powers of its digits add up to $n$.
Definition:Inconsummate Number
Let $m \in \Z_{>0}$ be a positive integer. Let $s_{10}$ denote the digit sum base $10$  . $m$ is an '''inconsummate number''' : :$\nexists n \in \Z_{>0}: n = m \times s_{10} \left({n}\right)$ That is,  there exists no positive integer $n \in \Z_{>0}$ such that $n$ equals $m$ multiplied by the digit sum of $n$.
Definition:Kaprekar Mapping
The '''Kaprekar mapping''' is the arithmetic function $K: \Z_{>0} \to \Z_{>0}$ defined on the positive integers  as follows: Let $n \in \Z_{>0}$ be expressed in some number base $b$ (where $b$ is usually $10$). Let $n'$ be the positive integer created by arranging the digits of $n$ into descending order of size. Let $n''$ be the positive integer created by arranging the digits of $n$ into ascending order of size. Then: :$K \left({n}\right) = n' - n''$ making sure to retain any leading zeroes to ensure that $K \left({n}\right)$ has the same number of digits as $n$.
Definition:Kaprekar's Process
'''Kaprekar's process''' is the repeated application of the '''Kaprekar mapping''' to a given positive integer.
Definition:Sixth Power
A '''sixth power''' is an integer which can be expressed as the $6$th power of an integer.
Definition:Epsilon Relation/Restriction
Let $S$ be a set. The '''restriction of the epsilon relation''' on $S$ is defined as the endorelation $\Epsilon {\restriction_S} = \left({S, S, \in_S}\right)$, where: :$\in_S \; := \left\{{\left({x, y}\right) \in S \times S: x \in y}\right\}$
Definition:Weird Number
A '''weird number''' is an abundant number which is not at the same time semiperfect.
Definition:Weird Number/Sequence
The sequence of weird numbers begins: :$70, 836, 4030, 5830, 7192, 7912, 9272, \ldots$
Definition:Heptagonal Number
'''Heptagonal numbers''' are those denumerating a collection of objects which can be arranged in the form of a regular heptagon.
Definition:Heptagonal Number/Definition 3
:$\forall n \in \N: H_n = P \left({7, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({7, n - 1}\right) + 5 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$ where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.
Definition:Heptagonal Number/Definition 1
:$H_n = \begin{cases} 0 & : n = 0 \\ H_{n-1} + 5 n - 4 & : n > 0 \end{cases}$
Definition:Heptagonal Number/Definition 2
:$\displaystyle H_n = \sum_{i \mathop = 1}^n \left({5 i - 4}\right) = 1 + 6 + \cdots + \left({5 \left({n - 1}\right) - 4}\right) + \left({5 n - 4}\right)$
Definition:Heptagonal Number/Sequence
The sequence of heptagonal numbers, for $n \in \Z_{\ge 0}$, begins: :$0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, \ldots$
Definition:Zero Digit
Let $x \in \R$ be a number. Let $b \in \Z$ such that $b > 1$ be a number base in which $x$ is represented. By the Basis Representation Theorem, $x$ can be expressed uniquely in the form: :$\displaystyle x = \sum_{j \mathop \in \Z}^m r_j b^j$ Any instance of $r_j$ being equal to $0$ is known as a '''zero (digit)''' of $n$.
Definition:Prime Gap
A '''prime gap''' is a difference between two consecutive prime numbers.
Definition:Sophie Germain Prime/Sequence
The sequence of Sophie Germain primes begins: :$2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, \ldots$
Definition:Reverse-and-Add
Let $m \in \Z_{>0}$ be a positive integer expressed in decimal notation. Let $r: \Z_{>0} \to \Z_{>0}$ be the mapping defined as: :For all $n \in \Z_{>0}$: reverse the digits of $m$ and add the result to $m$. The mapping $r$ is the '''reverse-and-add operation'''.