hcju/proofwikips huggingface dataset:
Definition:Multiplicative Relation
Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $\mathcal R$ be a relation on $S$. Then $\mathcal R$ is '''multiplicative (relation)'''  :$\forall a, x, y \in S: \left({\left({a, x}\right), \left({a, y}\right) \in \mathcal R \implies \left({a, x \wedge y}\right) \in \mathcal R}\right)$
Definition:Polynomial Evaluation Homomorphism
Let $R, S$ be commutative rings with unity. Let $\kappa : R \to S$ be a unital ring homomorphism. === Single Indeterminate === {{:Definition:Polynomial Evaluation Homomorphism/Single Indeterminate}} === Multiple Indeterminates === {{:Definition:Polynomial Evaluation Homomorphism/Multiple Indeterminates}}
Definition:Free Module on Set/Canonical Basis
Let $R$ be a ring with unity.
Definition:Homomorphism Ring between Modules
Let $R$ be a ring. Let $M, N$ be $R$-modules. The '''homomorphism ring between $M$ and $N$''' is defined as the set of all $R$-module homomorphisms from $M$ to $N$, denoted: :$\map {\operatorname {Hom}_R} {M, N}$ or just: :$\map {\operatorname {Hom} } {M, N}$ if the specific ring $R$ in question is clear from the context.
Definition:Group Ring
Let $R$ be a ring. Let $\struct {G, +}$ be a group. The '''group ring of $G$ over $R$''' is the monoid ring of $G$ over $R$.
Definition:Power (Algebra)/Complex Number/Principal Branch
The principal branch of a complex number raised to a complex power is defined as: :$z^k = e^{k \Ln z}$ where $\Ln z$ is the principal branch of the natural logarithm.
Definition:Power (Algebra)/Complex Number/Principal Branch/Positive Real Base
Let $t > 0$ be a real number and let $k$ be a complex number. The principal branch of a positive real number raised to a complex power is defined as: :$t^k = e^{k \ln t}$ where $\ln$ is the natural logarithm of a positive real number.
Definition:Z-Module Associated with Abelian Group
Let $\struct {G, *}$ be an abelian group with identity $e$. Let $\struct {\Z, +, \times}$ be the ring of integers. === Definition 1 === {{:Definition:Z-Module Associated with Abelian Group/Definition 1}} === Definition 2 === {{:Definition:Z-Module Associated with Abelian Group/Definition 2}}
Definition:Octagonal Number
'''Octagonal numbers''' are those denumerating a collection of objects which can be arranged in the form of a regular octagon.
Definition:Octagonal Number/Definition 1
:$O_n = \begin{cases} 0 & : n = 0 \\ O_{n - 1} + 6 n - 5 & : n > 0 \end{cases}$
Definition:Octagonal Number/Definition 2
:$\displaystyle O_n = \sum_{i \mathop = 1}^n \left({6 i - 5}\right) = 1 + 7 + \cdots + \left({6 \left({n - 1}\right) - 5}\right) + \left({6 n - 5}\right)$
Definition:Octagonal Number/Definition 3
:$\forall n \in \N: O_n = P \left({8, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({8, n - 1}\right) + 6 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$ where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.
Definition:Octagonal Number/Sequence
The sequence of octagonal numbers, for $n \in \Z_{\ge 0}$, begins: :$0, 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, \ldots$
Definition:Ring of Integers
The '''ring of integers''' $\struct {\Z, +, \times}$ is the set of integers under the two operations of integer addition and integer multiplication.
Definition:Perfect Magic Cube
A '''perfect magic cube''' is an arrangement of the first $n^3$ (strictly) positive integers into an $n \times n \times n$ cubic array such that: :the sum of the entries in each row in each of the $3$ dimensions :the sum of the entries along the main diagonal of each plane :the sum of the entries along the space diagonals are the same.
Definition:Magic Cube/Order
An $n \times n \times n$ magic cube is called an '''order $n$ magic cube'''.
Definition:Magic Cube
A '''magic cube''' is an arrangement of the first $n^3$ (strictly) positive integers into an $n \times n \times n$ cubic array such that: :the sum of the entries in each row in each of the $3$ dimensions :the sum of the entries along the space diagonals are the same. It is not guaranteed that the entries along each of the main diagonals of each plane also sum to the same constant.
Definition:Deltahedron
A '''deltahedron''' is a polyhedron each of whose faces are equilateral triangles.
Definition:Bipyramid
A '''bipyramid''' is a polyhedron formed by taking two pyramids with congruent bases and placing those bases together: :400px
Definition:Triangular Bipyramid
A '''triangular bipyramid''' is a polyhedron formed by taking two regular tetrahedra and placing their bases together: :300px The '''triangular bipyramid''' is an example of a deltahedron.
Definition:Pentagonal Bipyramid
A '''pentagonal bipyramid''' is a polyhedron formed by taking two pyramids whose bases are regular pentagons and whose other faces are equilateral triangles, and placing their bases together: :300px The '''pentagonal bipyramid''' is an example of a deltahedron.
Definition:Lucky Number
Start with the list of (strictly) positive integers: :$1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, \ldots$ Remove every $2$nd number from $2$ onwards (that is, all the even integers): :$1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, \ldots$ The $2$nd term is $3$. Starting from the $3$rd element in this list (which is $5$), remove every $3$rd element from what is left: :$1, 3, 7, 9, 13, 15, 19, 21, 25, \ldots$ The $3$rd number is now $7$. Starting from the $7$th element in this list (which is $19$), remove every $7$th element from what is left: :$1, 3, 7, 9, 13, 15, 21, 25, \ldots$ The numbers remaining are the '''lucky numbers'''.
Definition:Lucky Number/Sequence
The sequence of lucky numbers begins: :$1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, \ldots$
Definition:Kaprekar Number
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Suppose that $n^2$, when expressed in number base $b$, can be split into two parts that add up to $n$. Then $n$ is a '''Kaprekar number''' for base $b$.
Definition:Kaprekar Number/Sequence
The sequence of Kaprekar numbers begins: :$1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, \ldots$
Definition:Subfactorial
Let $n \in \Z_{\ge0}$ be a (strictly) positive integer. The '''subfactorial''' of $n$ is defined and denoted as: {{begin-eqn}} {{eqn | l = !n       | o = :=       | r = n! \sum_{k \mathop = 0}^n \frac {\left({-1}\right)^k} {k!}       | c =  }} {{eqn | r = n! \left({1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} + \cdots + \dfrac {\left({-1}\right)^n} {n!} }\right)       | c =  }} {{end-eqn}} It arises as the number of derangements of $n$ distinct objects.
Definition:Compact Subset of Lattice
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. '''Compact subset''' of $S$, denoted $K\left({L}\right)$, equals to the set of all compact elements of $S$.
Definition:Compact Closure
Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $x \in S$. Then '''compact closure''' of $x$, denoted $x^{\mathrm{compact}}$, is defined by :$x^{\mathrm{compact}} := \left\{ {y \in S: y \preceq x \land y}\right.$ is compact$\left.{}\right\}$
Definition:Axiom of K-Approximation
Let $\left({S, \preceq}\right)$ be an ordered set. '''Axiom of K-approximation''' says  :$\forall x \in S: x = \sup \left({x^{\mathrm{compact} } }\right)$ where $x^{\mathrm{compact} }$ denotes the compact closure of $x$.
Definition:Algebraic Ordered Set
Let $\left({S, \preceq}\right)$ be an ordered set. Then $\left({S, \preceq}\right)$ is '''algebraic'''  :(for all elements $x$ of $S$: $x^{\mathrm{compact} }$ is directed) and :$\left({S, \preceq}\right)$ is up-complete and satisfies axiom of K-approximation where $x^{\mathrm{compact} }$ denotes the compact closure of $x$.
Definition:Asymmetric Relation/Definition 1
$\RR$ is '''asymmetric''' : :$\tuple {x, y} \in \RR \implies \tuple {y, x} \notin \RR$
Definition:Asymmetric Relation/Definition 2
$\RR$ is '''asymmetric'''  it and its inverse are disjoint: :$\RR \cap \RR^{-1} = \O$
Definition:Idempotence/Relation
Let $\RR \subseteq S \times S$ be a relation on $S$. Then $\RR$ is '''idempotent''' : :$\RR \circ \RR = \RR$ where $\circ$ denotes composition of relations.
Definition:Idempotence/Algebraic Structure
Let $\left({S, \circ}\right)$ be an algebraic structure. Let $\circ$ be an idempotent operation on $S$. Then $\left({S, \circ}\right)$ is an '''idempotent algebraic structure'''.
Definition:Meet Closed
Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $X$ be a subset of $S$. Then $X$ is '''meet closed'''  :$\forall x, y \in X: x \wedge y \in X$
Definition:Arithmetic Ordered Set
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. $L$ is '''arithmetic'''  :$L$ is algebraic and $K\left({L}\right)$ is meet closed where $K\left({L}\right)$ denotes the compact subset of $L$.
Definition:Endomorphism Ring of Abelian Group
Let $\struct {G, +}$ be an abelian group. Let $\map {\mathrm {End} } G$ be the set of endomorphisms of $G$. The '''endomorphism ring of $G$''' is the algebraic structure: :$\struct {\map {\mathrm {End} } G, +, \circ}$ where: :$\circ$ denotes composition :$+$ denotes pointwise addition.
Definition:Support of Permutation
Let $S$ be a set. Let $f$ be a permutation on $S$. The '''support''' of $f$ is the subset of moved elements: :$\operatorname {supp} \left({f}\right) = \left\{ {x \in S: f \left({x}\right) \ne  x}\right\}$
Definition:Support of Element of Direct Product
Let $\left({S_i, \circ_i}\right)_{i \mathop \in I}$ be a family of algebraic structures with identity. Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be their direct product. Let $e_i$ be an identity of $S_i$ for all $i \in I$. Let $m = \left({m_i}\right)_{i \mathop \in I} \in S$. The '''support''' of $m$ is defined as: :$\operatorname {supp} \left\{ {i \in I: m_i \ne e_i}\right\}$ === Finite Support === The element is said to have '''finite support'''  its support is a finite set.
Definition:Liouville Function
Let $n \in \Z_{>0}$, that is, a strictly positive integer. The '''Liouville function''' is the function $\lambda: \Z_{>0} \to \Z_{>0}$ defined as: :$\lambda \left({n}\right) = \left({-1}\right)^{\Omega \left({n}\right)}$ where $\Omega \left({n}\right)$ is the number of prime factors of $n$, counted with multiplicity.
Definition:Module of All Mappings
Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {M, +_M, \circ}_R$ be an $R$-module. Let $S$ be a set. Let $M^S$ be the set of all mappings from $S$ to $M$. Let: :$+$ be the operation induced on $M^S$ by $+_M$ :$\forall \lambda \in R: \forall f \in M^S: \forall x \in S: \map {\paren {\lambda \circ f} } x = \lambda \circ \paren {\map f x}$ Then $\struct {M^S, +, \circ}_R$ is the '''module of all mappings from $S$ to $M$'''.
Definition:Module Direct Product/Finite Case
Let $R$ be a ring. Let $\left({M_1, +_1, \circ_1}\right)_R, \left({M_2, +_2, \circ_2}\right)_R, \ldots, \left({M_n, +_n, \circ_n}\right)_R$ be $R$-modules. Let: :$\displaystyle M = \prod_{k \mathop = 1}^n M_k$ be the cartesian product of $M_1$ to $M_n$. Let: : $+$ be the operation induced on $M$ by the operations $+_1, +_2, \ldots, +_n$ on $M_1, M_2, \ldots, M_n$ : $\circ$ be defined as $\lambda \circ \left({x_1, x_2, \ldots, x_n}\right) = \left({\lambda \circ_1 x_1, \lambda \circ_2 x_2, \ldots, \lambda \circ_n x_n}\right)$ In Finite Direct Product of Modules is Module, it is shown that $\left\langle{M, +, \circ}\right\rangle$ is an $R$-module. The module $\left({M, +, \circ}\right)_R$ is called the '''(external) direct product''' of $M_1$ to $M_n$.
Definition:Module Direct Product/General Case
Let $R$ be a ring. Let $\left\{ \left\langle M_i,+_i,\circ_i\right\rangle\right\}_{i \in I}$ be a family of $R$-modules. {{explain|Expand the notation of the module to specify the operations of the ring.}} Let: :$M = \displaystyle \prod_{i \mathop \in I} M_i$ be the cartesian product of these modules. The operation $+$ induced on $M$ by $(+_i)_{i\in I}$ is the operation defined by: :$\left\langle{a_i}\right\rangle_{i \mathop \in I} + \left\langle{b_i}\right\rangle_{i \mathop \in I} = \left\langle{a_i +_i b_i}\right\rangle_{i \mathop \in I}$ That is, the additive group of the module $M$ is the direct product of the groups $\left\{ \left(M_i,+_i\right)\right\}_{i \in I}$. The $R$-action $\circ$ induced on $M$ by $(\circ_i)_{i\in I}$ is the operation defined by: :$r \circ \left\langle{m_i}\right\rangle_{i \mathop \in I} = \left\langle{r \circ_i m_i}\right\rangle_{i \mathop \in I}$ In Direct Product of Modules is Module, it is shown that $\left\langle{M, +, \circ}\right\rangle$ is an $R$-module. The module $\left\langle{M, +, \circ}\right\rangle$  is called the '''(external) direct product''' of $\left\{ \left\langle M_i,+_i,\circ_i\right\rangle\right\}_{i \in I}$.
Definition:Direct Product of Vector Spaces/Finite Case
Let $K$ be a field. Let $V_1, V_2, \ldots, V_n$ be $K$-vector spaces. Let: :$\displaystyle V = \prod_{k \mathop = 1}^n V_k$ be their cartesian product. Let: :$+$ be the operation induced on $V$ by the operations $+_1, +_2, \ldots, +_n$ on $V_1, V_2, \ldots, V_n$ :$\circ$ be defined as $\lambda \circ \tuple {x_1, x_2, \ldots, x_n} := \tuple {\lambda \circ x_1, \lambda \circ x_2, \ldots, \lambda \circ x_n}$ $\struct {V, +, \circ}_K$ is called the '''direct product of $V_1, \ldots, V_n$'''.
Definition:Direct Product of Vector Spaces
Let $K$ be a field. Let $V, W$ be $K$-vector spaces. The '''direct product of $V$ and $W$''' is their module direct product. === Finite Case === {{:Definition:Direct Product of Vector Spaces/Finite Case}} === General Case === {{:Definition:Direct Product of Vector Spaces/General Case}}
Definition:Direct Product of Vector Spaces/General Case
Let $K$ be a field. Let $\family {V_i, +_i, \circ_i}_{i \mathop \in I}$ be a family of $K$-vector spaces. The '''(external) direct product''' of $\family {V_i, +_i, \circ_i}_{i \mathop \in I}$ is their module direct product.
Definition:Module of Homomorphisms Between Modules
Let $R$ be a commutative ring. Let $M$ and $N$ be $R$-modules. Let $\map {\operatorname {Hom}_{R - \operatorname {mod} } } {M, N}$ denote the set of $R$-module homomorphisms from $M$ to $N$. Let: :$+$ be the operation on $\map {\operatorname {Hom}_{R - \operatorname {mod} } } {M, N}$ defined by $f + g: m \mapsto \map f m + \map g m$ :$\circ$ be defined as $\lambda \circ f: m \mapsto \lambda \map f m$ Then $\struct {\map {\operatorname {Hom}_{R - \operatorname {mod} } } {M, N}, + , \circ}$ is called the '''module of homomorphisms''' between $M$ and $N$.
Definition:Dual Module
Let $R$ be a commutative ring. Let $M$ be an $R$-module. Then the '''dual module''' of $M$ is the module of homomorphisms $\operatorname{Hom}_{R\text{-mod}}(M,R)$. That is, the elements of the dual module are the linear forms on $M$.
Definition:Order Complete Set/Definition 1
$\struct {S, \preceq}$ is '''order complete''' : :Each non-empty subset $H \subseteq S$ which has an upper bound admits a supremum.
Definition:Order Complete Set/Definition 2
$\struct {S, \preceq}$ is '''order complete''' : :Each non-empty subset $H \subseteq S$ which has a lower bound admits an infimum.
Definition:Order Complete Set
Let $\left({S, \preceq}\right)$ be an ordered set.
Definition:Normal Subgroup/Also known as
It is usual to describe a '''normal subgroup of $G$''' as '''normal in $G$'''. Some sources refer to a '''normal subgroup''' as an '''invariant subgroup''' or a '''self-conjugate subgroup'''. This arises from Definition 6: {{:Definition:Normal Subgroup/Definition 6}} which is another way of stating that $N$ is '''normal'''  $N$ stays the same under all inner automorphisms of $G$. Some sources use '''distinguished subgroup'''.
Definition:Kepler-Poinsot Polyhedron
A '''Kepler-Poinsot polyhedron''' is any of the four regular star polyhedra.
Definition:Altitude of Triangle/Foot
The point at which $h_a$ meets $BC$ is the '''foot of the altitude $h_a$'''.
Definition:Feuerbach Circle
Let $\triangle ABC$ be a triangle. The '''Feuerbach circle''' of $\triangle ABC$ is the circle which passes through each of the $9$ points: : the feet of the altitudes of $\triangle ABC$ : the midpoints of the sides of $\triangle ABC$ : the midpoints of the lines from the vertices of $\triangle ABC$ to the orthocenter of $\triangle ABC$. :500px
Definition:Cosine Integral Function
The '''cosine integral function''' is the real function $\Ci: \R_{>0} \to \R$ defined as: :$\map \Ci x = \displaystyle \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {\cos t} t \rd t$
Definition:Magic Square/Magic Constant
The '''magic constant''' of a magic square is the number that each of the rows and columns adds up to.
Definition:Field Extension/Complex
Let $\left({F, +, \times}\right)$ be a subfield of $\left({\Bbb C, +, \times}\right)$, the field of complex numbers. Let $X_1, X_2, \ldots, X_n$ be complex numbers, in or not in $F$. Then $F \left({X_1, X_2, \ldots, X_n}\right)$ is the smallest field extension over $F$ containing $X_1, X_2, \ldots, X_n$.
Definition:Euclid Prime
A '''Euclid prime''' is a natural number which is both a Euclid number and a prime number.
Definition:Lambert W Function/Principal Branch/Real Valued
The principal branch of the Lambert W function is the real function $W_0: \hointr {-\dfrac 1 e} \to \to \hointr {-1} \to$ such that: :$x = \map {W_0} x e^{\map {W_0} x}$
Definition:Lambert W Function
Let $e^z: \C \to \C$ be the complex exponential function. Then $\map W z$ is defined to be the multifunction that satisfies:  :$\map W z e^{\map W z} = z$
Definition:Omega Constant
The '''omega constant''' $\Omega$ is the constant as the value of the principal branch of the Lambert W function at $1$: :$\Omega \, e^ \Omega = 1$ That is, it is the root of the equation: :$x e^x = 1$ where $e$ denotes Euler's number.
Definition:Euclid Prime/Sequence
The sequence of '''Euclid primes''' begins: :$2, 3, 7, 31, 211, 2311, 200 \, 560 \, 490 \, 131, \ldots$
Definition:Euclid Number/Sequence
The sequence of Euclid numbers begins as follows: {{begin-eqn}} {{eqn | ll= E_0       | l = = p_0\# + 1       | r = 1 + 1       | rr= = 2       | c =  }} {{eqn | ll= E_1       | l = = p_1\# + 1       | r = 2 + 1       | rr= = 3       | c =  }} {{eqn | ll= E_2       | l = = p_2\# + 1       | r = 2 \times 3 + 1       | rr= = 7       | c =  }} {{eqn | ll= E_3       | l = = p_3\# + 1       | r = 2 \times 3 \times 5 + 1       | rr= = 31       | c =  }} {{eqn | ll= E_4       | l = = p_4\# + 1       | r = 2 \times 3 \times 5 \times 7 + 1       | rr= = 211       | c =  }} {{eqn | ll= E_5       | l = = p_5\# + 1       | r = 2 \times 3 \times 5 \times 7 \times 11 + 1       | rr= = 2311       | c =  }} {{eqn | ll= E_6       | l = = p_6\# + 1       | r = 2 \times 3 \times 5 \times 7 \times 11 \times 13 + 1       | rr= = 30031       | c =  }} {{end-eqn}}
Definition:Compact Element
Let $\left({S, \preceq}\right)$ be an ordered set. Let $x \in S$. Then $x$ is '''compact (element)'''  $x \ll x$ where $\ll$ denotes the way below relation.
Definition:Lambert W Function/Lower Branch
The '''lower branch'''  of the Lambert W function is the real function $W_{-1}: \left[{-\dfrac 1 e \,.\,.\, 0}\right) \to \left({\gets \,.\,.\, -1}\right]$ such that: :$x = W_{-1} \left({x}\right) e^{W_{-1} \left({x}\right)}$
Definition:Foiaș Constant
=== First Foiaș Constant === {{:Definition:Foiaș Constant/First}} === Second Foiaș Constant === {{:Definition:Foiaș Constant/Second}}
Definition:Foiaș Constant/First/Decimal Expansion
The decimal expansion of the first Foiaș Constant starts: :$x_{\infty} = 2 \cdotp 29316 \, 62874 \, 11861 \, 03150 \, 80282 \, 91250 \, 80586 \, 43722 \, 57290 \, 32712 \, 12485 \, 37 \ldots$
Definition:Foiaș Constant/First
Let: :$x_{n+1} = \left({1 + \dfrac 1 {x_n} }\right)^{x_n}$ for $n = 1, 2, 3, \ldots$ The '''first Foiaș constant''' is the limit of $x_n$ as $n \to \infty$.
Definition:Foiaș Constant/Second/Decimal Expansion
The decimal expansion of the second Foiaș Constant starts: :$\alpha = 1 \cdotp 18745 \, 23511 \, 26501 \ldots$
Definition:Foiaș Constant/Second
Let $x_1 \in \R_{>0}$ be a (strictly) positive real number. Let: :$x_{n + 1} = \left({1 + \dfrac 1 {x_n} }\right)^n$ for $n = 1, 2, 3, \ldots$ The '''second Foiaș constant''' is defined as the unique real number $\alpha$ such that if $x_1 = \alpha$ then the sequence $\left\langle{x_{n + 1} }\right\rangle$ diverges to infinity.
Definition:Pandigital Fraction
A '''pandigital fraction''' is a fraction where the digits $1$ to $9$ appear exactly once in the decimal expansions of the numerator and the denominator of the fraction combined.
Definition:Tetrahedral Number
'''Tetrahedral numbers''' are those denumerating a collection of objects which can be arranged in the form of a regular tetrahedron.
Definition:Polyhedral Number/Examples
=== Tetrahedral Number === {{:Definition:Tetrahedral Number}}
Definition:Noncototient
A '''noncototient''' is a positive integer $n$ such that: :$\nexists m \in \Z_{>0}: m - \phi \left({m}\right) = n$ where $\phi \left({m}\right)$ denotes the Euler $\phi$ function. That is, a '''noncototient''' is a positive integer which is not the cototient of any positive integer.
Definition:Noncototient/Sequence
The sequence of '''noncototients''' begins: :$10, 26, 34, 50, 52, 58, 86, 100, 116, 122, 130, 134, 146, 154, 170, 172, 186, 202, \ldots$
Definition:Cototient
Let $n \in \Z_{>0}$ be a (strictly) positive integer. The '''cototient''' of $n$ is defined as: :$n - \phi \left({n}\right)$ where $\phi \left({n}\right)$ denotes the Euler $\phi$ function.
Definition:Latin Square/Order
Let $\mathbf L$ be an $n \times n$ Latin square. The '''order''' of $\mathbf L$ is $n$.
Definition:Latin Square/Row
Let $\mathbf L$ be a Latin square. The '''rows''' of $\mathbf L$ are the lines of elements reading '''across''' the page.
Definition:Latin Square/Column
Let $\mathbf L$ be a Latin square. The '''columns''' of $\mathbf L$ are the lines of elements reading '''down''' the page.
Definition:Latin Square/Element
Let $\mathbf L$ be a Latin square of order $n$. The individual $n \times n$ symbols that go to form $\mathbf L$ are known as the '''elements''' of $\mathbf L$. The element at row $i$ and column $j$ is called '''element $\left({i, j}\right)$ of $\mathbf L$''', and can be written $a_{i j}$, or $a_{i, j}$ if $i$ and $j$ are of more than one character. If the indices are still more complicated coefficients and further clarity is required, then the form $a \left({i, j}\right)$ can be used. Note that the first subscript determines the row, and the second the column, of the Latin square where the element is positioned.
Definition:Orthogonal Latin Squares
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\mathbf A$ and $\mathbf B$ be Latin squares of order $n$. Let $\left[{a}\right]_{i j}$ and $\left[{b}\right]_{i j}$ be the elements of $\mathbf A$ and $\mathbf B$ respectively whose indices are $i$ and $j$. Then $\mathbf A$ and $\mathbf B$ are '''orthogonal''' : :for all $i$ and $j$, the ordered pairs $\left({\left[{a}\right]_{i j}, \left[{b}\right]_{i j}}\right)$ are distinct. That is,  $\left({\left[{a}\right]_{i j}, \left[{b}\right]_{i j}}\right)$ appears no more than once for every ordered pair $\left({i, j}\right)$. === Mutually Orthogonal === {{:Definition:Orthogonal Latin Squares/Mutually Orthogonal}}
Definition:Orthogonal Latin Squares/Mutually Orthogonal
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $S$ be a set of Latin squares of order $n$. Then the $S$ is '''mutually orthogonal'''  every element of $S$ is orthogonal to every other element of $S$.
Definition:Multiplicative Persistence
Let $n \in \N$ be a natural number. Let $n$ be expressed in decimal notation. Multiply the digits of $n$ together. Repeat with the answer, and again until a single digit remains. The number of steps it takes to reach $1$ digit is called the '''multiplicative persistence''' of $n$.
Definition:Octave
An '''octave''' is a musical interval whose components have frequencies in the ratio $2 : 1$.
Definition:Music
'''Music''' is the branch of study which creates aesthetically pleasing combinations of sounds.
Definition:Fifth (Music)
A '''fifth''' is a musical interval whose components have frequencies in the ratio $3 : 2$.
Definition:Fourth (Music)
A '''fourth''' is a musical interval whose components have frequencies in the ratio $4 : 3$.