hcju/proofwikips huggingface dataset:
Definition:Divisor Counting Function
Let $n$ be an integer such that $n \ge 1$. The '''divisor counting function''' is defined on $n$ as being the total number of positive integer divisors of $n$. It is denoted on  as $\tau$ (the Greek letter '''tau'''). That is: :$\displaystyle \map \tau n = \sum_{d \mathop \divides n} 1$ where $\displaystyle \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.
Definition:Sum Over Divisors
Let $n$ be a positive integer. Let $f: \Z_{>0} \to \Z_{>0}$ be a mapping on the positive integers. Let $d \divides n$ denote that $d$ is a divisor of $n$. Then the '''sum of $\map f d$ over all the divisors of $n$''' is denoted: :$\displaystyle \sum_{d \mathop \divides n} \map f d$ Thus, for example: :$\displaystyle \sum_{d \mathop \divides 10} \map f d = \map f 1 + \map f 2 + \map f 5 + \map f {10}$
Definition:Sigma Function
Let $n$ be an integer such that $n \ge 1$. The '''sigma function''' $\map \sigma n$ is defined on $n$ as being the sum of all the positive integer divisors of $n$. That is: :$\displaystyle \map \sigma n = \sum_{d \mathop \divides n} d$ where $\displaystyle \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.
Definition:Abundancy Index
Let $n$ be a positive integer. Let $\map \sigma n$ be the sigma function of $n$. That is, let $\map \sigma n$ be the sum of all positive divisors of $n$. Then the '''abundancy index''' of $n$ is defined as $\dfrac {\map \sigma n} n$.
Definition:Abundance
Let $n \in \Z_{\ge 0}$ be a positive integer. Let $\sigma \left({n}\right)$ be the sigma function of $n$. That is, let $\sigma \left({n}\right)$ be the sum of all positive divisors of $n$. Then the '''abundance''' of $n$ is defined as $A \left({n}\right) = \sigma \left({n}\right) - 2 n$.
Definition:Multiplicative Order of Integer
Let $a$ and $n$ be integers. Let there exist a positive integer $c$ such that: :$a^c \equiv 1 \pmod n$ Then the least such integer is called '''order of $a$ modulo $n$'''.
Definition:Euler Phi Function
Let $n \in \Z_{>0}$, that is, a strictly positive integer. The '''Euler $\phi$ (phi) function''' is the arithmetic function $\phi: \Z_{>0} \to \Z_{>0}$ defined as: :$\map \phi n = $ the number of strictly positive integers less than or equal to $n$ which are prime to $n$ That is: :$\map \phi n = \card {S_n}: S_n = \set {k: 1 \le k \le n, k \perp n}$
Definition:Reduced Residue System/Least Positive
The '''least positive reduced residue system modulo $m$''' is the set of integers: :$\set {a_1, a_2, \ldots, a_{\map \phi m} }$ with the following properties: :$\map \phi m$ is the Euler $\phi$ function :$\forall i: 0 < a_i < m$ :each of which is prime to $m$ :no two of which are congruent modulo $m$.
Definition:Divisor Function
The '''divisor function''': :$\displaystyle \map {\sigma_\alpha} n = \sum_{m \mathop \divides n} m^\alpha$ (meaning the summation is taken over all $m \le n$ such that $m$ divides $n$).
Definition:Von Mangoldt Function
The '''von Mangoldt function''' $\Lambda: \N \to \R$ is defined as: :$\Lambda \left({n}\right) = \begin{cases} \ln \left({p}\right) & : \exists m \in \N, p \in \mathbb P: n = p^m \\ 0 & : \text{otherwise} \end{cases}$ where $\mathbb P$ is the set of all prime numbers.
Definition:Knot (Knot Theory)
Let $Y$ be a manifold and $X \subset Y$ a submanifold of $Y$. Let $i: X \to Y$ be an inclusion, that is a mapping such that $i \sqbrk X = X$. Then a '''knotted embedding''' is an embedding $\phi: X \to Y$ (or the image of such an embedding) such that $\phi \sqbrk X$ is not freely homotopic to $i \sqbrk X$.
Definition:Reidemeister Move
The '''Reidemeister moves''' are one of three operations on knots which encapsulate the intuitive notion of '''equivalent knots'''.   The three moves are shown below. :File:Reidemeister move 1.png  :File:Reidemeister move 2.png  :File:Reidemeister move 3.png {{namedfor|Kurt Reidemeister}}  768qxtheox2rkueocc9cn88uv7mu0ff
Definition:Alexander Polynomial
For a knot $K$ with Seifert matrix $V$, the '''Alexander polynomial''' of $K$ is defined as: :$\Delta_K \left({t}\right) = \det \left({V - t V^T}\right)$ {{NamedforDef|James Waddell Alexander II|cat=Alexander}}
Definition:Seifert Matrix
For a knot $K$ with Seifert surface $S$, the '''Seifert matrix''' $V$ of $K$ is defined by its entries as: :$v_{ij} = \operatorname{lk} \left({ x_i, x_k^* }\right)$ where: : the $x_a$ are the generators of the fundamental group $\pi_1(S)$ : $x_a^*$ is the positive push-off of $x_a$ : $\operatorname{lk}$ is the linking number of the two loops. {{NamedforDef|Karl Johannes Herbert Seifert}}
Definition:Completely Multiplicative Function
Let $K$ be a field. Let $f: K \to K$ be a function on $K$. Then $f$ is described as '''completely multiplicative''' : :$\forall m, n \in K: \map f {m n} = \map f m \map f n$
Definition:Embedding (Topology)
Let $A, B$ be topological spaces. Let $f: A \to B$ be a mapping. Let the image of $f$ be given the subspace topology. Let the restriction $f {\restriction_{A \times f \left({A}\right)}}$ of $f$ to its image be a homeomorphism. Then $f$ is an '''embedding''' (of $A$ into $B$).
Definition:Primitive Root
Let $a, n \in \Z_{>0}$, that is, let $a$ and $n$ be strictly positive integers. Let the multiplicative order of $a$ modulo $n$ be $\phi \left({n}\right)$, where $\phi \left({n}\right)$ is the Euler phi function of $n$. Then $a$ is a '''primitive root of $n$''' or a '''primitive root modulo $n$'''.
Definition:Square-Free Integer
Let $n \in \Z$. Then $n$ is '''square-free'''  $n$ has no divisor which is the square of a prime. That is,  the prime decomposition $n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}$ is such that: :$\forall i: 1 \le i \le r: k_i = 1$
Definition:Quadratic Residue
Let $p$ be an odd prime. Let $a \in \Z$ be an integer such that $a \not \equiv 0 \pmod p$. Then $a$ is a '''quadratic residue of $p$'''  $x^2 \equiv a \pmod p$ has a solution. That is, : :$\exists x \in \Z: x^2 \equiv a \pmod p$
Definition:Set of Residue Classes/Least Positive
Let $\eqclass a m$ be the residue class of $a$ (modulo $m$). Let $r$ be the smallest non-negative integer in $\eqclass a m$. Then from Integer is Congruent to Integer less than Modulus: :$0 \le r < m$ and: :$a \equiv r \pmod m$ Then $r$ is called the '''least positive residue''' of $a \pmod m$.
Definition:Lowest Common Multiple
=== Integral Domain === {{:Definition:Lowest Common Multiple/Integral Domain}} === Integers === {{:Definition:Lowest Common Multiple/Integers}}
Definition:Greatest Common Divisor
=== Integral Domain === {{:Definition:Greatest Common Divisor/Integral Domain}} === Integers === When the integral domain in question is the integers $\Z$, the GCD is often defined differently, as follows: {{:Definition:Greatest Common Divisor/Integers/Definition 1}} === Polynomial Ring over Field === {{:Definition:Greatest Common Divisor/Polynomial Ring over Field}}
Definition:Prime-Counting Function
The '''prime-counting function''' is the function $\pi: \R \to \Z$ which counts the number of primes less than or equal to some real number. That is: :$\displaystyle \forall x \in \R: \map \pi x = \sum_{\substack {p \mathop \in \mathbb P \\ p \mathop \le x} } 1$ where $\mathbb P$ denotes the set of prime numbers.
Definition:Primorial
There are two definitions for '''primorials''': one for primes and one for positive integers. === Definition for Primes === {{:Definition:Primorial/Prime}} === Definition for Positive Integers === {{:Definition:Primorial/Positive Integer}}
Definition:Euclid Number
A '''Euclid number''' is a natural number of the form: :$E_n := p_n\# + 1$ where $p_n\#$ is the primorial of the $n$th prime number.
Definition:O Notation
'''$\OO$ notation''' is a type of order notation, typically used in computer science for comparing 'run-times' of algorithms, or in analysis for comparing growth rates between two growth functions.
Definition:Residue (Complex Analysis)
Let $f: \C \to \C$ be a complex function. Let $z_0 \in U \subset \C$ such that $f$ is analytic in $U \setminus \left\{{z_0}\right\}$. Then by Existence of Laurent Series, there is a Laurent series: :$\displaystyle \sum_{j \mathop = -\infty}^\infty a_j \left({z - z_0}\right)^j$ such that the sum converges to $f$ in $U - \left\{{z_0}\right\}$.   The '''residue''' at a point $z = z_0$ of a function $f: \C \to \C$ is defined as $a_{-1}$ in that Laurent series. It is denoted $\operatorname{Res} \left({f, z_0}\right)$ or just $\operatorname{Res}\left({z_0}\right)$ when $f$ is understood.
Definition:Laurent Series
Let $f: \C \to \C$ be a complex function. Let $z_0 \in U \subset \C$ such that $f$ is analytic in $U \setminus \left\{{z_0}\right\}$. A '''Laurent series''' is a sum: :$\displaystyle \sum_{j \mathop = -\infty}^\infty a_j \left({z - z_0}\right)^j$ such that the sum converges to $f$ in $U \setminus \left\{{z_0}\right\}$. {{NamedforDef|Pierre Alphonse Laurent|cat = Laurent}}
Definition:Residue Class
Let $m \in \Z$. Let $\mathcal R_m$ be the congruence relation modulo $m$ on the set of all $a, b \in \Z$: :$\mathcal R_m = \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$ We have that congruence modulo $m$ is an equivalence relation. So for any $m \in \Z$, we denote the equivalence class of any $a \in \Z$ by $\eqclass a m$, such that: {{begin-eqn}} {{eqn | l = \eqclass a m       | r = \set {x \in \Z: a \equiv x \pmod m}       | c =  }} {{eqn | r = \set {x \in \Z: \exists k \in \Z: x = a + k m}       | c =  }} {{eqn | r = \set {\ldots, a - 2 m, a - m, a, a + m, a + 2 m, \ldots}       | c =  }} {{end-eqn}} The equivalence class $\eqclass a m$ is called the '''residue class of $a$ (modulo $m$)'''.
Definition:Integral Polynomial
An '''integral polynomial''' is a polynomial over the ring of integers $\Z$.
Definition:Polynomial Congruence
Let $P \left({x}\right)$ be an integral polynomial. Then the expression: :$P \left({x}\right) \equiv 0 \pmod n$ is known as a '''polynomial congruence'''. === Linear Congruence === {{:Definition:Linear Congruence}}
Definition:Linear Congruence
A '''linear congruence''' is a polynomial congruence of the form: :$a_0 + a_1 x \equiv 0 \pmod n$ That is, one where the degree of the integral polynomial is $1$. Such a congruence is frequently encountered in the equivalent form: :$a x \equiv b \pmod n$
Definition:Simultaneous Congruences
A '''system of simultaneous congruences''' is a set of polynomial congruences: :$\forall i \in \left[{1 \,.\,.\, r}\right]: P_i \left({x}\right) \equiv 0 \pmod {n_i}$ That is: {{begin-eqn}} {{eqn | l = P_1 \left({x}\right)       | r = 0       | rr= \pmod {n_1}       | c =  }} {{eqn | l = P_2 \left({x}\right)       | r = 0       | rr= \pmod {n_2}       | c =  }} {{eqn | o = \cdots       | c =  }} {{eqn | l = P_r \left({x}\right)       | r = 0       | rr= \pmod {n_r}       | c =  }} {{end-eqn}}
Definition:Jacobi Symbol
Let $m \in \Z$ be any integer and $n \in \Z$ be any odd integer such that $n \ge 3$. Let the prime decomposition of $n$ be: : $\displaystyle n = \prod_{i \mathop = 1}^r p_i^{k_i}$. Then the '''Jacobi symbol''' $\left({\dfrac m n}\right)$ is defined as: :$\displaystyle \left({\frac m n}\right) = \prod_{i \mathop = 1}^r \left({\frac m {p_i}}\right)^{k_i}$ where $\left({\dfrac m {p_i}}\right)$ is defined as the Legendre symbol.
Definition:Continued Fraction
{{:Definition:Continued Fraction/Finite}}
Definition:Exterior Derivative
Let an exact $n$-form $\omega$ be given on an $m$-manifold, with local coordinates $x_1, x_2, \dots, x_m$. Let a local coordinate expression for $\omega$ be given: :$\omega = \map f {x_1, \ldots, x_m} \rd x_{\map \phi 1} \wedge \d x_{\map \phi 2} \wedge \cdots \wedge \d x_{\map \phi n}$ where: :$\phi: \set {1, \ldots, n} \to \set {1, \ldots, m}$ is an injection which determines which coordinate vectors $\omega$ acts on. :$\wedge$ denotes the wedge product. The '''exterior derivative''' $\d \omega$ is the $\paren {n + 1}$-form defined as: :$\displaystyle \d \omega = \paren {\sum_{k \mathop = 1}^m \frac {\partial f} {\partial x_k} \rd x_k} \wedge \d x_{\map \phi 1} \wedge \d x_{\map \phi 2} \wedge \dots \wedge \d x_{\map \phi n}$ For inexact forms: :$\map \d {a + b} = \d a + \d b$
Definition:Wedge Product
Let $\alpha$ and $\beta$ be two differential forms. Let $\alpha$ be an $x$-form. Let $\beta$ be a $y$-form. The '''wedge product''' $\alpha \wedge \beta$ is defined as the linear antisymmetric map from $F^x \times F^y \to F^{x + y}$, where $F^a$ is the set of $a$-forms in some manifold. Let $x_0$ be a specific point in a manifold $X$. Let $\alpha$ be an $x$-form. Let $\phi$ be a $1$-form. Let there be a set of vectors $\mathbf u_1, \mathbf u_2, \dotsc, \mathbf u_x, \mathbf v \in \map {T_{x_0} } X$. The '''wedge product''' is defined as: :$\alpha \wedge \map \phi {\mathbf u_1, \mathbf u_2, \dotsc, \mathbf u_x, \mathbf v} := \sum_P \map \varepsilon P \, \map \alpha {P_1} \, \map \phi {P_2}$ where: :$P$ is some permutation of $\mathbf u_1, \mathbf u_2, \dots, \mathbf u_x, \mathbf v$ :$P_1$ is the first $x$ terms of the permutation $P$ :$P_2$ the final term of permutation $P$ :$\varepsilon$ is the permutation symbol of $P$. The sum is taken over all possible permutations. This definition extends to wedge products of arbitrary forms through the linearity and antisymmetric conditions.
Definition:Permutation Symbol
The '''permutation symbol''' $\varepsilon$ of a permutation $P$ of a set of elements is defined as: :$+1$ for even permutations (permutations that are an even number of pair swaps) :$-1$ for odd permutations :$0$ if the list of elements is not a permutation (that is, contains a repeated value). Frequently, the permutation will be explicit, for example: :$\varepsilon_{i j k \ldots}$ :$\varepsilon^{i j \ldots}_{k l \ldots}$ :$\varepsilon^{i j k \ldots}$ This notation is especially useful when raising and lowering indices (that is, converting between forms and vectors).
Definition:Local Coordinates
Let $X$ be an $n$-dimensional manifold. Let $p \in X$, and let $U \subset X$ be a neighbourhood of $p$. Then a set of mappings $x_i: U \to \R$, $1 \le i \le n$, satisfying: :$a = b \iff \forall i: \map {x_i} a = \map {x_i} b$ is called a set of '''local coordinates'''. When the neighbourhood $U$ is to be stressed, one may also say '''local coordinates for $U$'''. Similarly, when the element $p$ is to be stressed, one may also say '''local coordinates around $p$'''.
Definition:Partial Derivative
=== Real Analysis === {{:Definition:Partial Derivative/Real Analysis}} === Complex Analysis === {{:Definition:Partial Derivative/Complex Analysis}}
Definition:Riemann Surface
A '''Riemann surface''' is a connected complex manifold of dimension $1$.
Definition:Riemann Sphere
Let $f_1: \C \to \R^2$ be defined as: :$\forall z \in \C: \map {f_1} z = \tuple {\map \Re z, \map \Im z}$ Let $f_2: \R^2 \to \R^3$ be the inclusion map: :$\forall \tuple {a, b} \in \C^2: \map {f_2} {a, b} = \tuple {a, b, 0}$ Let $f = f_2 \circ f_1$. Let $F: \C \to \map {\PP} {\R^3}$ be defined as the mapping which takes $z$ to the closed line interval from $\tuple {0, 0, 1}$ to $\map f z$ for all $z \in \C$. Let $G = \set {x, y, z: x^2 + y^2 + z^2 = 1}$. Then the Riemann map $R: \C \to \mathbb S^2$ is defined as: :$\map R x = \map F z \cap G$ The set $R \sqbrk \C \cup \set {\tuple {0, 0, 1} } $ is called the '''Riemann sphere''', with the understanding that $\map f \infty = \tuple {0, 0, 1}$. {{expand|According to the definition in Clapham & Nicholson, include the definition as the extended complex plane under a stereographic projection}}
Definition:Measure (Measure Theory)
Let $\struct {X, \Sigma}$ be a measurable space. Let $\mu: \Sigma \to \overline \R$ be a mapping, where $\overline \R$ denotes the set of extended real numbers. Then $\mu$ is called a '''measure''' on $\Sigma$  $\mu$ has the following properties: {{begin-axiom}} {{axiom | n = 1         | q = \forall E \in \Sigma         | ml= \map \mu E         | mo= \ge         | mr= 0 }} {{axiom | n = 2         | q = \forall \sequence {S_n}_{n \mathop \in \N} \subseteq \Sigma: \forall i, j \in \N: S_i \cap S_j = \O         | ml= \map \mu {\bigcup_{n \mathop = 1}^\infty S_n}         | mo= =         | mr= \sum_{n \mathop = 1}^\infty \map \mu {S_n}         | rc= that is, $\mu$ is a countably additive function }} {{axiom | n = 3         | q = \exists E \in \Sigma         | ml= \map \mu E         | mo= \in         | mr= \R         | rc= that is, there exists at least one $E \in \Sigma$ such that $\map \mu E$ is finite }} {{end-axiom}}
Definition:Periodic Continued Fraction
Let $\left[{a_1, a_2, a_3, \ldots}\right]$ be a simple infinite continued fraction. Let the partial quotients be of the form: :$\left[{r_1, r_2, \ldots, r_m, s_1, s_2, \ldots, s_n, s_1, s_2, \ldots, s_n, s_1, s_2, \ldots, s_n, \ldots}\right]$ that is, ending in a block of partial quotients which repeats itself indefinitely. Such a SICF is known as a '''periodic continued fraction'''. The notation used for this is $\left[{r_1, r_2, \ldots, r_m, \left \langle{s_1, s_2, \ldots, s_n}\right \rangle}\right]$, where the repeating block is placed in angle brackets. === Purely Periodic Continued Fraction === {{:Definition:Periodic Continued Fraction/Purely Periodic}} === Cycle === {{:Definition:Periodic Continued Fraction/Cycle}}
Definition:Fractional Part
Let $x \in \R$ be a real number. Let $\floor x$ be the floor function of $x$. The '''fractional part''' of $x$ is the difference: :$\fractpart x := x - \floor x$
Definition:Analytic Continuation
Let $U \subset \C$ be an open set. Let $f: U \to \C$ be an analytic function. Let $V$ be an open subset of $\C$ such that $U \subset V$. An '''analytic continuation of $f$ to $V$''' is an analytic function $F: V \to \C$ such that $\map F z = \map f z$ for $z \in U$.
Definition:Gamma Function
=== Integral Form === {{:Definition:Gamma Function/Integral Form}} === Weierstrass Form === {{:Definition:Gamma Function/Weierstrass Form}} === Hankel Form === {{:Definition:Gamma Function/Hankel Form}} === Euler Form === {{:Definition:Gamma Function/Euler Form}}
Definition:Absolute Convergence of Product
=== Complex Numbers === Let $\sequence {a_n}$ be a sequence in $\C$. {{:Definition:Absolute Convergence of Product/Complex Numbers/Definition 1}} === General Definition === Let $\struct {\mathbb K, \norm{\,\cdot\,} }$ be a valued field. Let $\sequence {a_n}$ be a sequence in $\mathbb K$. {{:Definition:Absolute Convergence of Product/General Definition/Definition 1}}
Definition:Fourier Series
=== Formulation 1 === {{:Definition:Fourier Series/Formulation 1}} === Formulation 2 === {{:Definition:Fourier Series/Formulation 2}}
Definition:Quadratic Irrational
A '''quadratic irrational''' is an irrational number of the form: :$r + s \sqrt n$ where $r, s$ are rational and $n$ is a positive integer which is not a square.
Definition:Pythagorean Triple
A '''Pythagorean triple''' is an ordered triple of positive integers $\tuple {x, y, z}$ such that $x^2 + y^2 = z^2$. That is, a '''Pythagorean triple''' is a solution to the Pythagorean equation.
Definition:Projective Space
=== Real Projective Space === {{:Definition:Projective Space/Real Projective Space}} === Over a Field === {{:Definition:Projective Space/Over a Field}}
Definition:Dirichlet Series
Let $a_n: \N \to \C$ be an arithmetic function. Its '''Dirichlet series''' is a complex function $f: \C \to \C$ defined by a series: :$\displaystyle \map f s = \sum_{n \mathop = 1}^\infty a_n n^{-s}$
Definition:Pythagorean Equation
The '''Pythagorean equation''' is the Diophantine equation: :$x^2 + y^2 = z^2$ where $x, y, z$ are integers such that $x, y, z > 0$. Solutions of this equation are known as Pythagorean triples. If, in addition, $\left({x, y, z}\right)$ is a primitive Pythagorean triple, then $\left({x, y, z}\right)$ is known as a '''primitive solution''' of $x^2 + y^2 = z^2$.
Definition:Triangular Number
'''Triangular numbers''' are those denumerating a collection of objects which can be arranged in the form of an equilateral triangle.
Definition:Geometric Sequence
A '''geometric sequence''' is a sequence $\sequence {x_n}$ in $\R$ defined as: :$x_n = a r^n$ for $n = 0, 1, 2, 3, \ldots$ Thus its general form is: :$a, ar, ar^2, ar^3, \ldots$ and the general term can be defined recursively as: :$x_n = \begin{cases} a & : n = 0 \\ r x_n & : n > 0 \\ \end{cases}$
Definition:Square Number
'''Square numbers''' are those denumerating a collection of objects which can be arranged in the form of a square. They can be denoted: :$S_1, S_2, S_3, \ldots$ === Definition 1 === {{:Definition:Square Number/Definition 1}} === Definition 2 === {{:Definition:Square Number/Definition 2}} === Definition 3 === {{:Definition:Square Number/Definition 3}} === Definition 4 === {{:Definition:Square Number/Definition 4}}
Definition:Arithmetic Sequence
An '''arithmetic sequence''' is a finite sequence $\sequence {a_k}$ in $\R$ or $\C$ defined as: :$a_k = a_0 + k d$ for $k = 0, 1, 2, \ldots, n - 1$ Thus its general form is: :$a_0, a_0 + d, a_0 + 2 d, a_0 + 3 d, \ldots, a_0 + \paren {n - 1} d$
Definition:Polygonal Number
'''Polygonal numbers''' are those denumerating a collection of objects which can be arranged in the form of an regular polygon.
Definition:Pentagonal Number
'''Pentagonal numbers''' are those denumerating a collection of objects which can be arranged in the form of a regular pentagon.
Definition:Hexagonal Number
'''Hexagonal numbers''' are those denumerating a collection of objects which can be arranged in the form of a regular hexagon.
Definition:Pyramidal Number
The '''pyramidal numbers''' are integers defined as sums of sequences of polygonal numbers as follows. Let $P \left({k, n}\right)$ be the $n$th $k$-gonal number. Then the '''$n$th $k$gonal-based pyramidal number''' $Y \left({k, n}\right)$ is defined as: :$\displaystyle Y \left({k, n}\right) = \sum_{i \mathop = 1}^n P \left({k, i}\right)$
Definition:Set of Residue Classes/Least Absolute
Let $\eqclass a m$ be the residue class of $a$ (modulo $m$). Except when $r = \dfrac m 2$, we can choose $r$ to be the integer in $\eqclass a m$ which has the smallest absolute value. In that exceptional case we have: : $-\dfrac m 2 + m = \dfrac m 2$ and so: : $-\dfrac m 2 \equiv \dfrac m 2 \pmod m$ Thus $r$ is defined as the '''least absolute residue''' of $a$ (modulo $m$) : : $-\dfrac m 2 < r \le \dfrac m 2$
Definition:Balanced Incomplete Block Design
A '''Balanced Incomplete Block Design''' or '''BIBD''' with parameters $v, b, r, k, \lambda$ is a block design where: * $v$ is the number of points in the design * $b$ is the number of blocks * $k$ is the size of each block * $r$ is the number of blocks any point can be in * $\lambda$ is the number of times any two points can occur in the same block and has the following properties: * Each block is of size $k$ * All of the $\displaystyle \binom v 2$ pairs occur together in exactly $\lambda$ blocks. A BIBD with parameters $v, b, r, k, \lambda$ is commonly written several ways, for example: * $\operatorname{BIBD} \left({v, k, \lambda}\right)$ * $\left ({v, k, \lambda}\right)$-$\operatorname{BIBD}$
Definition:Odd Prime
Every even integer is divisible by $2$ (because this is the definition of '''even'''). Therefore, apart from $2$ itself, all primes are odd. So, referring to an '''odd prime''' is a convenient way of specifying that a number is a prime number, but not equal to $2$.
Definition:Combination
Let $S$ be a set containing $n$ elements. An '''$r$-combination of $S$''' is a subset of $S$ which has $r$ elements.
Definition:Generating Function
Let $A = \left \langle {a_n}\right \rangle$ be a sequence in $\R$. Then $\displaystyle G_A \left({z}\right) = \sum_{n \mathop \ge 0} a_n z^n$ is called the '''generating function''' for the sequence $A$.
Definition:Fermat Pseudoprime
Let $q$ be a composite number such that $\exists n \in N: n^q \equiv n \pmod q$. Then $q$ is a '''Fermat pseudoprime to base $n$'''.
Definition:Algorithm
An '''algorithm''' is a finite set of '''instructions''' (or '''rules''') that defines a sequence of '''operations''' for solving a particular '''computational problem''' for all '''problem instances''' for some '''problem set'''.
Definition:Algorism
'''Algorism''' is an archaic term which means '''the process of doing arithmetic using Arabic numerals'''. === Algorist === {{:Definition:Algorism/Algorist}}