hcju/proofwikips huggingface dataset:
Definition:Closed-Form Expression
An expression is a '''closed-form expression'''  it can be expressed in terms of a bounded number of elementary functions. Informally is it in apposition to a recurrence relation, which defines a sequence of terms from earlier terms in that sequence.
Definition:Closed-Form Solution
A system of equations has a '''closed-form solution'''  at least one solution can be expressed as a closed-form expression.
Definition:Lexicographic Order
Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be ordered sets. The '''lexicographic order on $S_1 \times S_2$''' is the relation $\preccurlyeq$ defined on $S_1 \times S_2$ as: :$\tuple {x_1, x_2} \preccurlyeq \tuple {y_1, y_2} \iff \tuple {x_1 \prec_1 y_1} \lor \paren {x_1 = y_1 \land x_2 \preceq_2 y_2}$
Definition:Basis Expansion
=== Positive Real Numbers === {{:Definition:Basis Expansion/Positive Real Numbers}} === Negative Real Numbers === {{:Definition:Basis Expansion/Negative Real Numbers}}
Definition:Decimal Notation
'''Decimal notation''' is the quotidian technique of expressing numbers in base $10$. Every number $x \in \R$ is expressed in the form: :$\displaystyle x = \sum_{j \mathop \in \Z} r_j 10^j$ where: :$\forall j \in \Z: r_j \in \set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$
Definition:Decimal Expansion
Let $x \in \R$ be a real number. The '''decimal expansion''' of $x$ is the expansion of $x$ in base $10$. $x = \floor x + \displaystyle \sum_{j \mathop \ge 1} \frac {d_j} {10^j}$: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_{10}$ where: :$s = \floor x$, the floor of $x$ :it is not the case that there exists $m \in \N$ such that $d_M = 9$ for all $M \ge m$. (That is, the sequence of digits does not end with an infinite sequence of $9$s.)
Definition:Neo-Pythagoreanism
'''Neo-Pythagoreanism''' was a school of philosophy which revived the teachings of {{AuthorRef|Pythagoras of Samos|Pythagoras}}. It flourished mainly in the 1st and 2nd centuries C.E. Its influence on mathematics was to associate the numbers with mystical properties, effectively adding a religious aspect to the subject. Its influence upon the Western mathematical tradition was considerably greater than it might have been, in part because of the popularity of {{AuthorRef|Nicomachus of Gerasa|Nicomachus's}} {{BookLink|Introduction to Arithmetic|Nicomachus of Gerasa}}.
Definition:Iverson's Convention
'''Iverson's Convention''' is a notation which allows a compact means of assigning a value of $1$ or $0$ to a proposition $P$, depending on whether $P$ is true or false: :$\sqbrk P = \begin{cases} 1 & : P \ \text { is true} \\ 0 & : P \ \text { is false} \end{cases}$ It is sometimes seen specified as: :$\sqbrk P = \begin{cases} 1 & : P \ \text { is true} \\ 0 & : P \ \text { otherwise} \end{cases}$ which can be useful in fields of mathematics where the Law of the Excluded Middle does not apply.
Definition:Combinatorial Matrix
A '''combinatorial matrix''' is a square matrix specified as: :$a_{ij} = y + \delta_{ij} x$ where $\delta_{ij}$ is the Kronecker delta.
Definition:Vandermonde Matrix
The '''Vandermonde matrix''' of order $n$ is a square matrix specified variously as: :$a_{ij} = x_i^{j - 1}$ :$a_{ij} = x_j^i$ :$a_{ij} = x_i^{n - j}$ etc.
Definition:Cauchy Matrix
The '''Cauchy matrix''', commonly denoted $C_n$, can be found defined in two forms. The '''Cauchy matrix''' is an $m \times n$ matrix whose elements are in the form: : either $a_{ij} = \dfrac 1 {x_i + y_j}$ : or $a_{ij} = \dfrac 1 {x_i - y_j}$. where $x_1, x_2, \ldots, x_m$ and $y_1, y_2, \ldots, y_n$ are elements of a field $F$.
Definition:Cauchy Determinant
A '''Cauchy determinant of order $n$''' is the determinant of a square Cauchy matrix of order $n$: :$\det \left({C_n}\right) = \begin{vmatrix} \dfrac 1 {x_1 + y_1} & \dfrac 1 {x_1 + y_2} & \cdots & \dfrac 1 {x_1 + y_n} \\ \dfrac 1 {x_2 + y_1} & \dfrac 1 {x_2 + y_2} & \cdots & \dfrac 1 {x_2 + y_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac 1 {x_n + y_1} & \dfrac 1 {x_n + y_2} & \cdots & \dfrac 1 {x_n + y_n} \\ \end{vmatrix}$ Its value is given by: :$\det \left({C_n}\right) = \dfrac {\displaystyle \prod_{1 \mathop \le i \mathop < j \mathop \le n} \left({x_j - x_i}\right) \left({y_j - y_i}\right)} {\displaystyle \prod_{1 \mathop \le i, \, j \mathop \le n} \left({x_i + y_j}\right)}$ If $C_n$ is given by: :$\det \left({C_n}\right) = \begin{bmatrix} \dfrac 1 {x_1 - y_1} & \dfrac 1 {x_1 - y_2} & \cdots & \dfrac 1 {x_1 - y_n} \\ \dfrac 1 {x_2 - y_1} & \dfrac 1 {x_2 - y_2} & \cdots & \dfrac 1 {x_2 - y_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac 1 {x_n - y_1} & \dfrac 1 {x_n - y_2} & \cdots & \dfrac 1 {x_n - y_n} \\ \end{bmatrix}$ then its value is given by: :$\det \left({C_n}\right) = \dfrac {\displaystyle \prod_{1 \mathop \le i \mathop < j \mathop \le n} \left({x_j - x_i}\right) \left({y_j - y_i}\right)} {\displaystyle \prod_{1 \mathop \le i, \, j \mathop \le n} \left({x_i - y_j}\right)}$
Definition:Ordered Semigroup
An '''ordered semigroup''' is an ordered structure $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a semigroup.
Definition:Ordered Monoid
An '''ordered monoid''' is an ordered structure $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a monoid.
Definition:Ordered Group
An '''ordered group''' is an ordered structure $\left({G, \circ, \preceq}\right)$ such that $\left({G, \circ}\right)$ is a group.
Definition:Unitary Module
Let $\struct {R, +_R, \times_R}$ be a ring with unity whose unity is $1_R$. Let $\struct {G, +_G}$ be an abelian group. A unitary module over $R$ is an $R$-algebraic structure with one operation $\struct {G, +_G, \circ}_R$  which satisfies the unitary module axioms: {{:Definition:Unitary Module Axioms}}
Definition:Scalar Ring
Let $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$ be an $R$-algebraic structure with $n$ operations, where: :$\struct {R, +_R, \times_R}$ is a ring :$\struct {S, *_1, *_2, \ldots, *_n}$ is an algebraic structure with $n$ operations :$\circ: R \times S \to S$ is a binary operation. Then the ring $\struct {R, +_R, \times_R}$ is called the scalar ring of $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$. If the scalar ring is understood, then $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$ can be rendered $\struct {S, *_1, *_2, \ldots, *_n, \circ}$. === Scalar === {{:Definition:Scalar/R-Algebraic Structure}} === Zero Scalar === {{:Definition:Scalar Ring/Zero Scalar}}
Definition:Zero Vector
The identity of $\struct {G, +_G}$ is usually denoted $\bszero$, or some variant of this, and called the '''zero vector''': :$\forall \mathbf a \in \struct {G, +_G, \circ}_R: \bszero +_G \mathbf a = \mathbf a = \mathbf a +_G \bszero$ Note that on occasion it is advantageous to denote the '''zero vector''' differently, for example by $e$, or $\bszero_V$ or $\bszero_G$, in order to highlight the fact that the '''zero vector''' is not the same object as the zero scalar.
Definition:Ones Matrix
A '''ones matrix''' is a matrix all of whose elements are $1$. === Square Ones Matrix === {{:Definition:Ones Matrix/Square}}
Definition:Infinite Hilbert Matrix
The '''infinite Hilbert matrix''' is the infinite matrix whose elements are defined as: :$a_{ij} = \dfrac 1 {i + j - 1}$ In full, it appears as: :$\begin{bmatrix} 1 & \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \ldots \\ \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \ldots \\ \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \ldots \\ \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \ldots \\ \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \tfrac 1 {10} & \ldots \\ \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \tfrac 1 {10} & \tfrac 1 {11} & \ldots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots  \end{bmatrix}$ {{NamedforDef|David Hilbert|cat = Hilbert}}
Definition:Submatrix
Let $\mathbf A$ be a matrix with $m$ rows and $n$ columns. A '''submatrix''' of $\mathbf A$ is a matrix formed by selecting from $\mathbf A$: :a subset of the rows and: :a subset of the columns and forming a new matrix by using those entries, in the same relative positions, that appear in both the rows and columns of those selected.
Definition:Infinite Matrix
An '''infinite matrix''' is a matrix whose order is infinite.
Definition:Hilbert Matrix
A '''Hilbert matrix''' is an order $n$ square submatrix of the infinite Hilbert matrix, consisting of the elements in the first $n$ rows and columns of that matrix. Thus it is an $n \times n$ matrix whose elements are defined as: :$a_{i j} = \dfrac 1 {i + j - 1}$
Definition:Closure Operator/Power Set
Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. A '''closure operator''' on $S$ is a mapping: :$\cl: \powerset S \to \powerset S$ which satisfies the following conditions for all sets $X, Y \subseteq S$: {{begin-axiom}} {{axiom | n = 1         | lc= $\cl$ is inflationary         | q = \forall X \subseteq S         | ml= X         | mo= \subseteq         | mr= \map \cl X }} {{axiom | n = 2         | lc= $\cl$ is increasing         | q = \forall X, Y \subseteq S         | ml= X \subseteq Y         | mo= \implies         | mr= \map \cl X \subseteq \map \cl Y }} {{axiom | n = 3         | lc= $\cl$ is idempotent         | q = \forall X \subseteq S         | ml= \map \cl {\map \cl X}         | mo= =         | mr= \map \cl X }} {{end-axiom}}
Definition:Prime Factor
Let $n \in \Z$ be an integer. Then $p$ is a '''prime factor''' of $n$ : : $(1): \quad p$ is a prime number : $(2): \quad p$ is a divisor (that is, factor) of $n$.
Definition:Modulo Operation
Let $x, y \in \R$ be real numbers. Then the '''modulo operation''' is defined and denoted as: :$x \bmod y := \begin{cases} x - y \floor {\dfrac x y} & : y \ne 0 \\ x & : y = 0 \end{cases}$ where $\floor {\, \cdot \,}$ denotes the floor function.
Definition:Integral Multiple
=== Rings and Fields === {{:Definition:Integral Multiple/Rings and Fields}} === Real Numbers === This concept is often seen when $F$ is the set of real numbers $\R$. {{:Definition:Integral Multiple/Real Numbers}}
Definition:Quotient (Algebra)
Let $a, b \in \Z$ be integers such that $b \ne 0$. From the Division Theorem: :$\forall a, b \in \Z, b \ne 0: \exists_1 q, r \in \Z: a = q b + r, 0 \le r < \size b$ The value $q$ is defined as the '''quotient of $a$ on division by $b$''', or the '''quotient of $\dfrac a b$'''. === Real Arguments === When $x, y \in \R$ the quotient is still defined: {{:Definition:Quotient (Algebra)/Real}}
Definition:Pre-Socratic
The term '''pre-Socratic''' refers to various schools of Greek philosophy before {{AuthorRef|Socrates}}.
Definition:Eleatic School
The '''Eleatic School''' was a pre-Socratic school of Greek philosophy. Its guiding tenet was the concept of Oneness. God was the eternal Unity, who permeated the universe and guided it with his thought.
Definition:Steiner Inellipse
Let $\triangle ABC$ be a triangle whose sides have midpoints $D, E, F$. Then the unique ellipse which can be drawn wholly inside $\triangle ABC$ tangent to $D, E, F$ is called the '''Steiner inellipse''': :400px {{namedfor|Jakob Steiner}}
Definition:Boundary (Graph Theory)
=== Simple Graph === Let $G = \tuple {V, E}$ be a simple graph. Let $v \in V$ be a vertex of $G$. Then the '''boundary''' of $v$ is the set of all vertices of $G$ which are adjacent to $v$: :$\map B v = \set {u \in V: \set {u, v} \in E}$
Definition:Rectifiable Curve
A '''rectifiable curve''' is a curve of finite length.
Definition:Ray (Topology)
Let $T = \struct {S, \tau}$ be a topological space. A '''ray''' in $T$ is an embedding $R_{>0} \to S$.
Definition:Weight Function
A '''weight function''' on a set $S$ is a mapping from $S$ to the real numbers: :$w: S \to \R$ It is common for the requirements of a specific application under discussion for the codomain of $w$ to be restricted to the positive reals: :$w: S \to \R_{\ge 0}$ The thing that determines whether a given mapping is a '''weight function''' depends more on how it is used.
Definition:Rising Factorial
Let $x$ be a real number (but usually an integer). Let $n$ be a positive integer. Then '''$x$ to the (power of) $n$ rising''' is defined as: :$\displaystyle x^{\overline n} := \prod_{j \mathop = 0}^{n - 1} \paren {x + j} = x \paren {x + 1} \cdots \paren {x + n - 1}$
Definition:Falling Factorial
Let $x$ be a real number (but usually an integer). Let $n$ be a positive integer. Then '''$x$ to the (power of) $n$ falling''' is: :$\displaystyle x^{\underline n} := \prod_{j \mathop = 0}^{n - 1} \left({x - j}\right) = x \left({x - 1}\right) \cdots \left({x - n + 1}\right)$
Definition:Stirling Numbers of the First Kind
=== Unsigned Stirling Numbers of the First Kind === {{:Definition:Stirling Numbers of the First Kind/Unsigned}} === Signed Stirling Numbers of the First Kind === {{:Definition:Stirling Numbers of the First Kind/Signed}}
Definition:Stirling Numbers
'''Stirling numbers''' come in various forms. In the below: : $\delta_{n k}$ is the Kronecker delta : $n$ and $k$ are non-negative integers. === Unsigned Stirling Numbers of the First Kind === {{:Definition:Stirling Numbers of the First Kind/Unsigned}} === Signed Stirling Numbers of the First Kind === {{:Definition:Stirling Numbers of the First Kind/Signed}} === Stirling Numbers of the Second Kind === {{:Definition:Stirling Numbers of the Second Kind|Stirling Numbers of the Second Kind}}
Definition:Stirling Numbers of the Second Kind
{{:Definition:Stirling Numbers of the Second Kind/Definition 1}}
Definition:Pascal's Triangle
:$\begin{array}{r|rrrrrrrrrr} n & \binom n 0 & \binom n 1 & \binom n 2 & \binom n 3 & \binom n 4 & \binom n 5 & \binom n 6 & \binom n 7 & \binom n 8 & \binom n 9 & \binom n {10} & \binom n {11} & \binom n {12} \\ \hline 0  & 1 &  0 &  0 &   0 &   0 &   0 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\ 1  & 1 &  1 &  0 &   0 &   0 &   0 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\ 2  & 1 &  2 &  1 &   0 &   0 &   0 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\ 3  & 1 &  3 &  3 &   1 &   0 &   0 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\ 4  & 1 &  4 &  6 &   4 &   1 &   0 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\ 5  & 1 &  5 & 10 &  10 &   5 &   1 &   0 &   0 &   0 &  0  &  0 &  0 & 0 \\ 6  & 1 &  6 & 15 &  20 &  15 &   6 &   1 &   0 &   0 &  0  &  0 &  0 & 0 \\ 7  & 1 &  7 & 21 &  35 &  35 &  21 &   7 &   1 &   0 &  0  &  0 &  0 & 0 \\ 8  & 1 &  8 & 28 &  56 &  70 &  56 &  28 &   8 &   1 &  0  &  0 &  0 & 0 \\ 9  & 1 &  9 & 36 &  84 & 126 & 126 &  84 &  36 &   9 &  1  &  0 &  0 & 0 \\ 10 & 1 & 10 & 45 & 120 & 210 & 252 & 210 & 120 &  45 &  10 &  1 &  0 & 0 \\ 11 & 1 & 11 & 55 & 165 & 330 & 462 & 462 & 330 & 165 &  55 & 11 &  1 & 0 \\ 12 & 1 & 12 & 66 & 220 & 495 & 792 & 924 & 792 & 495 & 220 & 66 & 12 & 1 \\ \end{array}$
Definition:Stirling's Triangles
'''Stirling's Triangles''' are the arrays formed by arranging Stirling's Numbers of the first and second kind, as follows:
Definition:Array
An '''array''' is an arrangement (usually rectangular) of objects (usually numbers) such that the relations between the entries borne by their relative positions in that arrangement have a particular significance.
Definition:Congruence (Number Theory)
Let $z \in \R$. === Definition by Remainder after Division === {{:Definition:Congruence (Number Theory)/Remainder after Division}} === Definition by Modulo Operation === {{:Definition:Congruence (Number Theory)/Modulo Operation}} === Definition by Integer Multiple === {{:Definition:Congruence (Number Theory)/Integer Multiple}}
Definition:Filter on Set
Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. === Definition 1 === {{:Definition:Filter on Set/Definition 1}} === Definition 2 === {{:Definition:Filter on Set/Definition 2}} === Filtered Set === {{:Definition:Filter on Set/Filtered Set}} === Trivial Filter === {{:Definition:Filter on Set/Trivial Filter}}
Definition:Ultrafilter on Set
=== Definition 1 === {{:Definition:Ultrafilter on Set/Definition 1}} === Definition 2 === {{:Definition:Ultrafilter on Set/Definition 2}} === Definition 3 === {{:Definition:Ultrafilter on Set/Definition 3}} === Definition 4 === {{:Definition:Ultrafilter on Set/Definition 4}}
Definition:Filter Basis
Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. === Definition 1 === {{:Definition:Filter Basis/Definition 1}} === Definition 2 === {{:Definition:Filter Basis/Definition 2}}
Definition:Addition/Modulo Addition
{{:Definition:Addition/Modulo Addition/Definition 1}}
Definition:Multiplication/Modulo Multiplication
{{:Definition:Multiplication/Modulo Multiplication/Definition 1}}
Definition:Topology Generated by Synthetic Sub-Basis
Let $X$ be a set. Let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a synthetic sub-basis on $X$. === Definition 1 === {{:Definition:Topology Generated by Synthetic Sub-Basis/Definition 1}} === Definition 2 === {{:Definition:Topology Generated by Synthetic Sub-Basis/Definition 2}}
Definition:Initial Topology
Let $X$ be a set. Let $I$ be an indexing set. Let $\left \langle {\left({Y_i, \tau_i}\right)} \right \rangle_{i \mathop \in I}$ be an indexed family of topological spaces indexed by $I$. Let $\left \langle {f_i: X \to Y_i} \right \rangle_{i \mathop \in I}$ be an indexed family of mappings indexed by $I$.
Definition:Image Filter
Let $X, Y$ be sets. Let $\mathcal P \left({X}\right)$ and $\mathcal P \left({Y}\right)$ be the power sets of $X$ and $Y$ respectively. Let $f: X \to Y$ a mapping. Let $\mathcal F \subset \mathcal P \left({X}\right)$ be a filter on $X$. Then :$f \left({\mathcal F}\right) := \left\{{U \subseteq Y: f^{-1} \left({U}\right) \in \mathcal F}\right\}$ is a filter on $Y$, called the '''image filter''' of $\mathcal F$ with respect to $f$.
Definition:Open Mapping
Let $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$ be topological spaces. Let $f: S_1 \to S_2$ be a mapping.  Then $f$ is said to be an '''open mapping''' : :$\forall U \in \tau_1: f \sqbrk U \in \tau_2$ where $f \sqbrk U$ denotes the image of $U$ under $f$.
Definition:Replicative Function
A '''replicative function''' is a real function $f$ such that: :$\displaystyle \forall n \in \Z_{\ge 0}: \sum_{k \mathop = 0}^{n - 1} f \left({x + \frac k n}\right) = f \left({n x}\right)$ where $\sum$ denotes indexed summation.
Definition:Dimension (Geometry)
The '''dimension''' of a (geometrical) space is the minimum number of coordinates needed to specify a point in it.
Definition:Infinity
Informally, the term '''infinity''' is used to mean '''some infinite number''', but this concept falls very far short of a usable definition. The symbol $\infty$ (supposedly invented by {{AuthorRef|John Wallis}}) is often used in this context to mean '''an infinite number'''. However, outside of its formal use in the definition of limits its use is strongly discouraged until you know what you're talking about. It is defined as having the following properties: :$\forall n \in \Z: n < \infty$ :$\forall n \in \Z: n + \infty = \infty$ :$\forall n \in \Z: n \times \infty = \infty$ :$\infty^2 = \infty$ Similarly, the quantity written as $-\infty$ is defined as having the following properties: :$\forall n \in \Z: -\infty< n$ :$\forall n \in \Z: -\infty + n = -\infty$ :$\forall n \in \Z: -\infty \times n = -\infty$ :$\paren {-\infty}^2 = -\infty$ The latter result seems wrong when you think of the rule that a negative number square equals a positive one, but remember that infinity is not exactly a number as such.
Definition:Veridical Paradox
A '''veridical paradox''' is a counter-intuitive result which can be demonstrated to be true.
Definition:Falsidical Paradox
A '''falsidical paradox''' is a result which, as well as being absurd on the surface, is the result of faulty reasoning and so is genuinely false. Such apparent paradoxes can be classified as fallacies.
Definition:Antinomy
An '''antinomy''' is a self-contradictory statement which arises as the result of a valid argument from a set of premises which contains an inherent ambiguity or contradiction. Such apparent paradoxes can also often be classified as fallacies.
Definition:Dialetheia
A '''dialetheia''' is a statement which is paradoxical through the fact that it is both true and false at the same time. As such, it is incompatible with {{AuthorRef|Aristotle}}'s Law of the Excluded Middle and Principle of Non-Contradiction, and therefore does not in general appear in the context of conventional mathematics.
Definition:Finite Difference Operator
Let $f: \R \to \R$ be a real function. The '''(finite) difference operator''' on $f$ comes in a number of forms, as follows.
Definition:Chen Prime
A '''Chen Prime''' is a prime number $p$ such that $p + 2$ is either a prime number or a semiprime. The even integer $2 p + 2$ then satisfies Chen's Theorem. The first few '''Chen primes''' are :$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, \ldots$ {{OEIS|A109611}} The first few '''non-Chen primes''' are :$43, 61, 73, 79, 97, 103, 151, 163, 173, 193, 223, 229, 241, \ldots$ {{OEIS|A102540}}
Definition:Semiprime Number
A '''semiprime (number)''' is an integer which is the product of two (not necessarily distinct) primes.
Definition:Sufficiently Large
Let $P$ be a property of real numbers. Then: : '''$P \left({x}\right)$ holds for all sufficiently large $x$''' : :$\exists a \in \R: \forall x \in \R: x \ge a: P \left({x}\right)$ That is: :''There exists a real number $a$ such that for every (real) number not less than $a$, the property $P$ holds.'' It is not necessarily the case, for a given property $P$ about which such a statement is made, that the value of $a$ actually needs to be known, merely that such a value can be demonstrated to exist.
Definition:Inverse Trigonometric Function
As there are six basic trigonometric functions, so each of these has its inverse functions.
Definition:Harmonic Numbers
The '''harmonic numbers''' are denoted $H_n$ and are defined for positive integers $n$: :$\displaystyle \forall n \in \Z, n \ge 0: H_n = \sum_{k \mathop = 1}^n \frac 1 k$
Definition:Vector Cross Product
Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $\mathbf V$ of $3$ dimensions: :$\mathbf a = a_i \mathbf i + a_j \mathbf j + a_k \mathbf k$ :$\mathbf b = b_i \mathbf i + b_j \mathbf j + b_k \mathbf k$ where $\left({\mathbf i, \mathbf j, \mathbf k}\right)$ is the standard ordered basis of $\mathbf V$.
Definition:Digit Sum
Let $n \in \Z: n \ge 0$. The '''digit sum of $n$ to base $b$''' is the sum of all the digits of $n$ when expressed in base $b$. That is, if: :$\displaystyle n = \sum_{k \mathop \ge 0} r_k b^k$ where $0 \le r_k < b$, then: :$\displaystyle s_b \left({n}\right) = \sum_{k \mathop \ge 0} r_k$
Definition:Digital Root
Let $n \in \Z: n \ge 0$. Let $n$ be expressed in base $b$ notation. Let $n_1 = \map {s_b} n$ be the digit sum of $n$ to base $b$. Then let $n_2 = \map {s_b} {n_1}$ be the digit sum of $n_1$ to base $b$. Repeat the process, until $n_m$ has only one digit, that is, that $1 \le n_m < b$. Then $n_m$ is the '''digital root of $n$ to the base $b$'''.
Definition:Gaussian Integer
A '''Gaussian integer''' is a complex number whose real and imaginary parts are both integers. That is, a '''Gaussian integer''' is a number in the form: :$a + b i: a, b \in \Z$ The set of all '''Gaussian integers''' can be denoted $\Z \sqbrk i$, and hence can be defined as: :$\Z \sqbrk i = \set {a + b i: a, b \in \Z}$
Definition:Gaussian Rational
A '''Gaussian rational''' is a complex number whose real and imaginary parts are both rational numbers. That is, a '''Gaussian rational''' is a number in the form: :$a + b i: a, b \in \Q$ The set of all '''Gaussian rationals''' can be denoted $\Q \sqbrk i$, and hence can be defined as: :$\Q \sqbrk i = \set {a + b i: a, b \in \Q}$
Definition:Field Automorphism
Let $\struct {F, +, \circ}$ be a field. Let $\phi: F \to F$ be a (field) isomorphism from $F$ to itself. Then $\phi$ is a field automorphism.