hcju/proofwikips huggingface dataset:
Definition:Borel Sigma-Algebra
=== Topological Spaces === Let $\struct {S, \tau}$ be a topological space The '''Borel sigma-algebra''' $\map {\mathcal B} {S, \tau}$ of $\struct {S, \tau}$ is the $\sigma$-algebra generated by $\tau$. That is, it is the $\sigma$-algebra generated by the set of open sets in $S$. === Metric Spaces === Let $\struct {X, \norm {\,\cdot\,} }$ be a metric space. The '''Borel sigma-algebra''' (or '''$\sigma$-algebra''') on $\struct {X, \norm {\,\cdot\,} }$ is the $\sigma$-algebra generated by the open sets in $\powerset X$. By the definition of a topology induced by a metric, this definition is a particular instance of the definition of a Borel $\sigma$-algebra on topological spaces. === Borel Set === {{:Definition:Borel Sigma-Algebra/Borel Set}}
Definition:Lebesgue Sigma-Algebra
Let $\tau$ be the Euclidean topology on $\R^n$. Let $\mathcal N$ be the collection of null sets in $\R^n$. The '''Lebesgue sigma-algebra''' on $\R^n$ is the sigma-algebra generated by the set $\tau \cup \mathcal N$. {{namedfor|Henri Léon Lebesgue}}   74sbdzjrem7wc3tly1t0hqfqiuwfr5y
Definition:Null Set
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. A set $N \in \Sigma$ is called a '''($\mu$-)null set'''  $\mu \left({N}\right) = 0$. === Family of Null Sets === The family of '''$\mu$-null sets''', $\left\{{N \in \Sigma: \mu \left({N}\right) = 0}\right\}$, is denoted $\mathcal N_{\mu}$.
Definition:Complex Modulus
Let $z = a + i b$ be a complex number, where $a, b \in \R$. Then the '''(complex) modulus of $z$''' is written $\cmod z$ and is defined as the square root of the sum of the squares of the real and imaginary parts: :$\cmod z := \sqrt {a^2 + b^2}$
Definition:Real-Valued Function
Let $f: S \to T$ be a function. Let $S_1 \subseteq S$ such that $\map f {S_1} \subseteq \R$. Then $f$ is said to be '''real-valued on $S_1$'''. That is, $f$ is defined as '''real-valued on $S_1$'''  the image of $S_1$ under $f$ lies entirely within the set of real numbers $\R$. A '''real-valued function''' is a function $f: S \to \R$ whose codomain is the set of real numbers $\R$. That is, $f$ is '''real-valued'''  it is '''real-valued''' over its entire domain.
Definition:Complex-Valued Function
Let $f: S \to T$ be a function. Let $S_1 \subseteq S$ such that $f \left({S_1}\right) \subseteq \C$. Then $f$ is defined as '''complex-valued on $S_1$'''. That is, $f$ is defined as '''complex-valued on $S_1$''' if the image of $S_1$ under $f$ lies entirely within the set of complex numbers $\C$. A '''complex-valued function''' is a function $f: S \to \C$ whose codomain is the set of complex numbers $\C$. That is $f$ is complex-valued iff it is complex-valued over its entire domain.
Definition:Usual Metric
Let $X$ be one of the standard number fields $\Q$, $\R$, $\C$. Let $X^n$ be a cartesian space on $X$. The '''usual metric''' on $X^n$ is the Euclidean metric on $X^n$: === Real Numbers === {{:Definition:Euclidean Metric/Real Vector Space}} === Rational Numbers === {{:Definition:Euclidean Metric/Rational Space}} === Complex Plane === {{:Definition:Euclidean Metric/Complex Plane}}
Definition:Absolute Continuity
Let $I \subseteq \R$ be a real interval. A real function $f: I \to \R$ is said to be '''absolutely continuous''' if it satisfies the following property: :For every $\epsilon > 0$ there exists $\delta > 0$ such that the following property holds: ::For every finite set of disjoint closed real intervals $\closedint {a_1} {b_1}, \dotsc, \closedint {a_n} {b_n} \subseteq I$ such that: :::$\displaystyle \sum_{i \mathop = 1}^n \size {b_i - a_i} < \delta$ ::it holds that: :::$\displaystyle \sum_{i \mathop = 1}^n \size {\map f {b_i} - \map f {a_i} } < \epsilon$
Definition:Uniform Continuity
=== Metric Spaces === {{:Definition:Uniform Continuity/Metric Space}} === Real Numbers === {{:Definition:Uniform Continuity/Real Numbers}}
Definition:Inner Product
Let $\C$ be the field of complex numbers. Let $\F$ be a subfield of $\C$. Let $V$ be a vector space over $\F$. An '''inner product''' is a mapping $\innerprod \cdot \cdot: V \times V \to \mathbb F$ that satisfies the following properties: {{begin-axiom}} {{axiom | n = 1         | lc= Conjugate Symmetry         | q = \forall x, y \in V         | m = \quad \innerprod x y = \overline {\innerprod y x} }} {{axiom | n = 2         | lc= Bilinearity         | q = \forall x, y \in V, \forall a \in \mathbb F         | m = \quad \innerprod {a x + y} z = a \innerprod x z + \innerprod y z }} {{axiom | n = 3         | lc= Non-Negative Definiteness         | q = \forall x \in V         | m = \quad \innerprod x x \in \R_{\ge 0} }} {{axiom | n = 4         | lc= Positiveness         | q = \forall x \in V         | m = \quad \innerprod x x = 0 \implies x = \mathbf 0_V }} {{end-axiom}} That is, an inner product is a semi-inner product with the additional condition $(4)$. If $\mathbb F$ is a subfield of the field of real numbers $\R$, it follows from Complex Number equals Conjugate iff Wholly Real that $\overline {\innerprod y x} = \innerprod y x$ for all $x, y \in V$. Then $(1)$ above may be replaced by: {{begin-axiom}} {{axiom | n = 1'         | lc= Symmetry         | q = \forall x, y \in V         | m = \innerprod x y = \innerprod y x }} {{end-axiom}} === Inner Product Space === {{:Definition:Inner Product Space}}
Definition:Inner Product Space
An '''inner product space''' is a vector space together with an associated inner product.
Definition:Isolated Point (Topology)
Let $T = \struct {S, \tau}$ be a topological space. === Isolated Point of Subset === {{:Definition:Isolated Point (Topology)/Subset}} === Isolated Point of Space === When $H = S$ the definition applies to the entire topological space $T = \struct {S, \tau}$: {{:Definition:Isolated Point (Topology)/Space}}
Definition:Everywhere Dense
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a subset. === Definition 1 === {{:Definition:Everywhere Dense/Definition 1}} === Definition 2 === {{:Definition:Everywhere Dense/Definition 2}}
Definition:Sequentially Compact Space
Let $T = \left({S, \tau}\right)$ be a topological space. Let $H \subseteq S$. Then $H$ is '''sequentially compact in $T$'''  every infinite sequence in $H$ has a subsequence which converges to a point in $S$.
Definition:Totally Bounded Metric Space
=== Definition 1 === {{:Definition:Totally Bounded Metric Space/Definition 1}} === Definition 2 === {{:Definition:Totally Bounded Metric Space/Definition 2}}
Definition:Interior (Topology)
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. === Definition 1 === {{:Definition:Interior (Topology)/Definition 1}} === Definition 2 === {{:Definition:Interior (Topology)/Definition 2}} === Definition 3 === {{:Definition:Interior (Topology)/Definition 3}}
Definition:Nowhere Dense
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. === Definition 1 === {{:Definition:Nowhere Dense/Definition 1}} === Definition 2 === {{:Definition:Nowhere Dense/Definition 2}}
Definition:Quotient Topology
Let $T = \left({S, \tau}\right)$ be a topological space. Let $\mathcal R \subseteq S \times S$ be an equivalence relation on $S$. Let $q_\mathcal R: S \to S / \mathcal R$ be the quotient mapping induced by $\mathcal R$. === Definition 1 === {{:Definition:Quotient Topology/Definition 1}} === Definition 2 === {{:Definition:Quotient Topology/Definition 2}}
Definition:Topological Property
Let $P$ be a property whose domain is the set of all topological spaces. Suppose that whenever $\map P T$ holds, then so does $\map P {T'}$, where $T$ and $T'$ are topological spaces which are homeomorphic. Then $P$ is known as a '''topological property'''. Loosely, a topological property is one which is preserved under homeomorphism.
Definition:Cover of Set
Let $S$ be a set. A '''cover for $S$''' is a set of sets $\CC$ such that: :$\displaystyle S \subseteq \bigcup \CC$ where $\bigcup \CC$ denotes the union of $\CC$.
Definition:Subcover
Let $S$ be a set. Let $\mathcal U$ be a cover for $S$. A '''subcover of $\mathcal U$ for $S$''' is a set $\mathcal V \subseteq \mathcal U$ such that $\mathcal V$ is also a cover for $S$. === Finite Subcover === {{:Definition:Subcover/Finite}} === Countable Subcover === {{:Definition:Subcover/Countable}}
Definition:Relatively Compact Subspace
Let $T = \left({S, \tau}\right)$ be a topological space. Let $T_H = \left({H, \tau_H}\right)$ be a subspace of $T$. Let $\operatorname{cl} \left({H}\right)$ be the closure of $H$ in $T$. Then $T_H$ is '''relatively compact in $T$'''  $\operatorname{cl} \left({H}\right)$ is compact.
Definition:Inscribe
Let a geometric figure $A$ be constructed in the interior of another geometric figure $B$ such that: :$(1): \quad$ $A$ and $B$ have points in common :$(2): \quad$ No part of $A$ is outside $B$. Then $A$ is '''inscribed''' inside $B$. === Circle in Polygon === {{:Definition:Inscribe/Circle in Polygon}} === Polygon in Circle === {{:Definition:Inscribe/Polygon in Circle}} === Polygon in Polygon === {{:Definition:Inscribe/Polygon in Polygon}}
Definition:Circumscribe
=== Circle around Polygon === {{:Definition:Circumscribe/Circle around Polygon}} === Polygon around Circle === {{:Definition:Circumscribe/Polygon around Circle}} === Polygon around Polygon === {{:Definition:Circumscribe/Polygon around Polygon}}
Definition:Fixed Point
Let $f: S \to T$ be a mapping. Then a '''fixed point''' (or '''fixed element''') '''of $S$ under $f$''' is an $x \in S$ such that $\map f x = x$.
Definition:Coefficient
A '''coefficient''' is a constant which is used in a particular context to be multiplied by a variable that is under consideration. === Binomial Coefficient === {{:Definition:Binomial Coefficient/Integers/Definition 3}} === Polynomial Coefficient === For the usage of this term in the context of polynomial theory: {{:Definition:Polynomial Coefficient}}
Definition:Lemma
A '''lemma''' is a statement which is proven during the course of reaching the proof of a theorem. Logically there is no qualitative difference between a '''lemma''' and a theorem. They are both statements whose value is either true or false. However, a '''lemma''' is seen more as a stepping-stone than a theorem in itself (and frequently takes a lot more work to prove than the theorem to which it leads). Some lemmas are famous enough to be named after the mathematician who proved them (for example: Abel's Lemma and Urysohn's Lemma), but they are still categorised as second-class citizens in the aristocracy of mathematics. :''Always the lemma, never the theorem.''
Definition:Euclidean Geometry
Euclidean geometry is the branch of geometry in which the parallel postulate applies. An assumption which is currently under question is whether or not ordinary space is itself Euclidean. '''Euclidean geometry''' adheres to Euclid's postulates.
Definition:Uniform Equivalence
=== Metric Spaces === {{:Definition:Uniform Equivalence/Metric Spaces}} === Metrics === {{:Definition:Uniform Equivalence/Metrics}}
Definition:A Priori
'''A priori''' knowledge is the sort of knowledge which comes from reason alone. That is, it does not require the exercise of experience to know it. For example: :''If Fred Bloggs has committed a crime, then he is guilty'' as opposed to: :''Fred Bloggs has committed the crime of usury.''
Definition:Clopen Set
Let $T = \struct {S, \tau}$ be a topological space. Let $X \subseteq S$ such that $X$ is both open in $T$ and closed in $T$. Then $X$ is described as '''clopen'''.
Definition:Connected (Topology)
=== Topological Space === {{Definition:Connected (Topology)/Topological Space}} === Set of Topological Space === {{Definition:Connected (Topology)/Set}} === Points in Topological Space === {{:Definition:Connected (Topology)/Points}}
Definition:Position
'''Position''' is an abstract geometrical concept that defines '''where something is'''. It is only possible to define the '''position''' of an object in relation to the '''position''' of another object. Even when one tries to consider an '''''absolute'' position''', one has to describe it in relation to a frame of reference, which itself needs to have a '''position''' (which ultimately is itself arbitrarily defined).
Definition:Angle Bisector
:300px Let $\angle ABC$ be an angle. The '''angle bisector''' of $\angle ABC$ is the straight line which bisects $\angle ABC$. In the above diagram, $BD$ is the '''angle bisector''' of $\angle ABC$. Thus $\angle ABD \cong \angle DBC$ and $\angle ABD + \angle DBC = \angle ABC$.
Definition:Darboux Function
Let $S \subseteq \R$. Let $f: S \to \R$ be a real function. Then $f$ is '''Darboux''' , given any $a, b \in S$ and $y \in \R$ such that $a < b$ and $y$ is between $\map f a$ and $\map f b$, there exists a $c \in S$ such that $a \le c \le b$ and $\map f c = y$. That is, for every '''intermediate value''' between $\map f a$ and $\map f b$, that value is the image of some '''intermediate value''' between $a$ and $b$.
Definition:Path (Topology)
Let $T = \struct {S, \tau}$ be a topological space. Let $I \subset \R$ be the closed real interval $\closedint a b$. A '''path in $T$''' is a continuous mapping $\gamma: I \to S$. The mapping $\gamma$ can be referred as: :a '''path (in $T$) joining $\map \gamma a$ and $\map \gamma b$''' or: :a '''path (in $T$) from $\map \gamma a$ to $\map \gamma b$'''. It is common to refer to a point $z = \map \gamma t$ as a '''point on the path $\gamma$''', even though $z$ is in fact on the image of $\gamma$. === Initial Point === {{:Definition:Path (Topology)/Initial Point}} === Final Point === {{:Definition:Path (Topology)/Final Point}} === Endpoint === {{:Definition:Path (Topology)/Endpoint}}
Definition:Congruence (Geometry)
In the field of Euclidean geometry, two geometric figures are '''congruent''' if they are, informally speaking, both "the same size and shape". That is, one figure can be overlaid on the other figure with a series of rotations, translations, and reflections. Specifically: : all corresponding angles of the '''congruent''' figures must have the same measurement : all corresponding sides of the '''congruent''' figures must be be the same length.
Definition:WLOG
Suppose there are several cases which need to be investigated. If the same argument can be used to dispose of two or more of these cases, then it is acceptable in a proof to pick just one of these cases, and announce this fact with the words: '''Without loss of generality, ...''', or just '''WLOG'''.
Definition:Sector
A '''sector''' of a circle is the area bounded by two radii and an arc.  :''{{:Definition:Euclid's Definitions - Book III/10 - Sector of Circle}}'' {{EuclidDefRefNocat|III|10|Sector of Circle}} In the diagram below, $BAC$ is a '''sector'''.  :300px In fact there are two '''sectors''', together making up the whole of the circle. When describing a sector by means of its three defining points (as in $BAC$ above), the convention is to report them in the following order: :$(1):\quad$ The point on the circumference at the end of the clockwise radius :$(2):\quad$ The point at the center of the circle :$(3):\quad$ The point on the circumference at the end of the anticlockwise radius Thus in the '''sector''' above, $BAC$ describes the '''sector''' indicated by $\theta$, while the '''sector''' comprising the rest of the circle is described by $CAB$. === Angle of Sector === {{:Definition:Sector/Angle}}
Definition:Component (Topology)
Let $T = \left({S, \tau}\right)$ be a topological space. Let the relation $\sim $ be defined on $T$ as follows: :$x \sim y$  $x$ and $y$ are connected in $T$. That is,  there exists a connected set of $T$ that contains both $x$ and $y$. === Definition 1 === {{:Definition:Component (Topology)/Definition 1}} === Definition 2 === {{:Definition:Component (Topology)/Definition 2}} === Definition 3 === {{:Definition:Component (Topology)/Definition 3}}
Definition:Totally Disconnected Space
A topological space $T = \struct {S, \tau}$ is a '''totally disconnected space'''  all components of $T$ are singletons.
Definition:Extremal Length
Let $\Gamma$ be a set of rectifiable curves in the complex plane $\C$. We consider conformal metrics of the form $\map \rho z \cmod {\d z}$, where: :$\rho: \C \to \hointr 0 \to$ is Borel measurable and: :the area $\displaystyle \map A \rho := \iint \rho^2 \paren {x + i y} \rd x \rd y$ is finite and positive. Every $\gamma \in \Gamma$ has a distance function with respect to such a metric, defined by: :$\displaystyle \map L {\gamma, \rho} := \int_\gamma \map \rho z \size {\d z}$ We define: :$\displaystyle \map L {\Gamma, \rho} := \inf_{\gamma \mathop \in \Gamma} \map L {\gamma, \rho}$ and: :$\displaystyle \map \lambda \Gamma := \sup_\rho \frac {\map L {\Gamma, \rho}^2} {\map A \rho}$ The quantity $\map \lambda \Gamma$ is called the '''extremal length''' of the curve family $\Gamma$. Its reciprocal: :$\mod \Gamma := \dfrac 1 {\map \lambda \Gamma}$ is called the '''modulus''' of $\Gamma$.
Definition:Converse Statement
The '''converse''' of the conditional: :$p \implies q$ is the statement: :$q \implies p$
Definition:Complete Metric Space
=== Definition 1 === {{:Definition:Complete Metric Space/Definition 1}} === Definition 2 === {{:Definition:Complete Metric Space/Definition 2}}
Definition:Golden Mean
=== Definition 1 === {{:Definition:Golden Mean/Definition 1}} === Definition 2 === {{:Definition:Golden Mean/Definition 2}}
Definition:Teichmüller Annulus
Let $R \in \R_{>0}$. The set: :$A := \C \setminus \paren {\closedint{-1} 0 \cup \hointr R {+\infty} }$ is a '''Teichmüller annulus'''. The modulus of $A$ is denoted $\map \Lambda R$.
Definition:Degenerate Connected Set
Let $T = \left({S, \tau}\right)$ be a topological space. Let $H \subseteq S$ be a subset of $T$. $H$ is a '''degenerate connected set''' of $T$  it is a connected set of $T$ containing exactly one element. === Non-Degenerate Connected Set === {{:Definition:Degenerate Connected Set/Non-Degenerate}} === Degenerate Connected Space === When $H = S$ itself, the entire space can be referred to in this way: {{:Definition:Degenerate Connected Set/Topological Space}}
Definition:Riemann Zeta Function
The '''Riemann Zeta Function''' $\zeta$ is the complex function defined on the half-plane $\map \Re s > 1$ as the series: :$\displaystyle \map \zeta s = \sum_{n \mathop = 1}^\infty \frac 1 {n^s}$
Definition:Cevian
Let $\triangle ABC$ be a triangle. :400px A '''cevian''' is a line segment from a vertex of the triangle to the opposite side. In the above diagram, $CD$ is a '''cevian'''.
Definition:Median of Triangle
Let $\triangle ABC$ be a triangle. :400px A '''median''' is a cevian which bisects the opposite.
Definition:Planar Graph
A '''planar graph''' is a graph which can be drawn in the plane (for example, on a piece of paper) without any of the edges crossing over, that is, meeting at points other than the vertices. This is a '''planar graph''': :300px === Face === {{:Definition:Planar Graph/Face}}
Definition:Graph (Graph Theory)/Vertex
Let $G = \struct {V, E}$ be a graph. The '''vertices''' (singular: '''vertex''') are the elements of $V$. Informally, the '''vertices''' are the points that are connected by the edges.
Definition:Area
'''Area''' is the measure of the extent of a surface. It has $2$ dimensions and is specified in units of length squared.
Definition:Euler Characteristic of Finite Graph
Let $X = \struct {V, E}$ be a graph. Let $X$ be embedded in a surface. The '''Euler characteristic''' of $X$ is written $\map \chi X$ and is defined as: :$\map \chi x = v - e + f$ where: :$v = \size V$ is the number of vertices :$e = \size E$ is the number of edges :$f$ is the number of faces. === Generalized Formula ===
Definition:Deleted Neighborhood
=== Real Analysis === {{:Definition:Deleted Neighborhood/Real Analysis}} === Complex Analysis === {{:Definition:Deleted Neighborhood/Complex Analysis}} === Metric Space === {{:Definition:Deleted Neighborhood/Metric Space}} === Normed Vector Space === {{:Definition:Deleted Neighborhood/Normed Vector Space}} === Topology === {{:Definition:Deleted Neighborhood/Topology}}
Definition:Depressed Polynomial
Let $f \left({x}\right)$ be a polynomial over a field $k$: :$f \left({x}\right) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots + a_1 x + a_0$ If $a_{n-1} = 0_k$, then we call $f$ a '''depressed polynomial'''. It has been suggested that a polynomial with further zero terms might be referred to as "downright despondent", though this convention has yet to gain widespread usage by the community. {{refactor|Put the below into its own page}}
Definition:Cubic Equation
A '''cubic equation''' is a polynomial equation of the form: : $a x^3 + b x^2 + c x + d = 0$
Definition:Distinct
=== Of Two or More Objects === {{:Definition:Distinct/Plural}} === Of a Single Object === {{:Definition:Distinct/Singular}}
Definition:Quartic Equation
A '''quartic equation''' is a polynomial equation of the form: :$a x^4 + b x^3 + c x^2 + d x + e$
Definition:Quintic Equation
Let $\map f x = a x^5 + b x^4 + c x^3 + d x^2 + e x + f$ be a polynomial function over a field $\mathbb k$ of degree $5$. Then the equation $\map f x = 0$ is the general '''quintic equation''' over $\mathbb k$.
Definition:Mersenne Prime
A '''Mersenne prime''' is a Mersenne number which happens to be prime. That is, it is a prime number of the form $2^p - 1$. The number $2^p - 1$ is, in this context, often denoted $M_p$.
Definition:Net (Metric Space)
Let $M = \struct {A, d}$ be a metric space. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. An '''$\epsilon$-net for $M$''' is a subset $S \subseteq A$ such that: :$\displaystyle A \subseteq \bigcup_{x \mathop \in S} \map {B_\epsilon} x$ where $\map {B_\epsilon} x$ denotes the open $\epsilon$-ball of $x$ in $M$.
Definition:Lebesgue Number
Let $M = \left({A, d}\right)$ be a metric space. Let $\mathcal U$ be an open cover of $M$. A fixed strictly positive real number $\epsilon \in \R_{>0}$ is called a '''Lebesgue number for $\mathcal U$''' : :$\forall x \in A: \exists U \left({x}\right) \in \mathcal U: B_\epsilon \left({x}\right) \subseteq U \left({x}\right)$ where $B_\epsilon \left({x}\right)$ is the open $\epsilon$-ball of $x$ in $M$.
Definition:Circular Definition
A '''circular definition''' is a definition of a concept $A$ in terms of another concept $B$ such that the definition of $B$ itself refers (either directly or indirectly) to the definition of $A$. It is a truism that '''circular definitions''' are always to be avoided.
Definition:Basic Null Sequence
The sequences defined as: :$\sequence {\dfrac 1 {n^r} }_{n \mathop \in \N}$ for $r \in \R: r > 0$ :$\sequence {\alpha^n}_{n \mathop \in \N}$ for $\alpha \in \C: \cmod {\alpha} < 1$ are known as the '''basic null sequences'''.
Definition:Dot Product
Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $\mathbf V$ of $n$ dimensions: :$\mathbf a = \displaystyle \sum_{k \mathop = 1}^n a_k \mathbf e_k$ :$\mathbf b = \displaystyle \sum_{k \mathop = 1}^n b_k \mathbf e_k$ where $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is the standard ordered basis of $\mathbf V$.
Definition:Legendre Symbol
Let $p$ be an odd prime. Let $a \in \Z$ be an integer.
Definition:Diophantine Equation
A '''Diophantine equation''' is an indeterminate polynomial equation that allows the variables to take integer values only.
Definition:Perfect Number
=== Definition 1 === {{:Definition:Perfect Number/Definition 1}} === Definition 2 === {{:Definition:Perfect Number/Definition 2}} === Definition 3 === {{:Definition:Perfect Number/Definition 3}} === Definition 4 === {{:Definition:Perfect Number/Definition 4}}
Definition:Fermat Prime
A '''Fermat prime''' is a Fermat number, that is a number of the form $2^{\paren {2^n} } + 1$, which happens to be prime.
Definition:Mersenne Number
A '''Mersenne number''' is a natural number of the form $2^p - 1$, where $p$ is prime. The number $2^p - 1$, in this context, can be denoted $M_p$.
Definition:Carmichael Number
An integer $n > 0$ is a '''Carmichael number''' : : $(1): \quad n$ is composite : $(2): \quad \forall a \in \Z: a \perp n: a^n \equiv a \pmod n$, or, equivalently, that $a^{n - 1} \equiv 1 \pmod n$. That is, a '''Carmichael number''' is a composite number $n$ which satisfies $a^n \equiv a \pmod n$ for all integers $a$ which are coprime to it.