hcju/proofwikips huggingface dataset:
Definition:Integral Element of Algebra/Definition 4
The element $b$ is '''integral''' over $A$  there exists a [[Definition:Faithful Module|faithful $A[b]$-module]] whose restriction of scalars to $A$ is finitely generated.
Definition:Presheaf on Topological Space
Let $T = \left({S, \tau}\right)$ be a topological space. Let $\mathbf C$ be a category. === Definition 1 === {{:Definition:Presheaf on Topological Space/Definition 1}} === Definition 2 === {{:Definition:Presheaf on Topological Space/Definition 2}}
Definition:Sheaf on Topological Space
Let $T = \struct {S, \tau}$ be a topological space. Let $\mathbf C$ be a category. === Definition 1 === {{:Definition:Sheaf on Topological Space/Definition 1}} === Definition 2 === {{:Definition:Sheaf on Topological Space/Definition 2}} === Definition 3 === {{:Definition:Sheaf on Topological Space/Definition 3}}
Definition:Morphism of Presheaves on Topological Space
Let $X$ be a topological space. Let $\mathbf C$ be a category. Let $\mathcal F$ and $\mathcal G$ be $\mathbf C$-valued presheaves on $X$. === Definition 1 === {{definition wanted|using restrictions}} === Definition 2 === A '''morphism of presheaves''' from $\mathcal F$ to $\mathcal G$ is a natural transformation $\mathcal F \to \mathcal G$.
Definition:Abelian Sheaf
Let $X$ be a topological space. === Definition 1 === An '''abelian sheaf''' on $X$ is a sheaf on $X$ with values in the category of abelian groups. === Definition 2 === An '''abelian sheaf''' on $X$ is a sheaf of $\underline{\mathbb{Z}}$-modules on $X$, where $\underline{\mathbb{Z}}$ is the constant sheaf with value the ring of integers $\Z$.
Definition:Stalk of Presheaf
Let $T = \left({S, \tau}\right)$ be a topological space. Let $\mathbf C$ be a category which has all small inductive limits. Let $\mathcal F$ be a $\mathbf C$-valued presheaf on $T$. Let $x \in S$. The '''stalk''' $\mathcal F_x$ of $\mathcal F$ at $x$ is the inductive limit: :$\displaystyle\varinjlim_{U \ni x} \mathcal F \left({U}\right)$ over all open $U \subseteq S$ containing $x$.
Definition:Morphism of Sheaves
Let $X$ be a topological space. Let $\mathbf C$ be a category. Let $\mathcal F$ and $\mathcal G$ be $\mathbf C$-valued sheaves on $X$. A '''sheaf morphism''' from $\mathcal F$ to $\mathcal G$ is a presheaf morphism $\mathcal F \to\mathcal G$.
Definition:Category of Sheaves on Topological Space
Let $X$ be a topological space.  Let $\mathbf C$ be a category. The '''category of $\mathbf C$-valued sheaves on $X$''' is the category with: {{DefineCategory |ob = $\mathbf C$-valued sheaves on $X$ |mor = morphisms of presheaves |id = identity morphisms on presheaves |comp = composition of morphisms of presheaves }}
Definition:Category of Presheaves on Topological Space
Let $X$ be a topological space.  Let $\mathbf C$ be a category. The '''category of $\mathbf C$-valued presheaves on $X$''' is the category with: {{DefineCategory |ob = $\mathbf C$-valued presheaves on $X$ |mor = morphisms of presheaves |id = identity morphisms on presheaves |comp = composition of morphisms of presheaves }}
Definition:Ringed Space
A '''ringed space''' is a pair $\struct {X, \OO_X}$ where: :$X$ is a topological space :$\OO_X$ is a sheaf of rings with unity on $X$.
Definition:Locally Ringed Space
A '''locally ringed space''' is a ringed space $\struct {X, \OO_X}$ such that the stalks of the structure sheaf $\OO_X$ are local rings.
Definition:Natural Transformation/Covariant Functors
Let $F, G : \mathbf C \to \mathbf D$ be covariant functors. A '''natural transformation''' $\eta$ from $F$ to $G$ is a mapping on $\mathbf C$ such that: :$(1): \quad$ For all $x \in \mathbf C$, $\eta_x$ is a morphism from $\map F x$ to $\map G x$. :$(2): \quad$ For all $x, y \in C$ and morphism $f: x \to y$, the following diagram commutes: ::$\xymatrix{ \map F x \ar[d]^{\eta_x} \ar[r]^{\map F f} & \map F y \ar[d]^{\eta_y} \\ \map G x \ar[r]^{\map G f}                 & \map G y}$
Definition:Natural Transformation/Contravariant Functors
Let $F, G: \mathbf C \to \mathbf D$ be contravariant functors. A '''natural transformation''' $\eta$ from $F$ to $G$ is a mapping on $\mathbf C$ such that: :$(1): \quad$ For all $x \in \mathbf C$, $\eta_x$ is a morphism from $\map F x$ to $\map G x$. :$(2): \quad$ For all $x, y \in C$ and morphism $f: x \to y$, the following diagram commutes: ::$\xymatrix{ \map F x \ar[d]^{\eta_x} & \map F y \ar[d]^{\eta_y} \ar[l]^{\map F f}  \\ \map G x                 & \map G y \ar[l]^{\map G f} }$
Definition:Natural Transformation
Let $\mathbf C$ and $\mathbf D$ be categories. === Covariant Functors === {{Definition:Natural Transformation/Covariant Functors}} === Contravariant Functors === {{Definition:Natural Transformation/Contravariant Functors}}
Definition:Natural Isomorphism
Let $\mathbf C$ and $\mathbf D$ be categories. === Covariant Functors === {{Definition:Natural Isomorphism between Covariant Functors}} === Contravariant Functors === {{Definition:Natural Isomorphism between Contravariant Functors}}
Definition:Natural Isomorphism between Covariant Functors
Let $F, G : \mathbf C \to \mathbf D$ be covariant functors. === Definition 1 === A '''natural isomorphism''' from $F$ to $G$ is a natural transformation $\eta : F \to G$ such that for all $x\in \mathbf C$, $\eta_x : F(x) \to G(x)$ is an isomorphism. === Definition 2 === A '''natural isomorphism''' from $F$ to $G$ is an isomorphism in the functor category $\operatorname{Funct}(\mathbf C, \mathbf D)$, that is, a natural transformation $\eta : F \to G$ for which there exists a natural transformation $\xi : G \to F$ such that the compositions $\xi \circ \eta = 1_F$ and $\eta \circ \xi = 1_G$ are identity natural transformations.
Definition:Hom Functor
Let $\mathbf{Set}$ be the category of sets. Let $\mathbf C$ be a locally small category. === Hom Bifunctor === {{Definition:Hom Bifunctor}} === Covariant Hom Functor === {{Definition:Covariant Hom Functor}} === Contravariant Hom Functor === {{Definition:Contravariant Hom Functor}} {{definition wanted|for enriched categories}}
Definition:Covariant Hom Functor
Let $C \in \mathbf C_0$ be an object of $\mathbf C$. The '''covariant hom functor based at $C$''', $\operatorname{Hom}_{\mathbf C} \left({C, \cdot}\right): \mathbf C \to \mathbf{Set}$, is the covariant functor defined by: {{begin-axiom}} {{axiom|lc= Object functor:        |m = \operatorname{Hom}_{\mathbf C} \left({C, D}\right) = \operatorname{Hom}_{\mathbf C} \left({C, D}\right) }} {{axiom|lc= Morphism functor:        |m = \operatorname{Hom}_{\mathbf C} \left({C, f}\right): \operatorname{Hom}_{\mathbf C} \left({C, A}\right) \to \operatorname{Hom}_{\mathbf C} \left({C, B}\right), g \mapsto f \circ g        |rc= for $f: A \to B$ }} {{end-axiom}} where $\operatorname{Hom}_{\mathbf C} \left({C, D}\right)$ denotes a hom set. Thus, the morphism functor is defined to be postcomposition.
Definition:Hom Bifunctor
The '''hom bifunctor''' on $\mathbf C$ is the covariant functor $\map {\operatorname {Hom} } {-, -} : \mathbf C^{\operatorname {op} } \times \mathbf C \to \mathbf {Set}$ from the product with the opposite category to the category of sets such that: :$(1): \quad \map {\operatorname {Hom} } {a, b}$ is the hom class :$(2): \quad$ If $\tuple {f^{\operatorname {op} }, g}: \tuple {a, b} \to \tuple {c, d}$ is a morphism, $\map {\operatorname {Hom} } {f^{\operatorname{op} }, g}: \map {\operatorname {Hom} } {a, b} \to \map {\operatorname {Hom} } {c, d}$ is $f_* \circ g^*$, the postcomposition with $g$ composed with the precomposition with $f$.
Definition:Lehmer's Electromechanical Sieve
rightLehmer's Electromechanical Sieve '''Lehmer's electromechanical sieve''' is a device which was invented for the purpose of solving number theory problems, such as: : determining the prime decomposition of integers : determining whether an integer is prime. It consists of: : a mechanical assemblage of gearwheels : photo detector : an electronic amplifier. It is currently located at the [http://www.computerhistory.org/ Computer History Museum], where its lot number is $\text X 85.82$. {{NamedforDef|Derrick Henry Lehmer|cat = Lehmer D H}}
Definition:Category of Abelian Groups
The '''category of abelian groups''' is the category $\mathbf{Ab}$ with: {{DefineCategory |ob = abelian groups |mor = group homomorphisms |id = identity mappings |comp = composition of mappings }}
Definition:Category of Open Sets
Let $T = \left({S, \tau}\right)$ be a topological space. === Definition 1 === {{:Definition:Category of Open Sets/Definition 1}} === Definition 2 === {{:Definition:Category of Open Sets/Definition 2}}
Definition:Direct Image of Presheaf
Let $\mathbf C$ be a category. Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces. Let $f : T_1 \to T_2$ be continuous. Let $\mathcal F$ be a $\mathbf C$-valued presheaf on $T_1$. === Definition 1 === {{:Definition:Direct Image of Presheaf/Definition 1}} === Definition 2 === {{:Definition:Direct Image of Presheaf/Definition 2}}
Definition:Direct Image of Sheaf
Let $X$ be a topological space. Let $\mathbf C$ be a category. Let $\mathcal F$ be a $\mathbf C$-valued sheaf on $X$. The '''direct image''' of $\mathcal F$ is the direct image of the presheaf $\mathcal F$.
Definition:Étalé Space of Presheaf
Let $X$ be a topological space. Let $\mathbf C$ be a category. Let $\mathcal F$ be a $\mathbf C$-valued presheaf on $X$. The '''étalé space''' of $\mathcal F$ is the pair $\struct {\map {\operatorname {\acute Et} } {\mathcal F}, \pi}$ where: :$\map {\operatorname {\acute Et} } {\mathcal F}$ is the disjoint union $\displaystyle \bigsqcup_{x \mathop \in X} \mathcal F_x$ of stalks of $\mathcal F$ :$\pi: \map {\operatorname {\acute Et} } {\mathcal F} \to X$ is the canonical projection. === Topology on Étalé Space === {{Definition:Topology on Étalé Space of Presheaf}}
Definition:Associated Section of Étalé Space
Let $T = \struct {S, \tau}$ be a topological space. Let $\mathcal F$ be a presheaf of sets. Let $\map {\operatorname {\acute Et} } {\mathcal F}$ be its étalé space. Let $U \subseteq S$ be open in $T$. Let $s \in \map {\mathcal F} U$ be a section. The '''associated section''' of $\map {\operatorname {\acute Et} } {\mathcal F}$ is the mapping: :$\overline s : U \to \map {\operatorname {\acute Et} } {\mathcal F} : x \mapsto \tuple {x, s_x}$ where $s_x$ is the image in the stalk at $x$.
Definition:Topology on Étalé Space of Presheaf
The '''topology on $\map {\operatorname {\acute Et} } {\mathcal F}$''' is the final topology with respect to the sections associated to elements of $\map {\mathcal F} U$ with $U \subseteq S$ open.
Definition:Trapdoor Function
A '''trapdoor function''' is a mapping $f$ which is: :for a given $x$ in the domain of $f$ is easy to calculate  :for a given $y$ in the image of $f$ is hard to calculate unless a specific piece of information $t$ is supplied, in which case it is easy. That piece of information is known as the '''trapdoor'''.
Definition:Section of Étalé Space
Let $T = \left({S, \tau}\right)$ be a topological space. Let $\mathcal F$ be a presheaf of sets on $T$. Let $\left({\operatorname{\acute Et} \left({\mathcal F}\right), \pi}\right)$ be its étalé space. Let $U \subseteq S$ be open. === Definition 1 === {{:Definition:Section of Étalé Space/Definition 1}} === Definition 2 === {{:Definition:Section of Étalé Space/Definition 2}}
Definition:Sheafification
Let $X$ be a topological space.  Let $\mathbf C$ be a category. Let $\mathcal F$ be a $\mathbf C$-valued presheaf on $X$. === Definition 1 === {{definition wanted|as a representation of a functor}} === Definition 2 === Let $\mathcal F$ be a presheaf of sets. The '''sheafification''' of $\mathcal F$ is the sheaf of continuous sections of its étalé space $\map {\operatorname {\acute Et} } {\mathcal F}$.
Definition:Inverse Image Presheaf
Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces. Let $f: T_1 \to T_2$ be continuous. Let $\mathbf C$ be a category which has all small inductive limits. Let $\mathcal F$ be a $\mathbf C$-valued presheaf on $T_2$. The '''inverse image presheaf''' of $\mathcal F$ via $f$ is the presheaf $f^{-1}_{\operatorname{Psh}} \mathcal F$ on $T_1$ with: : $\left({f^{-1}_{\operatorname{Psh}} \mathcal F}\right) \left({U}\right) = \displaystyle \varinjlim_{V \supseteq f \left({U}\right)} \mathcal F \left({V}\right)$ where the inductive limit goes over open $V \subseteq Y$ : $\operatorname{res}^U_W$ is the induced map on the inductive limit of the subset $\left\{ {V: V \supseteq f \left({U}\right)}\right\} \subseteq \left\{ {V : V \supseteq f \left({W}\right)}\right\}$
Definition:Unique up to Unique Isomorphism
Let $\mathbf C$ be a category. Let $S \subseteq \operatorname{Ob}(\mathbf C)$ be a subclass of its objects. The class $S$ is '''unique up to unique isomorphism'''  for all objects $s,t \in S$ there is a unique isomorphism from $s$ to $t$.
Definition:Zero Object
Let $\mathbf C$ be a category. A '''zero object''' of $\mathbf C$ is an object which is initial and terminal.
Definition:RSA 129
'''RSA $129$''' is the name given to the semiprime: :$114 \, 381 \, 625 \, 757 \, 888 \, 867 \, 669 \, 235 \, 779 \, 976 \, 146 \, 612 \, 010 \, 218 \, 296 \, 721 \, 242 \, 362 \, 562 \, 561 \, 842 \, 935 \, 706 \, 935 \, 245 \, 733 \, 897 \, 830 \, 597 \, 123 \, 563 \, 958 \, 705 \, 058 \, 989 \, 075 \, 147 \, 599 \, 290 \, 026 \, 879 \, 543 \, 541$ Its factors are: :$3 \, 490 \, 529 \, 510 \, 847 \, 650 \, 949 \, 147 \, 849 \, 619 \, 903 \, 898 \, 133 \, 417 \, 764 \, 638 \, 493 \, 387 \, 843 \, 990 \, 820 \, 577$ and: :$32 \, 769 \, 132 \, 993 \, 266 \, 709 \, 549 \, 961 \, 988 \, 190 \, 834 \, 461 \, 413 \, 177 \, 642 \, 967 \, 992 \, 942 \, 539 \, 798 \, 288 \, 533$
Definition:Tychonoff Topology/Natural Sub-Basis
The '''natural sub-basis on $X$''' is defined as: :$\mathcal S = \set {\pr_i^{-1} \sqbrk U: i \in I, \, U \in \tau_i}$
Definition:Tychonoff Topology/Natural Basis
The '''natural basis on $X$''' is defined as the basis generated by $\SS$.
Definition:RSA 130
'''RSA $130$''' is the name given to the semiprime: :$1 \, 807 \, 082 \, 088 \, 687 \, 404 \, 805 \, 951 \, 656 \, 164 \, 405 \, 905 \, 566 \, 278 \, 102 \, 516 \, 769 \, 401 \, 349 \, 170 \, 127 \, 021 \, 450 \, 056 \, 662 \, 540 \, 244 \, 048 \, 387 \, 341 \, 127 \, 590 \, 812 \, 303 \, 371 \, 781 \, 887 \, 966 \, 563 \, 182 \, 013 \, 214 \, 880 \, 557$ Its factors are: :$39 \, 685 \, 999 \, 459 \, 597 \, 454 \, 290 \, 161 \, 126 \, 162 \, 883 \, 786 \, 067 \, 576 \, 449 \, 112 \, 810 \, 064 \, 832 \, 555 \, 157 \, 243$ and: :$45 \, 534 \, 498 \, 646 \, 735 \, 972 \, 188 \, 403 \, 686 \, 897 \, 274 \, 408 \, 864 \, 356 \, 301 \, 263 \, 205 \, 069 \, 600 \, 999 \, 044 \, 599$
Definition:Mersenne Prime/Index
The '''index''' of the '''Mersenne prime''' $M_p = 2^p - 1$ is the (prime) number $p$.
Definition:Direct Limit
Let $\mathbf C$ be a category. Let $I$ be an upward directed set. Let $\mathbf I$ be its order category. Let $D : \mathbf I \to \mathbf C$ be a diagram. The '''direct limit''' of $D$ is the colimit of the diagram $D$.
Definition:Contravariant Hom Functor
Let $C \in \mathbf C_0$ be an object of $\mathbf C$. The '''contravariant hom functor based at $C$''': : $\operatorname{Hom}_{\mathbf C} \left({\cdot, C}\right): \mathbf C \to \mathbf{Set}$ is the covariant functor defined by: {{begin-axiom}} {{axiom | lc= Object functor:         | m = \operatorname{Hom}_{\mathbf C} \left({B, C}\right) = \operatorname{Hom}_{\mathbf C} \left({B, C}\right) }} {{axiom | lc= Morphism functor:         | m = \operatorname{Hom}_{\mathbf C} \left({f, C}\right): \operatorname{Hom}_{\mathbf C} \left({B, C}\right) \to \operatorname{Hom}_{\mathbf C} \left({A, C}\right), g \mapsto g \circ f         | rc= for $f: A \to B$ }} {{end-axiom}} where $\operatorname{Hom}_{\mathbf C} \left({B, C}\right)$ denotes a hom set. Thus, the morphism functor is defined to be precomposition.
Definition:Representable Functor
Let $\mathbf C$ be a locally small category. Let $\mathbf{Set}$ be the category of sets. Let $F : \mathbf C \to \mathbf{Set}$ be a covariant functor. Then $F$ is '''representable''' if there exists an object $a\in \mathbf C$ such that $F$ is naturally isomorphic to the covariant hom functor $\operatorname{Hom}(a,-)$. {{DefinitionWanted|contravariant}}
Definition:Yoneda Functor/Yoneda Embedding
The '''Yoneda embedding''' of $C$ is the covariant functor $h_- : C \to \left[{C^{\operatorname{op}}, \mathbf{Set} }\right]$ which sends: : an object $X\in C$ to the contravariant hom-functor $h_X = \operatorname{Hom} \left({-, X}\right)$ : a morphism $f : X \to Y$ to the postcomposition natural transformation $h_f : \operatorname{Hom} \left({-, X}\right) \to \operatorname{Hom} \left({-, Y}\right)$
Definition:Yoneda Functor/Contravariant
The '''contravariant Yoneda functor''' of $C$ is the contravariant functor $h^- : C \to \left[{C, \mathbf{Set} }\right]$ which sends : an object $X \in C$ to the covariant hom-functor $h^X = \operatorname{Hom} \left({X, -}\right)$ : a morphism $f : X \to Y$ to the precomposition natural transformation $h^f : \operatorname{Hom} \left({Y, -}\right) \to \operatorname{Hom} \left({X, -}\right) : g \mapsto g \circ f$
Definition:Yoneda Functor
Let $C$ be a locally small category. Let $C^{\operatorname{op}}$ be its opposite category. Let $\mathbf{Set}$ be the category of sets. Let $\left[{C^{\operatorname{op}}, \mathbf{Set} }\right]$ be the functor category between them. === Yoneda Embedding === {{Definition:Yoneda Embedding}} === Contravariant Yoneda Functor === {{Definition:Contravariant Yoneda Functor}}
Definition:Whiskering
Let $B,C,D,E$ be categories. Let $F, G : C \to D$ be covariant functors. Let $\eta : F \to G$ be a natural transformation. === Natural Transformation followed by Functor === Let $H : D \to E$ be a covariant functor. The '''right whiskering''' of $H$ and $\eta$ is the natural transformation $H\eta : H \circ F \to H \circ G$ between compositions of functors defined by $(H\eta)_A = H(\eta_A)$ for $A\in C$. === Functor followed by Natural Transformation === Let $K : B \to C$ be a covariant functor. The '''left whiskering''' of $\eta$ and $K$ is the natural transformation $\eta K : F \circ K \to G \circ K$ between compositions of functors defined by $(\eta K)_A = \eta_{K(A)}$ for $A\in B$.
Definition:Functor Evaluation Bifunctor
Let $C$ and $D$ be categories. Let $\operatorname{Funct}(C, D)$ be their covariant functor category. Let $\operatorname{Funct}(C, D) \times C$ be the product category. The '''evaluation bifunctor''' $\operatorname{ev} : \operatorname{Funct}(C, D) \times C \to D$ is the covariant functor that sends: *an object $(F, a)$ to $F(a)$ *a morphism $(\eta, f) : (F, a) \to (G, b)$ to $G(f) \circ \eta_a = \eta_b \circ F(f)$
Definition:Vertical Composition of Natural Transformations
Let $C$ and $D$ be categories. Let $F,G,H : C \to D$ be covariant functors. Let $\eta : F \to G$ and $\xi : G \to H$ be natural transformations. The '''vertical composition''' of $\eta$ and $\xi$ is the natural transformation $\xi \circ \eta : F \Rightarrow H$ with $(\xi \circ \eta)_A = \xi_A \circ \eta_A$ for $A\in C$.
Definition:Titanic Prime
A '''titanic prime''' is a prime number with $1000$ digits or more.
Definition:Representation of Functor
Let $\mathbf C$ be a locally small category. Let $\mathbf{Set}$ be the category of sets. Let $F : \mathbf C \to \mathbf{Set}$ be a covariant functor. A '''representation''' of $F$ is a pair $(A, \eta)$ where $\eta : \operatorname{Hom}(A, -) \to F$ is a natural isomorphism with the covariant hom functor of $A$.
Definition:Fully Faithful Functor
Let $C$ and $D$ be categories. Let $F : C \to D$ be a covariant functor. The functor $F$ is '''fully faithful'''  it is full and faithful.
Definition:Embedding of Categories
Let $C$ and $D$ be categories. Let $F : C \to D$ be a functor. === Definition 1 === The functor $F$ is an '''embedding'''  it is: * injective on objects * faithful === Definition 2 === The functor $F$ is an '''embedding'''  it is injective on morphisms. === Definition 3 === The functor $F$ is an '''embedding'''  it is a monomorphisms in the category of categories.
Definition:Group Category
Let $G$ be a group. The '''group category''' of $G$ is the monoid category of $G$.
Definition:Primorial Prime
A '''primorial prime''' is a prime number $p$ such that $p + 1$ is a primorial. That is, a prime number which is $1$ less than a primorial.
Definition:Covariant Power Set Functor
The '''(covariant) power set functor''' $\mathcal P : \mathbf{Set} \to \mathbf{Set}$ is the covariant functor which sends: * An object $x$ to its power set $\mathcal P(x)$. * A morphism $f : x \to y$ to the direct image mapping $\mathcal P(f) : \mathcal P(x) \to \mathcal P(y)$.
Definition:Contravariant Power Set Functor
The '''contravariant power set functor''' $\overline{\mathcal P} : \mathbf{Set} \to \mathbf{Set}$ is the contravariant functor which sends: * An object $x$ to its power set $\mathcal P(x)$. * A morphism $f : x \to y$ to the inverse image mapping $\overline{\mathcal P}(f) : \mathcal P(y) \to \mathcal P(x)$.
Definition:Power Set Functor
=== Covariant === {{Definition:Covariant Power Set Functor}} === Contravariant === {{Definition:Contravariant Power Set Functor}} {{definition wanted|Create the analogous definition for relations.}}
Definition:Gigantic Prime
A '''gigantic prime''' is a prime number with $10 \, 000$ digits or more.
Definition:Mapping Induced on Powerset
The '''mapping induced on a powerset''' can be either of two concepts: == Direct Image Mapping == {{:Definition:Direct Image Mapping}} == Inverse Image Mapping == {{:Definition:Inverse Image Mapping}}
Definition:Proth Prime
A '''Proth prime''' is a Proth number which is also a prime.
Definition:Proth Number
A '''Proth number''' is a natural number of the form: :$N = k \times 2^n + 1$ where: :$k$ is an odd integer :$n$ is a positive integer :$2^n > k$ The condition is included, otherwise every odd integer greater than $1$ would be a '''Proth number'''.
Definition:Hyperoperation
=== Hyperoperation Sequence === {{:Definition:Hyperoperation/Sequence}} === $n$th Hyperoperation === {{:Definition:Hyperoperation/Nth Hyperoperation}}
Definition:Hyperoperation/Sequence
The '''hyperoperation sequence''' is the sequence $\left\langle{H_n}\right\rangle$ of binary operations $H_n : \Z_{\ge 0} \times \Z_{\ge 0} \to \Z_{\ge 0}$, defined as: $\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases} y + 1 & : n = 0 \\ x & : n = 1, y = 0 \\ 0 & : n = 2, y = 0 \\ 1 & : n > 2, y = 0 \\ H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \end{cases}$
Definition:Hyperoperation/Nth Hyperoperation
The $n$th term of $\left\langle{H_n}\right\rangle$, which is the binary operation $H_n : \Z_{\ge 0} \times \Z_{\ge 0} \to \Z_{\ge 0}$, is known as the '''$n$th hyperoperation'''.
Definition:Power (Algebra)/Integer/Knuth Notation
In certain contexts in number theory, the symbol $\uparrow$ is used to denote the (usually) integer power operation: :$x \uparrow y := x^y$ This notation is usually referred to as '''Knuth (uparrow) notation'''.
Definition:Gigaplex
A '''gigaplex''' is defined to be $10$ to the power of a (short scale) billion, that is $10^{1 \, 000 \, 000 \, 000}$.
Definition:Billion
=== Short Scale === {{:Definition:Billion/Short Scale}} === Long Scale === {{:Definition:Billion/Long Scale}}
Definition:Billion/Short Scale
'''Billion''' is a name for $10^9$ in the short scale system: :'''One billion''' $= 10^{3 \times 2 + 3}$
Definition:Billion/Long Scale
'''Billion''' is a name for $10^{12}$ in the long scale system: :'''One billion''' $= 10^{6 \times 2}$
Definition:Skewes' Number
'''Skewes' number''' is: :$10^{10^{10^{34} } }$ In Knuth notation this can be presented as: :$10 \uparrow \paren {10 \uparrow \paren {10 \uparrow 34} }$