hcju/proofwikips huggingface dataset:
Sufficient Condition for Twice Differentiable Functional to have Minimum
Let $J$ be a twice differentiable functional. Let $J$ have an extremum for $y=\hat y$. Let the second variation $\delta^2 J \sqbrk {\hat y; h}$ be strongly positive  $h$. Then $J$ acquires the minimum for $y = \hat y$ .
Discrete Category is Order Category
Let $\mathbf{Dis} \left({S}\right)$ be a discrete category. Then $\mathbf{Dis} \left({S}\right)$ is also an order category.
Category of Monoids is Category
Let $\mathbf{Mon}$ be the category of monoids. Then $\mathbf{Mon}$ is a metacategory.
Cayley's Representation Theorem/General Case
Let $\struct {G, \cdot}$ be a group. Then there exists a permutation group $P$ on some set $S$ such that: :$G \cong P$ That is, such that $G$ is isomorphic to $P$.
Permutation of Cosets/Corollary 1
Let $G$ be a group. Let $H \le G$ such that $\index G H = n$ where $n \in \Z$. Then: : $\exists N \lhd G: N \lhd H: n \divides \index G H \divides n!$
Cartesian Product is Small iff Inverse is Small
Let $A$ and $B$ be classes. Then the Cartesian product $A \times B$ is a small class  $B \times A$ is small.
Order Isomorphism on Foundational Relation preserves Foundational Structure
Let $A_1$ and $A_2$ be classes. Let $\prec_1$ and $\prec_2$ be relations. Let $\phi: \left({A_1, \prec_1}\right) \to \left({A_2, \prec_2}\right)$ be an order isomorphism. Then $\left({A_1, \prec_1}\right)$ is a foundational structure iff $\left({A_2, \prec_2}\right)$ is also a foundational structure.
Kleene Closure is Free Monoid
Let $S$ be a set. Let $S^*$ be its Kleene closure, and let $i: S \to S^*$ be the insertion of generators. Then $\left({S^*, i}\right)$ is a free monoid over $S$.
Transitive Closure Always Exists (Set Theory)
Let $S$ be a set. Let $G$ be a mapping such that $\map G x = x \cup \bigcup x$. {{explain|Domain and range of $G$ needed}} Let $F$ be defined using the Principle of Recursive Definition: :$\map F 0 = S$ :$\map F {n^+} = \map G {\map F n}$ Let $\displaystyle T = \bigcup_{n \mathop \in \omega} \map F n$. Then: :$T$ is a set and is transitive :$S \subseteq T$ :If $R$ is transitive and $S \subseteq R$, then $T \subseteq R$. That is, given any set $S$, there is an explicit construction for its transitive closure.
Stabilizer of Polynomial
Let $n \in \Z: n > 0$. Let $\map f {x_1, x_2, \ldots, x_n}$ be a polynomial in $n$ variables $x_1, x_2, \ldots, x_n$. Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi, \rho \in S_n$. Let the group action $\pi * f$ be defined as the permutation on the polynomial $f$ by $\pi$. Then the stabilizer of $f$ is the set of permutations on $n$ letters which fix $f$.
Stabilizer of Element of Group Acting on Itself is Trivial
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $*$ be the group action of $\struct {G, \circ}$ on itself by the rule: :$\forall g, h \in G: g * h = g \circ h$ Then the stabilizer of an element $x \in G$ is given by: :$\Stab x = \set e$
Even Perfect Number except 6 is Congruent to 1 Modulo 9
Let $n$ be an even perfect number, but not $6$. Then: :$n \equiv 1 \pmod 9$
Divisibility of Sum of 3 Fourth Powers
Let $n \in \Z_{\ge 0}$ be the sum of three $4$th powers. Then: :$n$ is divisible by $5$  all three addends are also divisible by $5$ :$n$ is divisible by $29$  all three addends are also divisible by $29$.
Prime-Generating Quadratics of form 2 a squared plus p
The quadratic form: :$2 a^2 + p$ yields prime numbers for $a = 0, 1, \ldots, p - 1$ for values of $p$: :$3, 5, 11, 29$
Trivial Group is Initial Object
Let $\mathbf{Grp}$ be the category of groups. Let $1 = \left\{{e}\right\}$ be the trivial group. Then $1$ is an initial object of $\mathbf{Grp}$.
Rank is Ordinal
Let $S$ be a small class The rank of $S$ is an ordinal.
Group Direct Product of Cyclic Groups/Corollary
Let $n_1, n_2, \ldots, n_s$ be a finite sequence of integers, all greater than $1$, such that for any pair of them $n_i$ and $n_j$, $n_1 \perp n_j$. Let $G_i$ be a cyclic group of order $n_i$ for each $i: 1 \le i \le s$. Then $G_1 \times G_2 \times \cdots \times G_s$ is cyclic of order $n_1 n_2 \ldots n_s$.
Compact Closure of Element is Principal Ideal on Compact Subset iff Element is Compact
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below algebraic join semilattice. Let $P = \left({K \left({L} \right), \precsim}\right)$ be an ordered subset of $L$ where $K \left({L} \right)$ denotes the compact subset of $L$. Let $x \in S$. Then $x^{\mathrm{compact} }$ is principal ideal in $P$  $x$ is a compact element.
Arithmetic Sequence of 4 Terms with 3 Distinct Prime Factors
The arithmetic sequence: :$30, 66, 102, 138$ is the smallest of $4$ terms which consists entirely of positive integers each with $3$ distinct prime factors.
Identity Morphism is Terminal Object in Slice Category
Let $\mathbf C$ be a metacategory, and let $C \in \mathbf C_0$ be an object of $\mathbf C$. Let $\operatorname{id}_C: C \to C$ be the identity morphism for $C$. Then $\operatorname{id}_C$ is a terminal object in the slice category $\mathbf C \mathop / C$.
Identity Morphism is Initial Object in Coslice Category
Let $\mathbf C$ be a metacategory, and let $C \in \mathbf C_0$ be an object of $\mathbf C$. Let $\operatorname{id}_C: C \to C$ be the identity morphism for $C$. Then $\operatorname{id}_C$ is an initial object in the coslice category $C \mathop / \mathbf C$.
Ordinal Subset is Well-Ordered
Let $S$ be a class. Let every element of $S$ be an ordinal. Then $\struct {S, \in}$ is a strict well-ordering.
Real Symmetric Matrix is Hermitian
Every real symmetric matrix is Hermitian.
Sufficient Conditions for Weak Extremum
Let $J$ be a functional such that: :$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$ :$\map y a = A$ :$\map y b = B$ Let $y = \map y x$ be an extremum. Let the strengthened Legendre's Condition hold. Let the strengthened Jacobi's Necessary Condition hold. {{explain|specific links to those strengthened versions}} Then the functional $J$ has a weak minimum for $y = \map y x$.
Equivalence Class of Fixed Element/Corollary
:$i \notin \Fix \sigma$  $\eqclass i {\RR_\sigma}$ contains more than one element
Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence
There are an infinite number of rows of Pascal's triangle which contain $3$ integers in arithmetic sequence.
Rows in Pascal's Triangle containing Numbers in Geometric Sequence
There exist no rows of Pascal's triangle which contain $3$ integers in geometric sequence.
Rows in Pascal's Triangle containing Numbers in Harmonic Sequence
There exist no rows of Pascal's triangle which contain $3$ integers in harmonic sequence.
Element of Leibniz Harmonic Triangle is Sum of Numbers Below
The elements in the Leibniz harmonic triangle are the sum of the elements immediately below them. {{refactor|Rework this as another definition of LHT, establishing that column and diagonal $0$ are defined as the reciprocals.}}
Characterization of Prime Element in Inclusion Ordered Set of Topology
Let $T = \struct {S, \tau}$ be a topological space. Let $L = \struct {\tau, \preceq}$  be an inclusion ordered set of $\tau$. Let $Z \in \tau$. Then $Z$ is prime element in $L$ : :$\forall X, Y \in \tau: X \cap Y \subseteq Z \implies X \subseteq Z \lor Y \subseteq Z$
Permutation is Cyclic iff At Most One Non-Trivial Orbit
Let $S$ be a set. Let $\rho: S \to S$ be a permutation on $S$. Then: :$\rho$ is a cyclic permutation : :$S$ has no more than one orbit under $\rho$ with more than one element.
Class of Cardinals Contains Minimal Infinite Successor Set
Let $\mathcal N$ denote the class of all cardinal numbers. Then: :$\omega \subseteq \mathcal N$ Where $\omega$ denotes the minimal infinite successor set.
Nonlimit Ordinal Cofinal to One
Let $x$ be a nonlimit non-empty ordinal. Let $\operatorname{cof}$ denote the cofinal relation.  Let $1$ denote the ordinal one. Then: :$\operatorname{cof} \left({x, 1}\right)$
GCD of Polynomials does not depend on Base Field
Let $E / F$ be a field extension. Let $P, Q \in F \sqbrk X$ be polynomials. Let: :$\gcd \set {P, Q} = R$ in $F \sqbrk X$ :$\gcd \set {P, Q} = S$ in $E \sqbrk X$. Then $R = S$. In particular, $S \in F \sqbrk X$.
Sequences of 4 Consecutive Integers with Falling Sigma
The following ordered quadruple of consecutive integers have sigma values which are strictly decreasing: :$44, 45, 46, 47$ :$104, 105, 106, 107$
Pairs of Consecutive Integers with 6 Divisors
The following sequence of integers are those $n$  which fulfil the equation: :$\tau \left({n}\right) = \tau \left({n + 1}\right) = 6$ where $\tau \left({n}\right)$ denotes the $\tau$ function. That is, they are the first of pairs of consecutive integers which each have $6$ divisors: :$44, 75, 98, 116, 147, 171, 242, 243, 244, 332, \ldots$ {{OEIS|A049103}}
Intersection of Subsemigroups/General Result
Let $\mathbb S$ be a set of subsemigroups of $\left({S, \circ}\right)$, where $\mathbb S \ne \varnothing$. Then the intersection $\bigcap \mathbb S$ of the members of $\mathbb S$ is itself a subsemigroup of $\left({S, \circ}\right)$. Also, $\bigcap \mathbb S$ is the largest subsemigroup of $\left({S, \circ}\right)$ contained in each member of $\mathbb S$.
Abelian Group of Order Twice Odd has Exactly One Order 2 Element
Let $G$ be an abelian group whose identity element is $e$. Let the order of $G$ be $2 n$ such that $n$ is odd. Then there exists exactly one $g \in G$ with $g \ne e$ such that $g = g^{-1}$.
Numbers which Multiplied by 2 are the Reverse of when Added to 2
{{begin-eqn}} {{eqn | l = 47 + 2       | r = 49 }} {{eqn | l = 47 \times 2       | r = 94 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 497 + 2       | r = 499 }} {{eqn | l = 497 \times 2       | r = 994 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 4997 + 2       | r = 4999 }} {{eqn | l = 4997 \times 2       | r = 9994 }} {{end-eqn}} ... and so on:
Third Isomorphism Theorem/Groups/Corollary 1
Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $q: G \to \dfrac G N$ be the quotient epimorphism from $G$ to the quotient group $\dfrac G N$. Let $K$ be the kernel of $q$. Then: :$\dfrac G N \cong \dfrac {G / K} {N / K}$
Morphisms-Only Metacategory Induces Metacategory
Let $\mathbf C$ be a morphisms-only metacategory. Then $\mathbf C$ induces a metacategory $\mathbf C'$, as follows (phrased to fit with Characterization of Metacategory via Equations): :Define $\mathbf C'_1$ to be the collection $\mathbf C_1$ of morphisms of $\mathbf C$. :Define $\mathbf C'_0$ to be the image of the operation $\operatorname{dom}$ on $\mathbf C_1$. :Define $\operatorname{id}$ to be the identity on $\mathbf C'_0$, and take $\operatorname{dom}$ and $\operatorname{cod}$ as in $\mathbf C$. :Define $\circ$ to be as in $\mathbf C$; i.e., $g \circ f$ is the unique element of $\mathbf C_1$ with $R_\circ \left({g, f, g \circ f}\right)$.
Smallest Pair of Quasiamicable Numbers
The smallest pair of quasiamicable numbers is $48$ and $75$.
Equivalence of Definitions of Quasiamicable Numbers
Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers. {{TFAE|def = Quasiamicable Numbers}}
Duality Principle (Category Theory)/Formal Duality
=== Morphisms-Only Category Theory === Let $\Sigma$ be a statement in the language of category theory. Suppose $\Sigma$ is provable from the axioms for morphisms-only category theory $\mathrm{MOCT}$: :$\mathrm{MOCT} \vdash \Sigma$ Then the dual statement $\Sigma^*$ is also provable from these axioms, i.e.: :$\mathrm{MOCT} \vdash \Sigma^*$ === Object Category Theory === Let $\mathrm{CT}$ be the collection of seven axioms on Characterization of Metacategory via Equations. Suppose a statement $\Sigma$ about metacategories follows from the axioms $\mathrm{CT}$. Then so does its dual statement $\Sigma^*$.
Logarithm of Divergent Product of Real Numbers/Zero
The following are equivalent: * The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$. * The series $\displaystyle \sum_{n \mathop = 1}^\infty \log a_n$ diverges to $-\infty$.
Disjoint Union is Coproduct in Category of Sets
Let $\mathbf{Set}$ be the category of sets. Let $S$ and $T$ be sets. Then their disjoint union $S \sqcup T$ is a coproduct in $\mathbf{Set}$.
Continuous Implies Locally Bounded
Let $X$ be a topological space. Let $M$ be a metric space. Let $f: X \to M$ be continuous. Then $f$ is locally bounded.
Image of Canonical Injection is Normal Subgroup
Let $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ be groups with identity elements $e_1$ and $e_2$ respectively. Let $\struct {G_1 \times G_2, \circ}$ be the group direct product of $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$. Let: :$\inj_1: \struct {G_1, \circ_1} \to \struct {G_1 \times G_2, \circ}$ be the canonical injection from $\struct {G_1, \circ_1}$ to $\struct {G_1 \times G_2, \circ}$ :$\inj_2: \struct {G_2, \circ_2} \to \struct {G_1 \times G_2, \circ}$ be the canonical injection from $\struct {G_2, \circ_2}$ to $\struct {G_1 \times G_2, \circ}$. Then: :$(1): \quad \Img {\inj_1} \lhd \struct {G_1 \times G_2, \circ}$ :$(2): \quad \Img {\inj_2} \lhd \struct {G_1 \times G_2, \circ}$ That is, the images of the canonical injections are normal subgroups of the group direct product of $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$.
Period of Reciprocal of 53 is of Quarter Maximal Length
The decimal expansion of the reciprocal of $53$ has $\dfrac 1 4$ the maximum period, that is: $13$: :$\dfrac 1 {53} = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3$ {{OEIS|A007450}}
Pointwise Addition on Real-Valued Functions is Associative
Let $f, g, h: S \to \R$ be real-valued functions. Let $f + g: S \to \R$ denote the pointwise sum of $f$ and $g$. Then: :$\paren {f + g} + h = f + \paren {g + h}$
Pointwise Addition on Integer-Valued Functions is Associative
Let $f, g, h: S \to \Z$ be integer-valued functions. Let $f + g: S \to \Z$ denote the pointwise sum of $f$ and $g$. Then: :$\paren {f + g} + h = f + \paren {g + h}$
Pointwise Addition on Rational-Valued Functions is Associative
Let $f, g, h: S \to \Q$ be rational-valued functions. Let $f + g: S \to \Q$ denote the pointwise sum of $f$ and $g$. Then: :$\paren {f + g} + h = f + \paren {g + h}$
Dirichlet Convolution Preserves Multiplicativity
Let $f, g: \N \to \C$ be multiplicative arithmetic functions. Then their Dirichlet convolution $f * g$ is again multiplicative.
Closed Form for Pentatope Numbers
The closed-form expression for the $n$th pentatope number is: :$P_n = \dfrac {n \paren {n + 1} \paren {n + 2} \paren {n + 3} } {24}$
Necessary and Sufficient Condition for First Order System to be Field for Second Order System
Let $\mathbf y$, $\mathbf f$, $\boldsymbol \psi$ be N-dimensional vectors. Let $\boldsymbol\psi$ be continuously differentiable. Then $\forall x \in \closedint a b$ the first-order system of differential equations: :$\mathbf y' = \map {\boldsymbol \psi} {x, \mathbf y}$ is a field for the second-order system :$\mathbf y'' = \map {\mathbf f} {x, \mathbf y, \mathbf y'}$  $\boldsymbol \psi$ satisfies : :$\displaystyle \frac {\partial \boldsymbol \psi} {\partial x} + \sum_{i \mathop = 1}^N \frac {\partial \boldsymbol \psi} {\partial y_i} \psi_i = \map {\mathbf f} {x, \mathbf y, \boldsymbol \psi}$ That is, every solution to Hamilton-Jacobi system is a field for the original system.
Tetrahedral Number as Sum of Squares
:$H_n = \displaystyle \sum_{k \mathop = 0}^{n / 2} \paren {n - 2 k}^2$ where $H_n$ denotes the $n$th tetrahedral number.
Smooth Homotopy is an Equivalence Relation
Let $X$ and $Y$ be smooth manifolds. Let $K \subseteq X$ be a (possibly empty) subset of $X$. Let $\mathcal C^\infty \left({X, Y}\right)$ be the set of all smooth mappings from $X$ to $Y$. Define a relation $\sim$ on $\mathcal C \left({X, Y}\right)$ by $f \sim g$ if $f$ and $g$ are smoothly homotopic relative to $K$. Then $\sim$ is an equivalence relation.
Positive Even Integers as Sum of 2 Composite Odd Integers in 2 Ways
Let $n \in \Z_{>0}$ be a positive even integer. Let $n$ be such that it cannot be expressed as the sum of $2$ odd positive composite integers in at least $2$ different ways. Then $n$ belongs to the set: :$\left\{ {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 38, 40, 44, 46, 52, 56, 62, 68}\right\}$ {{OEIS|A284788}}
Continuous Image of Path-Connected Set is Path-Connected
Let $M_1, M_2$ be metric spaces whose metrics are $d_1, d_2$ respectively. Let $f: M_1 \to M_2$ be a continuous mapping. Let $S \subseteq M_1$ be a path-connected subspace of $M_1$. Then $f \sqbrk S$ is a a path-connected subspace of $M_2$.
Quotient Mapping is Coequalizer
Let $\mathbf{Set}$ be the category of sets. Let $S$ be a Set, and let $\mathcal R \subseteq S \times S$ be an equivalence relation on $S$. Let $r_1, r_2: \mathcal R \to S$ be the projections corresponding to the inclusion mapping $\mathcal R \hookrightarrow S \times S$. Let $q: S \to S / \mathcal R$ be the quotient mapping induced by $\mathcal R$. Then $q$ is a coequalizer of $r_1$ and $r_2$ in $\mathbf{Set}$.
Reciprocal of 81
The decimal expansion of the reciprocal of $81$ has a particularly interesting pattern: :$\dfrac 1 {81} = 0 \cdotp \dot 01234 \, 567 \dot 9$
Open Real Interval is not Closed Set/Corollary
Let: :$I_a = \openint \gets a$ :$I_b = \openint b \to$ be unbounded open real intervals. Then neither $I_a$ nor $I_b$ are closed sets of $\R$.
Necessary and Sufficient Condition for First Order System to be Mutually Consistent
Let $\mathbf y$, $\boldsymbol \psi$ be N-dimensional vectors. Let $g$ be a twice differentiable mapping. Let :$(1): \quad \map {\boldsymbol \psi} {x, \mathbf y} = \map {\mathbf y'} {x, \mathbf y}$ :$(2): \quad \mathbf p \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } = \map {g_{\mathbf y} } {x, \mathbf y}$ where $\mathbf p$ is a momentum. Then the boundary conditions defined by $(1)$ are mutually consistent  the mapping $\map g {x, \mathbf y}$ satisfies the Hamilton-Jacobi equation: :$(3): \quad \dfrac {\partial g} {\partial x} + \map H {x, \mathbf y, \dfrac {\partial g} {\partial \mathbf y} } = 0$
Reciprocal of 89
The decimal expansion of the reciprocal of $89$ contains the Fibonacci sequence: :$\dfrac 1 {89} = 0 \cdotp \dot 01123 \, 59550 \, 56179 \, 77528 \, 08988 \, 76404 \, 49438 \, 20224 \, 719 \dot 1$
Reciprocal as Sum of Fibonacci Numbers by Negative Powers of 10
:$\displaystyle \sum_{k \mathop \ge 0} \dfrac {F_k} {10^{k + 1} } = \dfrac 1 {89}$ where $F_k$ is the $k$th Fibonacci number: :$F_0 = 0, F_1 = 1, F_k = F_{k - 1} + F_{k - 2}$ That is: <pre> 1 / 89 = 0.0        + 0.01        + 0.001        + 0.0002        + 0.00003        + 0.000005        + 0.0000008        + 0.00000013        + 0.000000021        + 0.0000000034        + 0.00000000055        + .............. </pre>
Continuous Function on Compact Space is Uniformly Continuous
Let $\R^n$ be the $n$-dimensional Euclidean space. Let $S \subseteq \R^n$ be a compact subspace of $\R^n$. Let $f: S \to \R$ be a continuous function. Then $f$ is uniformly continuous in $\R$.
91 is Pseudoprime to 35 Bases less than 91
$91$ is a Fermat pseudoprime in $35$ bases less than itself: :$3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 27, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 64, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88, 90$
Reciprocal of 97
The decimal expansion of the reciprocal of $97$ has the maximum period, that is: $96$: :$\dfrac 1 {97} = 0 \cdotp \dot 01030 \, 92783 \, 50515 \, 46391 \, 75257 \, 73195 \, 87628 \, 86597 \, 93814 \, 43298 \, 96907 \, 21649 \, 48453 \, 60824 \, 74226 \, 80412 \, 37113 \, 40206 \, 18556 \, \dot 7$ {{OEIS|A007450}}
Reciprocal of 98
The decimal expansion of the reciprocal of $98$ starts with the powers of $2$: :$\dfrac 1 {98} = 0 \cdotp 0 \dot 1020 \, 40816 \, 32653 \, 06122 \, 44897 \, 95918 \, 36734 \, 69387 \, 75 \dot 5$ {{OEIS|A021102}}
Reciprocal of 103
:$\dfrac 1 {103} = 0 \cdotp \dot 00970 \, 87378 \, 64077 \, 66990 \, 29126 \, 21359 \, 223 \dot 3$
Infinite Set in Compact Space has Omega-Accumulation Point
Let $\struct {X, \tau}$ be a compact topological space. Let $A \subseteq X$ be infinite. Then $A$ has an $\omega$-accumulation point in $X$.
Cube of 11 is Palindromic
::$11^3 = 1331$
Fourth Power of 11 is Palindromic
::$11^4 = 14641$
Local Membership of Equalizer
Let $\mathbf C$ be a metacategory. Let $e: E \to C$ be the equalizer of $f,g : C \to D$. Then a variable element $z: Z \to C$ is a local member of $e$ iff $f \circ z = g \circ z$: :$z \in_C e \iff f \circ z = g \circ z$
Topological Equivalence is Equivalence Relation
Let $A$ be a set. Let $\mathcal D$ be the set of all metrics on $A$. Let $\sim$ be the relation on $\mathcal D$ defined as: :$\forall d_1, d_2 \in \mathcal D: d_1 \sim d_2 \iff d_1$ is topologically equivalent to $d_2$ Then $\sim$ is an equivalence relation.
Prime Gaps of 10
The following pairs of consecutive prime numbers are those whose difference is $10$: :$\tuple {139, 149}, \tuple {181, 191}, \tuple {241, 251}, \tuple {283, 293}, \ldots$ {{OEIS|A031928|order = lower}}
Euclidean Plane is Abstract Geometry
The Euclidean plane $\left({\R^2, L_E}\right)$ is an abstract geometry.
Factorions Base 10
The following positive integers are the only factorions base $10$: :$1, 2, 145, 40 \, 585$
Strong Separation Theorem
Let $C \subset \R^\ell$ be closed and convex. Let $D = \set {\mathbf v} \subset C^c$. Then $C$ and $D$ can be strongly separated.
Pairs of Integers whose Product with Tau Value are Equal
Let $\tau \left({n}\right)$ denote the $\tau$ function: the number of divisors of $n$. The following pairs of integers $T$ have the property that $m \tau \left({m}\right)$ is equal for each $m \in T$: :$\left\{ {18, 27}\right\}$ :$\left\{ {24, 32}\right\}$ :$\left\{ {56, 64}\right\}$
Numbers that cannot be made Prime by changing 1 Digit
The following positive integers cannot be made into prime numbers by changing just one digit: :$200, 202, 204, 205, 206, 208, \ldots$ {{OEIS|A192545}}
Triangular Numbers which are also Pentagonal
The sequence of triangular numbers which are also pentagonal begins: :$1, 210, 40 \, 755, 7 \, 906 \, 276, 1 \, 533 \, 776 \, 805, 297 \, 544 \, 793 \, 910, \ldots$ {{OEIS|A014979}}
For Complete Ritz Sequence Continuous Functional approaches its Minimal Value
Let $J$ be a continuous functional. Let $\sequence {\phi_n}$ be a complete Ritz sequence. {{explain|The concept of $\sequence {\phi_n}$ does not appear to be related in any way to the statement of the theorem.}} Then: :$\displaystyle \lim_{n \mathop \to \infty} \mu_n = \mu$ where $\displaystyle \mu = \inf_y J \sqbrk y$.
17 Wallpaper Groups
There are $17$ wallpaper groups.
Closure of Infinite Union may not equal Union of Closures
Let $T$ be a topological space. Let $I$ be an infinite indexing set. Let $\family {H_i}_{i \mathop \in I}$ be an indexed family of subsets of a set $S$. Let $\displaystyle H = \bigcup_{i \mathop \in I} H_i$ be the union of $\family {H_i}_{i \mathop \in I}$. Then it is not always the case that: :$\displaystyle \bigcup_{i \mathop \in I} \map \cl {H_i} = \map \cl {\bigcup_{i \mathop \in I} H_i}$ where $\map \cl {H_i}$ denotes the closure of $H_i$.
Set of Reciprocals of Positive Integers is Nowhere Dense in Reals
Let $N$ be the set defined as: :$N := \set {\dfrac 1 n: n \in \Z_{>0} }$ where $\Z_{>0}$ is the set of (strictly) positive integers. Let $\R$ denote the real number line with the usual (Euclidean) metric. Then $N$ is nowhere dense in $\R$.
Numbers in Even-Even Amicable Pair are not Divisible by 3
Let $\tuple {m_1, m_2}$ be an amicable pair such that both $m_1$ and $m_2$ are even. Then neither $m_1$ nor $m_2$ is divisible by $3$.
Construction of Regular 257-Gon
It is possible to construct a regular polygon with $257$ sides) using a compass and straightedge construction.
Young's Inequality for Increasing Functions/Equality
Let $a_0$ and $b_0$ be strictly positive real numbers. Let $f: \closedint 0 {a_0} \to \closedint 0 {b_0}$ be a strictly increasing bijection. Let $a$ and $b$ be real numbers such that $0 \le a \le a_0$ and $0 \le b \le b_0$. Then $b = \map f a$ : :$\displaystyle a b = \int_0^a \map f u \rd u + \int_0^b \map {f^{-1} } v \rd v$ where $\displaystyle \int$ denotes the Darboux integral.