hcju/proofwikips huggingface dataset:
Sigma Function of Non-Square Semiprime
Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$. Then: :$\map \sigma n = \paren {p + 1} \paren {q + 1}$ where $\map \sigma n$ denotes the sigma function.
Sigma Function of Square-Free Integer
Let $n$ be an integer such that $n \ge 2$. Let $n$ be square-free. Let the prime decomposition of $n$ be: :$\displaystyle n = \prod_{1 \mathop \le i \mathop \le r} p_i =  p_1 p_2 \cdots p_r$ Let $\sigma \left({n}\right)$ be the sigma function of $n$. That is, let $\sigma \left({n}\right)$ be the sum of all positive divisors of $n$. Then: :$\displaystyle \sigma \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} p_i + 1$
Spheres in 24 Dimensions in Leech Lattice
Let a set of identical spheres in a $24$-dimensional space be arranged in a Leech lattice. Then each sphere will touch $196 \, 560$ other spheres. This is believed to be the densest possible sphere packing in $24$ dimensions.
Cross-Sections of Leech Lattice
Except for dimensions $10$, $11$ and $13$, the densest possible sphere packing in all dimensions lower than $24$ can be obtained by cross-sections of the $24$-dimensional space Leech lattice arrangement.
Difference Triangle for Sequence of Fifth Powers
The difference triangle for the sequence of fifth powers ends on the fifth line with instances of $5!$, where $!$ denotes factorial.
Ideals form Arithmetic Lattice
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below lattice. Let $I = \left({\mathit{Ids}\left({L}\right), \precsim}\right)$ be an inclusion ordered set where :$\mathit{Ids}\left({L}\right)$ denotes the set of all ideals in $L$, :$\mathord\precsim = \mathord\subseteq \cap \left({\mathit{Ids}\left({L}\right) \times \mathit{Ids}\left({L}\right)}\right)$ Then $I$ is an arithmetic lattice.
Power of n equalling (n - 1)! + 1
There is exactly one solution to the equation in the integers: :$\paren {n - 1}! + 1 = n^k$ for $k > 1$, and that is: :$n = 5$ :$k = 2$
Square which is 2 Less than Cube
$25$ is the only square number which is $2$ less than a cube: :$5^2 + 2 = 3^3$
Ideals form Algebraic Lattice
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $I = \left({\mathit{Ids}\left({L}\right), \precsim}\right)$ be an inclusion ordered set where :$\mathit{Ids}\left({L}\right)$ denotes the set of all ideals in $L$, :$\mathord\precsim = \mathord\subseteq \cap \left({\mathit{Ids}\left({L}\right) \times \mathit{Ids}\left({L}\right)}\right)$ Then $I$ is algebraic lattice.
Definition:Powerful Number
A '''powerful number''' is a positive integer such that each of its prime factors appears with multiplicity at least $2$. That is, each of its prime factors occurs at least squared.
Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite
Let   :$\map P x : \closedint a b \to \R$  :$\map h x : \closedint a b \to \R$. Let $\map h x$ be continuously differentiable $\forall x \in \closedint a b$. Suppose: :$\forall x \in \closedint a b: \map P x > 0$ Then: :$\ds \forall \map h x : \map h a = \map h b = 0 : \int_a^b \paren {P h'^2 + Q h^2} \rd x > 0$  the interval $\closedint a b$ contains no points conjugate to $a$.
Consecutive Odd Powerful Numbers
$25$ and $27$ are the only known pair of consecutive odd powerful numbers. It is not known whether there are any more.
Square Cullen Numbers
The numbers: :$1, 9, 25$ are Cullen numbers which are also square.
Numbers Partitioned into Six Hexagonal Numbers
The integers $11$ and $26$ cannot be represented by the sum of less than $6$ hexagonal numbers.
Continuous Lattice Subframe of Algebraic Lattice is Algebraic Lattice
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below algebraic lattice. Let $P = \struct {T, \precsim}$ be a continuous lattice subframe of $L$. Then $P$ is algebraic lattice.
Period of Reciprocal of 27 is Smallest with Length 3
$27$ is the smallest positive integer the decimal expansion of whose reciprocal has a period of $3$: :$\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$ {{OEIS|A021027}}
Period of Reciprocal of 7 is of Maximal Length
$7$ is the smallest integer $n$ the decimal expansion of whose reciprocal has the maximum period $n - 1$, that is: $6$: :$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$ {{OEIS|A020806}} :300px
Sequence of Smallest Numbers whose Reciprocal has Period n
Let $\sequence {s_n}$ be the sequence defined as: :$s_n$ is the smallest positive integer the decimal expansion of whose reciprocal has a period of $n$ for $n = 0, 1, 2, \ldots$ Then $\sequence {s_n}$ begins: :$1, 3, 11, 27, 101, 41, 7, 239, 73, 81, 451, \ldots$ {{OEIS|A003060}}
Sequence of Successive Longest Collatz Sequence Generators
The sequence of integers which generate a Collatz process which is longer than that of any smaller integers begins: {{begin-eqn}} {{eqn | l = 1       | o = :       | c = $0$ steps }} {{eqn | l = 2       | o = :       | c = $1$ step }} {{eqn | l = 3       | o = :       | c = $7$ steps }} {{eqn | l = 6       | o = :       | c = $8$ steps }} {{eqn | l = 7       | o = :       | c = $16$ steps }} {{eqn | l = 9       | o = :       | c = $19$ steps }} {{eqn | l = 18       | o = :       | c = $20$ steps }} {{eqn | l = 25       | o = :       | c = $23$ steps }} {{eqn | l = 27       | o = :       | c = $111$ steps }} {{end-eqn}} {{OEIS|A006877}}
Dissection of Nonagon into Triangles with Chords
There are $27$ different ways to dissect a convex nonagon into triangles using $6$ chords, not counting reflections and rotations as different.
Image of Directed Suprema Preserving Closure Operator is Algebraic Lattice
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below algebraic lattice. Let $c:S \to S$ be a closure operator that preserves directed suprema. Let $C = \left({c\left[{S}\right], \precsim}\right)$ be an ordered subset of $L$. Then $C$ is algebraic lattice.
Dissection of Polygon into Triangles with Chords
The number of different ways $k$ a convex $n$-sided polygon can be divided into triangles using chords, not counting reflections and rotations as different, is given for the first few $n$ as follows: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="right" style = "padding: 2px 10px" | $k$ |- | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $5$ | align="right" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="right" style = "padding: 2px 10px" | $3$ |- | align="right" style = "padding: 2px 10px" | $7$ | align="right" style = "padding: 2px 10px" | $4$ |- | align="right" style = "padding: 2px 10px" | $8$ | align="right" style = "padding: 2px 10px" | $12$ |- | align="right" style = "padding: 2px 10px" | $9$ | align="right" style = "padding: 2px 10px" | $27$ |- | align="right" style = "padding: 2px 10px" | $10$ | align="right" style = "padding: 2px 10px" | $82$ |- | align="right" style = "padding: 2px 10px" | $11$ | align="right" style = "padding: 2px 10px" | $228$ |- | align="right" style = "padding: 2px 10px" | $12$ | align="right" style = "padding: 2px 10px" | $733$ |- | align="right" style = "padding: 2px 10px" | $13$ | align="right" style = "padding: 2px 10px" | $2282$ |} {{OEIS|A000207}}
Dissection of Polygon into Triangles with Chords counting Isometries
The number of different ways $k$ a convex $n$-sided polygon can be divided into triangles using chords, counting reflections and rotations as different, is given for the first few $n$ as follows: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="right" style = "padding: 2px 10px" | $k$ |- | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $2$ |- | align="right" style = "padding: 2px 10px" | $5$ | align="right" style = "padding: 2px 10px" | $5$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="right" style = "padding: 2px 10px" | $14$ |- | align="right" style = "padding: 2px 10px" | $7$ | align="right" style = "padding: 2px 10px" | $42$ |- | align="right" style = "padding: 2px 10px" | $8$ | align="right" style = "padding: 2px 10px" | $132$ |- | align="right" style = "padding: 2px 10px" | $9$ | align="right" style = "padding: 2px 10px" | $429$ |- | align="right" style = "padding: 2px 10px" | $10$ | align="right" style = "padding: 2px 10px" | $1430$ |- | align="right" style = "padding: 2px 10px" | $11$ | align="right" style = "padding: 2px 10px" | $4862$ |} {{OEIS|A000108}} These are the Catalan numbers.
Boundary of Compact Closed Set is Compact
Let $X$ be a topological space. Let $K\subset X$ be a compact subspace of $X$. Let $K$ be closed in $X$. Then its boundary $\partial K$ is compact.
Discrete Subgroup of Hausdorff Group is Closed
Let $G$ be a Hausdorff topological group. Let $H$ be a discrete subgroup of $G$. Then $H$ is closed in $G$.
Group is Hausdorff iff Discrete Subgroups are Closed
A topological group is Hausdorff  its discrete subgroups are closed.
Closure of Subgroup is Group
Let $G$ be a topological group. Let $H\leq G$ be a subgroup. Let $\overline H$ denote its closure. Then $\overline H$ is a subgroup of $G$.
Image of Group Homomorphism is Hausdorff Implies Kernel is Closed
Let $G$ and $H$ be topological groups. Let $f: G \to H$ be a morphism. Let its image $\Img f$ be Hausdorff. Then its kernel $\map \ker f$ is closed in $G$.
Group is Connected iff Subgroup and Quotient are Connected
Let $G$ be a topological group. Let $H \le G$ be a subgroup.  :$(1):\quad$ $G$ is connected :$(2):\quad$ $H$ is connected and the left quotient space $G / H$ is connected :$(3):\quad$ $H$ is connected and the right quotient space $G / H$ is connected.
Group Acts by Homeomorphisms Implies Projection on Quotient Space is Open
Let $G$ be a group acting by homeomorphisms on a topological space $X$. Then the projection map $\pi: X \to X / G$ is open.
Neighborhood iff Contains Neighborhood
Let $X$ be a topological space. Let $x\in X$. Let $V\subset X$ be a subset. Then the following are equivalent: :$V$ is a neighborhood of $x$ in $X$ :$V$ contains a neighborhood of $x$ in $X$
Perfect Number ends in 6 or 28 preceded by Odd Digit
Let $n$ be an even perfect number. Then $n$ ends either in $6$ or $28$ preceded by an odd digit.
Open Subset of Locally Connected Space is Locally Connected
Let $T - \left({S, \tau}\right)$ be a locally connected topological space. Let $U \subset S$ be open in $T$. Then $U$ is locally connected.
Component of Locally Connected Space is Open
Let $T = \left({S, \tau}\right)$ be a locally connected topological space. Let $G$ be a component of $T$. Then $G$ is open.
Path Component of Locally Path-Connected Space is Open
Let $T = \left({S, \tau}\right)$ be a locally path-connected topological space. Let $G$ be a path component of $T$. Then $G$ is open in $T$.
Connected and Locally Path-Connected Implies Path Connected
Let $T$ be a connected and locally path-connected topological space. Then $T$ is path-connected.
Components are Open iff Union of Open Connected Sets
Let $T = \struct {S, \tau}$ be a topological space.  :$(1): \quad$ The components of $T$ are open. :$(2): \quad S$ is a union of open connected sets of $T$.
Path Components are Open iff Union of Open Path-Connected Sets
Let $T = \left({S, \tau}\right)$ be a topological space. : :(1) $\quad$ The path components of $T$ are open. :(2) $\quad S$ is a union of open path-connected sets of $T$.
Open Subset of Locally Path-Connected Space is Locally Path-Connected
Let $T = \struct {S, \tau}$ be a locally path-connected topological space. Let $U \subset S$ be open in $T$. Then $U$ is locally path-connected in $T$.
Even Perfect Number is Hexagonal
All perfect numbers which are even are hexagonal.
Hexagonal Number is Triangular Number
Let $H_n$ be the $n$th hexagonal number. Then $H_n$ is the $2 n - 1$th triangular number.
Perfect Number which is Sum of Equal Powers of Two Numbers
$28$ is the only perfect number which is the sum of equal powers of exactly $2$ positive integers: :$28 = 1^3 + 3^3$
Congruence Modulo 3 of Power of 2
Let $n \in \Z_{\ge 0}$ be a positive integer. Then: :$2^n \equiv \paren {-1}^n \pmod 3$ That is: :$\exists q \in \Z: 2^n = 3 q + \paren {-1}^n$
Odd Power of 2 is Congruent to 2 Modulo 3
Let $n \in \Z_{\ge 0}$ be an odd positive integer. Then: :$2^n \equiv 2 \pmod 3$
Open Set in Open Subspace
Let $X$ be a topological space. Let $U\subset X$ be an open subset. Let $V\subset U$ be a subset. Then $V$ is open in $U$  $V$ is open in $X$.
Equivalence of Definitions of Locally Compact Hausdorff Space
{{TFAE|def = Locally Compact Hausdorff Space}} Let $T = \struct{S, \tau}$ be a Hausdorff topological space.
Neighborhood in Compact Hausdorff Space Contains Compact Neighborhood
Let $X$ be a compact Hausdorff topological space. Let $x\in X$. Let $U$ be a neighborhood of $x$. Then $U$ contains a compact neighborhood of $x$.
Are All Perfect Numbers Even?/Progress/Form
An odd perfect number $n$ is of the form: :$n = p^a q^b r^c \cdots$ where: :$p, q, r, \ldots$ are prime numbers of the form $4 k + 1$ for some $k \in \Z_{>0}$ :$a$ is also of the form $4 k + 1$ for some $k \in \Z_{>0}$ :$b, c, \ldots$ are all even.
Image of Idempotent and Directed Suprema Preserving Mapping is Complete Lattice
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Let $f:S \to S$ be a mapping that is idempotent and preserves directed suprema. Let $F = \struct {f \sqbrk S, \precsim}$ be an ordered subset of $L$. Then $F$ inherits directed suprema and is complete lattice.
Are All Perfect Numbers Even?/Progress/Minimum Size
It had been established by $1986$ that an odd perfect number, if one were to exist, would have over $200$ digits. By $1997$ that lower bound had been raised to $300$ digits. By $2012$ that lower bound had been raised again to $1500$ digits.
Are All Perfect Numbers Even?/Progress/Prime Factors
An odd perfect number has: :at least $8$ distinct prime factors :at least $11$ distinct prime factors if $3$ is not one of them :at least $101$ prime factors (not necessarily distinct) :its greatest prime factor is greater than $1 \, 000 \, 000$ :its second largest prime factor is greater than $1000$ :at least one of the prime powers factoring it is greater than $10^{62}$ :if less than $10^{9118}$ then it is divisible by the $6$th power of some prime.
Prime-Generating Quadratics of form 2 a squared plus p/3
The quadratic form: :$2 a^2 + 3$ yields prime numbers for $a = 0, 1, 2$.
Prime-Generating Quadratics of form 2 a squared plus p/5
The quadratic form: :$2 a^2 + 5$ yields prime numbers for $a = 0, 1, \ldots, 4$.
Prime-Generating Quadratics of form 2 a squared plus p/11
The quadratic form: :$2 a^2 + 11$ yields prime numbers for $a = 0, 1, \ldots, 10$.
Prime-Generating Quadratics of form 2 a squared plus p/29
The quadratic form: :$2 a^2 + 29$ yields prime numbers for $a = 0, 1, \ldots, 28$.
Directed Suprema Preserving Mapping is Increasing
Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice. Let $f:S \to S$ be a mapping that preserves directed suprema. Then $f$ is an increasing mapping.
Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma 1
Let the function $\map h x$ satisfy the equation: :$-\map {\dfrac \d {\d x} } {P h'} + Q h = 0$ Let $\map h x$ have the boundary conditions: :$\map h a = \map h b = 0$ Then: :$\displaystyle \int_a^b \paren {P h'^2 + Q h^2} \rd x = 0$
Sequence of 4 Consecutive Square-Free Triplets
The following sets of $4$ consecutive triplets of integers, with one integer between each triplet, are square-free: :$29, 30, 31; 33, 34, 35; 37, 38, 39; 41, 42, 43$ :$101, 102, 103; 105, 106, 107; 109, 110, 111; 113, 114, 115$
Sequence of Prime Primorial minus 1
For prime $p$, let $p \#$ denote the $p$th primorial, defined in the sense that $p \#$ is the product of all primes less than or equal to $p$. The sequence $\left\langle{p}\right\rangle$ such that $p \# - 1$ is prime begins: :$3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, \ldots$ {{OEIS|A006794}}
Image of Pair under Mapping
Let $S, T$ be sets. Let $f: S \to T$ be a mapping. Then: :$\forall x, y \in S: f \sqbrk {\set {x, y} } = \set {\map f x, \map f y}$
Image under Increasing Mapping equal to Special Set is Complete Lattice
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Let $f: S \to S$ be an increasing mapping. Let $P = \struct {M, \precsim}$ be an ordered subset of $L$ such that :$M = \set {x \in S: x = \map f x}$ Then $P$ is complete lattice.
Hensel's Lemma/First Form
Let $p$ be a prime number. Let $k > 0$ be a positive integer. Let $f \left({X}\right) \in \Z \left[{X}\right]$ be a polynomial. Let $x_k \in \Z$ such that: :$f \left({x_k}\right) \equiv 0 \pmod {p^k}$ :$f' \left({x_k}\right) \not \equiv 0 \pmod p$ Then for every integer $l \ge 0$, there exists an integer $x_{k + l}$ such that: :$f \left({x_{k + l} }\right) \equiv 0 \pmod {p^{k + l} }$ :$x_{k + l}\equiv x_k \pmod {p^k}$ and any two integers satisfying these congruences are congruent modulo $p^{k + l}$. Moreover, for all $l\geq0$ and any solutions $x_{k + l}$ and $x_{k + l + 1}$: :$x_{k + l + 1} \equiv x_{k + l} - \dfrac {f \left({x_{k + l} }\right)} {f' \left({x_{k + l} }\right)} \pmod {p^{k + l + 1} }$ :$x_{k + l + 1} \equiv x_{k + l} \pmod {p^{k + l} }$
Sigma Function of Power of 2
Let $n \in \Z_{>0}$ be a power of $2$. Then: :$\map \sigma n = 2 n - 1$
Equivalence of Definitions of Saturation Under Equivalence Relation
Let $\sim$ be an equivalence relation on a set $S$. Let $T \subset S$ be a subset. {{TFAE|def = Saturation (Equivalence Relation)|view = saturation}}
Equivalence of Definitions of Saturated Set Under Equivalence Relation
Let $\sim$ be an equivalence relation on a set $S$. Let $T\subset S$ be a subset. {{TFAE|def = Saturated Set (Equivalence Relation)|view = saturated set|context = Equivalence Relation}}
Compact Subset is Bounded Below Join Semilattice
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below 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 $L$. Then $P$ is a bounded below join semilattice.
Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma 2
Let $\map h x : \closedint a b \to \R$ be continuously differentiable $\forall x \in \closedint a b$. Suppose the function $\map h x$ satisfies the equation: :$-\map {\dfrac \d {\d x} } {\paren {t P + \paren {1 - t} } h'} + t Q h = 0$ subject to the boundary conditions: :$\map h {a, t} = \map h {b, t} = 0$ Then: :$\displaystyle \int_a^b \paren {\paren {P h'^2 + Q h^2} t + \paren {1 - t} h'^2} \rd x = 0$
Jacobi's Necessary Condition
Let $J$ be a functional, such that: :$J \sqbrk y = \ds \int_a^b \map F {x, y, y'} \rd x$ Let $\map y x$ correspond to the minimum of $J$. Let: :$F_{y'y'}>0$ along $\map y x$. Then the open interval $\openint a b$ contains no points conjugate to $a$.
Stolz-Cesàro Theorem
Let $\sequence {a_n}$ be a sequence. {{explain|Domain of $\sequence {a_n}$ -- $\R$ presumably but could it be $\C$?}} Let $\sequence {b_n}$ be a sequence of (strictly) positive real numbers such that: :$\displaystyle \sum_{i \mathop = 0}^\infty b_n = \infty$ If: :$\displaystyle \lim_{n \mathop \to \infty} \dfrac {a_n} {b_n} = L \in \R$ then also: :$\displaystyle \lim_{n \mathop \to \infty} \dfrac {a_1 + a_2 + \cdots + a_n} {b_1 + b_2 + \cdots + b_n} = L$
Bottom is Compact
Let $L$ be a bounded below ordered set. Then $\bot$ is a compact element where $\bot$ is the smallest element in $L$.
Saturation Under Equivalence Relation in Terms of Graph
Let $\RR \subset S \times S$ be an equivalence relation on a set $S$. Let $\pr_1, \pr_2 : S \times S \to S$ denote the projections. Let $T\subset S$ be a subset. Let $\overline T$ denote its saturation. Then the following hold: :$\overline T = \map {\pr_1} {\RR \cap \map {\pr_2^{-1} } T}$ :$\overline T = \map {\pr_2} {\RR \cap \map {\pr_1^{-1} } T}$
Nonnegative Quadratic Functional implies no Interior Conjugate Points
If the quadratic functional  :$\ds \int_a^b \paren {P h'^2 + Q h^2} \rd x$ where: :$\forall x \in \closedint a b: \map P x > 0$ is nonnegative for all $\map h x$: :$\map h a = \map h b = 0$ then the closed interval $\closedint a b$ contains no inside points conjugate to $a$. In other words, the open interval $\openint a b$ contains no points conjugate to $a$. {{explain|Rewrite the above so it makes better sense. For example, should the "nonnegative" comment be above the condition on $\map P x$?}}
Projection of Subset is Open iff Saturation is Open
Let $\sim$ be an equivalence relation on a topological space $X$. Let $X / \sim$ be the quotient space. Let $p$ denote the quotient mapping. Let $U \subset X$. Then the following are equivalent: :$\map p U$ is open in $X / \sim$ :The saturation of $U$ is open in $X$
Multiple of 6 is Semiperfect
Let $n \in \Z_{>0}$ be a multiple of $6$. Then $n$ is semiperfect.
Open Projection and Closed Graph Implies Quotient is Hausdorff
Let $\mathcal R\subset X\times X$ be an equivalence relation on a topological space $X$. Let $X/\mathcal R$ be the quotient space. Let $p$ denote the quotient mapping. Let: :$\mathcal R$ be closed in $X\times X$ :$p$ be open Then $X/\mathcal R$ is Hausdorff.
Subgroup is Closed iff Quotient is Hausdorff
Let $G$ be a topological group. Let $H \le G$ be a subgroup. Let $G / H$ be their quotient. Then the following are equivalent: :$H$ is closed in $G$ :$G / H$ is Hausdorff
Higher Homotopy Groups are Abelian
Let $T = \left({S, \tau}\right)$ be a topological space. Let $x_0 \in S$. Let $n \ge 2$ be a integer. Let $\pi_n \left({T, x_0}\right)$ be the $n$th homotopy group with base point $x_0$. Then $\pi_n \left({T, x_0}\right)$ is abelian.
Multiple of Semiperfect Number is Semiperfect
Let $n \in \Z_{>0}$ be a semiperfect number. Let $k \in \Z_{>0}$ be a (strictly) positive integer. Then $k n$ is also a semiperfect number.
Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset
Let $L = \left({S, \preceq}\right)$ be an algebric lattice. Let $c:S \to S$ be a closure operator that preserves directed suprema. Then $c\left[{K\left({L}\right)}\right] \subseteq K\left({\left({c\left[{S}\right], \precsim}\right)}\right)$ where :$K\left({L}\right)$ denotes the compact subset of $L$, :$c\left[{S}\right]$ denotes the image of $S$ under $c$, :$\mathord\precsim = \mathord\preceq \cap \left({c\left[{S}\right] \times c\left[{S}\right]}\right)$
Integers such that all Coprime and Less are Prime
The following positive integers have the property that all positive integers less than and coprime to it, excluding $1$, are prime: :$1, 2, 3, 4, 6, 8, 12, 18, 24, 30$ {{OEIS|A048597}} There are no other positive integers with this property.
Separable Extension is Contained in Galois Extension
Let $E/F$ be a separable finite field extension. Then there exists a finite field extension $L/E$ such that $L/F$ is Galois.
Galois Extension is Galois over Intermediate Field
Let $L / F$ be a Galois Extension. Let $E$ be an intermediate field. Then $L / E$ is Galois.
Frobenius Endomorphism on Field is Injective
Let $p$ be a prime number. Let $F$ be a field of characteristic $p$. Then the Frobenius endomorphism $\Frob: F \to F$ is injective.
Galois Field is Perfect
Let $\GF$ be a Galois field. Then $\GF$ is perfect.
Algebraically Closed Field is Perfect
Let $F$ be an algebraically closed field. Then $F$ is perfect.
Area of Smallest Rectangle accommodating Re-Entrant Knight's Tour
The area of the smallest rectangular chessboard on which a re-entrant knight's tour is possible is $30$ squares. This can be configured either as a $5 \times 6$ chessboard or a $3 \times 10$ chessboard.
Area of Smallest Square accommodating Re-Entrant Knight's Tour
The area of the smallest square chessboard on which a re-entrant knight's tour is possible is $6 \times 6 = 36$ squares.
Subgroup of Index Least Prime Divisor is Normal
Let $G$ be a finite group of order $n>1$. Let $p$ be the least prime divisor of $n$. Let $H$ be a subgroup of index $p$. Then $H$ is normal.