hcju/proofwikips huggingface dataset:
Coset Product on Non-Normal Subgroup is not Well-Defined
Let $\struct {G, \circ}$ be a group. Let $H$ be a subgroup of $G$ which is not normal. Let $a, b \in G$. Then it is not necessarily the case that the coset product: :$\paren {a \circ H} \circ \paren {b \circ H} = \paren {a \circ b} \circ H$ is well-defined.
Klein Four-Group is Normal in A4
Let $A_4$ denote the alternating group on $4$ letters, whose Cayley table is given as: {{:Alternating Group on 4 Letters/Cayley Table}}  The subsets of $A_4$ which form subgroups of $A_4$ are as follows: Consider the order $4$ subgroup $V$ of $A_4$, presented by Cayley table: :$\begin{array}{c|cccc} \circ & e & t & u & v \\ \hline e & e & t & u & v \\ t & t & e & v & u \\ u & u & v & e & t \\ v & v & u & t & e \\ \end{array}$ Then $V$ is normal in $A_4$. Its index is: :$\index {A_4} V = \dfrac {\order {A_4} } {\order V} = \dfrac {12} 4 = 3$ The (left) cosets of $V$ are: :$V$ :$A := a V$ :$P := p V$ and the Cayley table of the quotient group $A_4 / V$ is given by: :$\begin{array}{c|ccc} \circ & V & A & P \\ \hline V & V & A & P \\ A & A & P & V \\ P & P & V & A \\ \end{array}$ Note that while $A_4 / V$ is Abelian, $A_4$ is not.
Coset of Trivial Subgroup is Singleton
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $E := \struct {\set e, \circ}$ denote the trivial subgroup of $\struct {G, \circ}$. Let $g \in G$. Then the left coset and right coset of $E$ by $g$ is $\set g$.
Inverse Elements of Right Transversal is Left Transversal
Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $S \subseteq G$ be a right transversal for $H$ in $G$. Let $T$ be the set defined as: :$T := \set {x^{-1}: x \in S}$ where $x^{-1}$ is the inverse of $x$ in $G$. Then $T$ is a left transversal for $H$ in $G$.
Condition for Subset of Group to be Right Transversal
Let $G$ be a group. Let $H$ be a subgroup of $G$ whose index in $G$ is $n$: :$\index G H = n$ Let $S \subseteq G$ be a subset of $G$ of cardinality $n$. Then $S$ is a right transversal for $H$ in $G$ : :$\forall x, y \in S: x \ne y \implies x y^{-1} \notin H$
Group Action on Coset Space is Transitive
Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$. Let $*: G \times G / H \to G / H$ be the action on the (left) coset space: :$\forall g \in G, \forall g' H \in G / H: g * \paren {g' H} := \paren {g g'} H$ Then $G$ is a transitive group action.
Stabilizer of Coset under Group Action on Coset Space
Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$. Let $*: G \times G / H \to G / H$ be the action on the (left) coset space: :$\forall g \in G, \forall g' H \in G / H: g * \paren {g' H} := \paren {g g'} H$ Then the stabilizer of $a H$ under $*$ is given by: :$\Stab {a H} = a H a^{-1}$
Index of Subgroup equals Index of Conjugate
Let $G$ be a group. Let $H$ be a subgroup of $G$. Then: :$\index G H = \index G {a H a^{-1} }$ where $\index G H$ denotes the index of $H$ in $G$.
Normality Relation is not Transitive
Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $K$ be a normal subgroup of $N$. Then it is not necessarily the case that $K$ is a normal subgroup of $G$.
Power of Coset Product is Coset of Power
Let $\struct {G, \circ}$ be a group. Let $N$ be a normal subgroup of $G$. Let $a \in G$. Then: :$\forall n \in \Z: \paren {a \circ N} = \paren {a^n} \circ N$
Condition for Power of Element of Quotient Group to be Identity
Let $G$ be a group whose identity is $e$. Let $N$ be a normal subgroup of $G$. Let $a \in G$. Then: :$\paren {a N}^n$ is the identity of the quotient group $G / N$ : :$a^n \in N$
Additive Group of Integers is Normal Subgroup of Reals
Let $\struct {\Z, +}$ be the additive group of integers. Let $\struct {\R, +}$ be the additive group of real numbers. Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\R, +}$.
Mapping from Additive Group of Integers to Powers of Group Element is Homomorphism
Let $\struct {G, \circ}$ be a group. Let $g \in G$. Let $\struct {\Z, +}$ denote the additive group of integers. Let $\phi_g: \struct {\Z, +} \to \struct {G, \circ}$ be the mapping defined as: :$\forall k \in \Z: \map {\phi_g} k = g^k$ Then $\phi_g$ is a (group) homomorphism.
Inner Automorphisms form Subgroup of Symmetric Group
Let $G$ be a group. Let $\struct {\map \Gamma G, \circ}$ be the symmetric group on $G$. Let $\Inn G$ denote the inner automorphism group of $G$. Then: :$\Inn G \le \struct {\map \Gamma G, \circ}$ where $\le$ denotes the relation of being a subgroup.
Order is Preserved by Group Isomorphism
Let $G$ and $H$ be groups. Let $\phi: G \to H$ be a (group) isomorphism. Then: :$\order G = \order H$ where $\order {\, \cdot \,}$ denotes the order of a group.
Additive Group of Reals is Subgroup of Complex
Let $\struct {\R, +}$ be the additive group of real numbers. Let $\struct {\C, +}$ be the additive group of complex numbers. Then $\struct {\R, +}$ is a subgroup of $\struct {\C, +}$.
Mapping to Power is Endomorphism iff Abelian
Let $\struct {G, \circ}$ be a group. Let $n \in \Z$ be an integer. Let $\phi: G \to G$ be defined as: :$\forall g \in G: \map \phi g = g^n$ Then $\struct {G, \circ}$ is abelian  $\phi$ is a (group) endomorphism.
Additive Groups of Integers and Integer Multiples are Isomorphic
Let $n \in \Z_{> 0}$ be a strictly positive integer. Let $\struct {n \Z, +}$ denote the additive group of integer multiples. Let $\struct {\Z, +}$ denote the additive group of integers. Then $\struct {n \Z, +}$ is isomorphic to $\struct {\Z, +}$.
Additive Group of Real Numbers is Not Isomorphic to Multiplicative Group of Real Numbers
Let $\struct {\R, +}$ denote the additive group of real numbers. Let $\struct {\R_{\ne 0}, \times}$ denote the multiplicative group of real numbers. Then $\struct {\R, +}$ is not isomorphic to $\struct {\R_{\ne 0}, \times}$.
Normal Subgroup is Kernel of Group Homomorphism
Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Then there exists a group homomorphism of which $N$ is the kernel.
Homomorphic Image of Cyclic Group is Cyclic Group
Let $G$ be a cyclic group with generator $g$. Let $H$ be a group. Let $\phi: G \to H$ be a (group) homomorphism. Let $\Img G$ denote the homomorphic image of $G$ under $\phi$. Then $\Img G$ is a cyclic group with generator $\map \phi g$. That is: :$\phi \sqbrk {\gen g} = \gen {\map \phi g}$
Kernel of Homomorphism on Cyclic Group
Let $G = \gen g$ be a cyclic group with generator $g$. Let $H$ be a group. Let $\phi: G \to H$ be a (group) homomorphism. Let $\map \ker \phi$ denote the kernel of $\phi$. Let $\Img G$ denote the homomorphic image of $G$ under $\phi$. Then: :$\map \ker \phi = \gen {g^m}$ where: :$m = 0$ if $\Img \phi$ is an infinite cyclic group :$m = \order {\Img \phi}$ if $\Img \phi$ is a finite cyclic group.
Mapping from Group Element to Inner Automorphism is Homomorphism
Let $G$ be a group. Let $\kappa: G \to \Aut G$ be the mapping from $G$ to the automorphism group of $G$ defined as: :$\forall x \in G: \map \kappa x := \kappa_x$ where $\kappa_x$ is the inner automorphism on $x$: :$\forall g \in G: \map {\kappa_x} g = x g x^{-1}$ Then $\kappa$ is a homomorphism.
Image of Mapping from Group Element to Inner Automorphism is Inner Automorphism Group
Let $G$ be a group. Let $\kappa: G \to \Aut G$ be the mapping from $G$ to the automorphism group of $G$ defined as: :$\forall x \in G: \map \kappa x := \kappa_x$ where $\kappa_x$ is the inner automorphism on $x$: :$\forall g \in G: \map {\kappa_x} g = x g x^{-1}$ Then $\Img \kappa$ is the inner automorphism group of $G$.
Order of Monomorphic Image of Group Element
Let $G$ and $H$be groups whose identities are $e_G$ and $e_H$ respectively. Let $\phi: G \to H$ be a monomorphism. Let $g \in G$ be of finite order. Then: :$\forall g \in G: \order {\map \phi g} = \order g$
Image under Epimorphism of Center is Subset of Center
Let $G$ and $H$ be groups. Let $\theta: G \to H$ be an epimorphism. Let $\map Z G$ denote the center of $G$. Then: :$\theta \sqbrk {\map Z G} \subseteq \map Z H$
Metric Subspace Induces Subspace Topology
Let $M = \struct {A,d}$ be a metric space. Let $H \subseteq A$. Let $\tau$ be the topology induced by the metric $d$. Let $\tau_H$ be the subspace topology induced by $\tau$ on $H$. Let $d_H$ be the subspace metric induced by $d$ on $H$. Let $\tau_{d_H}$ be the topology induced by the metric $d_H$. Then: :$\tau_{d_H} = \tau_H$
Quotient of Cauchy Sequences is Metric Completion/Lemma 1
:$\quad \mathcal C \,\big / \mathcal N = \tilde {\mathcal C}$
Quotient of Cauchy Sequences is Metric Completion/Lemma 2
:$\quad d' = \tilde d$
Generator for Quaternion Group
The Quaternion Group can be generated by the matrices: :$\mathbf a = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf b = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$ where $i$ is the imaginary unit: :$i^2 = -1$
Condition for Nu Function to be 1
Let: :$n = \displaystyle \prod_{i \mathop = 1}^s p_i^{m_i}$ where $p_1, p_2, \ldots, p_s$ are distinct primes. Then: :$(1): \quad m_1, m_2, \ldots, m_s = 1$, that is, $n$ is square-free :$(2): \quad \forall i, j \in \set {1, 2, \ldots, s}: p_i \not \equiv 1 \pmod {p_j}$ : :every group $G$ of order $n$ is cyclic and so $\map \nu n = 1$.
Group of Order 15 is Cyclic Group
Let $G$ be a group whose order is $15$. Then $G$ is cyclic.
Number of Abelian Groups
Let $n \in \Z_{\ge 1}$ be a (strictly) positive integer. Let: :$n = \displaystyle \prod_{i \mathop = 1}^s p_i^{m_i}$ where the $p_i$ are distinct primes. Let $\map {\nu_a} n$ denote the number of abelian groups of order $n$. Then: :$\map {\nu_a} n = \displaystyle \prod_{i \mathop = 1}^s \map {\nu_a} {p_i^{m_i} }$ where: :$\map {\nu_a} {p_i^{m_i} }$ is the number of integer partitions of $m_i$.
Order of Quotient Group
Let $G$ be a finite group. Let $N$ be a normal subgroup of $G$. Let $G / N$ be the quotient group of $G$ by $N$. Then: :$\dfrac {\order G} {\order N} = \order {G / N}$ where $\order G$ denotes the order of $G$.
Definition:Composition Series/Composition Length
Let $\HH$ be a composition series for $G$. The '''composition length''' of $G$ is the length of $\HH$.
Definition:Composition Series/Composition Factor
Let $\HH = \set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G$ be a composition series for $G$. Each of the quotient groups: :$G_1 / G_0, G_2 / G_1, \ldots, G_n / G_{n - 1}$ are the '''composition factors''' of $G$.
Non-Abelian Simple Finite Groups are Infinitely Many
There exist infinitely many types of group which are non-abelian and finite.
Upper Bound of Order of Non-Abelian Finite Simple Group
Let $G$ be a non-abelian finite simple group. Let $t \in G$ be a self-inverse element of $G$. Let $\map {C_G} t$ denote the centralizer of $t$ in $G$. Let $m = \order {\map {C_G} t}$ be the order of $\map {C_G} t$. Then: :$\order G \le \paren {\dfrac {m \paren {m + 1} } 2}!$
Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 1
:$\norm {\, \cdot \,}_1$ is well-defined. That is,  :$(1): \quad \forall \eqclass {x_n}{}: \lim_{n \mathop \to \infty} \norm{x_n}$ exists. :$(2): \quad \displaystyle \forall \eqclass {x_n}{}, \eqclass {y_n}{} \in \mathcal C \,\big / \mathcal N: \eqclass {x_n}{} = \eqclass {y_n}{} \implies \lim_{n \mathop \to \infty} \norm{x_n} = \lim_{n \mathop \to \infty} \norm{y_n}$
Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 2
:$\norm {\, \cdot \,}_1$ satisfies {{NormAxiom|1}} That is: :$\forall \eqclass {x_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} }_1 = 0 \iff \eqclass {x_n} {} = \eqclass {0_R} {} $
Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 3
:$\norm {\, \cdot \,}_1$ satisfies the {{NormAxiom|2}}. That is: :$\forall \eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} \eqclass {y_n} {} }_1 = \norm {\eqclass {x_n} {} }_1 \times \norm {\eqclass {y_n} {} }_1$
Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 4
:$\norm {\, \cdot \,}_1$ satisfies the {{NormAxiom|3}}. That is: :$\forall \eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} +  \eqclass {y_n} {} }_1 \le \norm {\eqclass {x_n} {} }_1 + \norm {\eqclass {y_n} {} }_1$
Invertible Elements of Monoid form Subgroup
Let $\struct {S, \circ}$ be a monoid whose identity element is $e$. Let $U \subseteq S$ be the subset of $S$ consisting of the invertible elements of $S$. Then $\struct {U, \circ}$ forms a subgroup of $S$.
Combination Theorem for Sequences/Normed Division Ring/Inverse Rule/Lemma
:$\displaystyle \lim_{n \mathop \to \infty} y_n^{-1} = l^{-1}$
Division Ring Norm is Continuous on Induced Metric Space
Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring. Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$. The mapping $\norm {\,\cdot\,} : \struct {R, d} \to \R$ is continuous.
Properties of Norm on Division Ring
Let $\struct {R, +, \circ}$ be a division ring with zero $0_R$ and unity $1_R$. Let $\norm {\,\cdot\,}$ be a norm on $R$. Let $x, y \in R$. Then the following hold:
Properties of Norm on Division Ring/Norm of Negative
:$\norm {-x} = \norm {x}$
Properties of Norm on Division Ring/Norm of Unity
:$\norm {1_R} = 1$.
Properties of Norm on Division Ring/Norm of Negative of Unity
:$\norm{-1_R} = 1$
Repunit is Zuckerman Number
Let $n$ be a repunit. Then $n$ is also a Zuckerman number.
Properties of Norm on Division Ring/Norm of Difference
:$\norm {x - y} \le \norm x + \norm y$
Properties of Norm on Division Ring/Norm of Inverse
:$x \ne 0_R \implies \norm {x^{-1} } = \dfrac 1 {\norm x}$
Properties of Norm on Division Ring/Norm of Quotient
:$y \ne 0_R \implies \norm{xy^{-1}} = \norm{y^{-1}x} = \dfrac {\norm{x}}{\norm{y}}$
Properties of Norm on Division Ring/Norm of Power Equals Unity
:$\forall n \in \N_{\gt 0}: \norm {x^n} = 1 \implies \norm x = 1$
Conversion from Hexadecimal to Binary
Let $n$ be a (positive) integer expressed in hexadecimal notation as: :$n = \sqbrk {a_r a_{r - 1} \dotso a_1 a_0}_H$ Then $n$ can be expressed in binary notation as: :$n = \sqbrk {b_{r 3} b_{r 2} b_{r 1} b_{r 0} b_{\paren {r - 1} 3} b_{\paren {r - 1} 2} b_{\paren {r - 1} 1} b_{\paren {r - 1} 0} \dotso b_{1 3} b_{1 2} b_{1 1} b_{1 0} b_{0 3} b_{0 2} b_{0 1} b_{0 0} }_2$ where $\sqbrk {b_{j 3} b_{j 2} b_{j 1} b_{j 0} }_2$ is the expression of the hexadecimal digit $a_j$ in binary notation. That is, you take the binary expression of each hexadecimal digit, padding them out with zeroes to make them $4$ bits long, and simply concatenate them.
Birthday Paradox/General/3
Let $n$ be a set of people. Let the probability that at least $3$ of them have the same birthday be greater than $50 \%$. Then $n \ge 88$.
Convergent Sequence is Cauchy Sequence/Normed Division Ring
Let $\struct {R, \norm {\,\cdot\,}} $ be a normed division ring. Every convergent sequence in $R$ is a Cauchy sequence.
Sum of 3 Squares in 2 Distinct Ways
$27$ is the smallest positive integer which can be expressed as the sum of $3$ square numbers in $2$ distinct ways: {{begin-eqn}} {{eqn | l = 27       | r = 3^2 + 3^2 + 3^2 }} {{eqn | r = 5^2 + 1^2 + 1^2 }} {{end-eqn}}
Triangular Numbers which are Sum of Two Cubes
The sequence of triangular numbers which are the sum of $2$ cubes begins: :$28, 91, 351, 2926, 8001, 46971, 58653, 93528, 97461, \dots$ {{OEIS|A113958}}
Product of Factors of Perfect Number
Let $P$ be the perfect number $2^{n - 1} \paren {2^n - 1}$. Then: :$\displaystyle \prod_{d \mathop \divides P} d = P^n$
Sequence is Bounded in Norm iff Bounded in Metric
Let $\struct {R, \norm {\,\cdot\,} } $ be a normed division ring. Let $d$ be the metric induced on $R$ be the norm $\norm {\,\cdot\,}$. Let $\sequence {x_n}$ be a sequence in $R$. Then: :$\sequence {x_n} $ is a bounded sequence in the normed division ring $\struct {R, \norm {\,\cdot\,} }$  $\sequence {x_n} $ is a bounded sequence in the metric space $\struct {R, d}$.
Sequence is Bounded in Norm iff Bounded in Metric/Necessary Condition
Let $\struct {R, \norm {\,\cdot\,}} $ be a normed division ring. Let $d$ be the metric induced on $R$ be the norm $\norm {\,\cdot\,}$. Let $\sequence {x_n}$ be a sequence in $R$. Let $\sequence {x_n} $ be a bounded sequence in the normed division ring $\struct {R, \norm {\,\cdot\,}}$  Then: :$\sequence {x_n} $ is a bounded sequence in the metric space $\struct {R, d}$
Sequence is Bounded in Norm iff Bounded in Metric/Sufficient Condition
Let $\struct {R, \norm {\,\cdot\,}} $ be a normed division ring. Let $d$ be the metric induced on $R$ be the norm $\norm {\,\cdot\,}$. Let $\sequence {x_n}$ be a sequence in $R$. Let $\sequence {x_n} $ be a bounded sequence in the metric space $\struct {R, d}$ Then: :$\sequence {x_n} $ is a bounded sequence in the normed division ring $\struct {R, \norm {\,\cdot\,} }$
Magic Constant of Magic Cube
The magic constant of a magic cube of order $n$ is given by: :$C_n = \dfrac {n \paren {n^3 + 1} } 2$
Sum of Terms of Magic Cube
The total of all the entries in a magic cube of order $n$ is given by: :$T_n = \dfrac {n^3 \paren {n^3 + 1} } 2$
Smallest Magic Cube is of Order 3
Apart from the trivial order $1$ magic cube: {{:Magic Cube/Examples/Order 1}} the smallest magic cube is the order $3$ magic cube: {{:Magic Cube/Examples/Order 3}}
Fourth Power as Summation of Groups of Consecutive Integers
Take the positive integers and group them in sets such that the $m$th set contains the next $m$ positive integers: :$\set 1, \set {2, 3}, \set {4, 5, 6}, \set {7, 8, 9, 10}, \set {11, 12, 13, 14, 15}, \ldots$ Remove all the sets with an even number of elements. Then the sum of all the integers in the first $n$ sets remaining equals $n^4$.
Smallest Number with 2^n Divisors
The smallest positive integer with $2^n$ divisors is found by multiplying together the first $n$ numbers in this sequence: :$2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, \ldots$ which consists of all the primes and powers of primes.
Convergent Subsequence of Cauchy Sequence/Normed Division Ring
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring. Let $\sequence{x_n}_{n \mathop \in \N}$ be a Cauchy sequence in $\struct {R, \norm {\,\cdot\,} }$.  Let $x \in R$. Then $\sequence {x_n}$ converges to $x$  $\sequence {x_n}$ has a subsequence that converges to $x$.
Null Sequences form Maximal Left and Right Ideal
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\CC$ be the ring of Cauchy sequences over $R$. Let $\NN$ be the set of null sequences. That is: :$\NN = \set {\sequence {x_n}: \displaystyle \lim_{n \mathop \to \infty} x_n = 0 }$ Then $\NN$ is a ring ideal of $\CC$ that is a maximal left ideal and a maximal right ideal.
Null Sequences form Maximal Left and Right Ideal/Lemma 1
:$\mathcal N$ is an ideal of $\mathcal C$.
Null Sequences form Maximal Left and Right Ideal/Lemma 2
:$\mathcal N$ is a maximal left ideal.
Null Sequences form Maximal Left and Right Ideal/Lemma 3
:$\NN$ is a maximal right ideal.
Prime Decomposition of 8th Fermat Number
The prime decomposition of the $8$th Fermat number is given by: {{begin-eqn}} {{eqn | l = 2^{\paren {2^8} } + 1       | r = 115 \, 792 \, 089 \, 237 \, 316 \, 195 \, 423 \, 570 \, 985 \, 008 \, 687 \, 907 \, 853 \, 269 \, 984 \, 665 \, 640 \, 564 \, 039 \, 457 \, 584 \, 007 \, 913 \, 129 \, 639 \, 937       | c =  }} {{eqn | r = 1 \, 238 \, 926 \, 361 \, 552 \, 897       | c =  }} {{eqn | o =       | ro= \times       | r = 93 \, 461 \, 639 \, 715 \, 357 \, 977 \, 769 \, 163 \, 558 \, 199 \, 606 \, 896 \, 584 \, 051 \, 237 \, 541 \, 638 \, 188 \, 580 \, 280 \, 321       | c =  }} {{eqn | r = \paren {2 \times 157 \times 3 \, 853 \, 149 \, 761 \times 2^{10} + 1}       | c =  }} {{eqn | o =       | ro= \times       | r = \paren {2 \times 3 \times 5 \times 7 \times 13 \times 31 \, 618 \, 624 \, 099 \, 079 \times 1 \, 057 \, 372 \, 046 \, 781 \, 162 \, 536 \, 274 \, 034 \, 354 \, 686 \, 893 \, 329 \, 625 \, 329 \times 2^{10} + 1}       | c =  }} {{end-eqn}}
Product of Sequence of Fermat Numbers plus 2
Let $F_n$ denote the $n$th Fermat number. Then: {{begin-eqn}} {{eqn | ll= \forall n \in \Z_{>0}:       | l = F_n       | r = \prod_{j \mathop = 0}^{n - 1} F_j + 2       | c =  }} {{eqn | r = F_0 F_1 \dotsm F_{n - 1} + 2       | c =  }} {{end-eqn}}
492 is Sum of 3 Cubes in 3 Ways
$492$ can be expressed as the sum of $3$ cubes, either positive or negative in $3$ known ways. {{begin-eqn}} {{eqn | l = 492       | r = 50^3 + \paren {-19}^3 + \paren {-49}^3 }} {{eqn | r = 123 \, 134^3 + 9179^3 + \paren {-123 \, 151}^3 }} {{eqn | r = 1 \, 793 \, 337 \, 644^3 + \paren {-81 \, 3701 \, 167}^3 + \paren {-1 \, 735 \, 662 \, 109}^3 }} {{end-eqn}}
Smallest n needing 6 Numbers less than n so that Product of Factorials is Square
Let $n \in \Z_{>0}$ be a positive integer. Then it is possible to choose at most $6$ positive integers less than $n$ such that the product of their factorials is square. The smallest $n$ that actually requires $6$ numbers to be chosen is $527$.
Null Sequences form Maximal Left and Right Ideal/Lemma 1/Lemma 1.1
:$\mathcal N \ne \O$
Null Sequences form Maximal Left and Right Ideal/Lemma 1/Lemma 1.2
:$\forall \sequence {x_n}, \sequence {y_n} \in \NN: \sequence {x_n} + \paren {-\sequence {y_n} } \in \NN$
Null Sequences form Maximal Left and Right Ideal/Lemma 1/Lemma 1.3
:$\quad \forall \sequence {x_n} \in \NN, \sequence {y_n} \in \CC: \sequence {x_n} \sequence {y_n} \in \NN, \sequence {y_n} \sequence {x_n} \in \NN$
Null Sequences form Maximal Left and Right Ideal/Lemma 2/Lemma 2.1
:$\NN \subsetneq \CC$.
Null Sequences form Maximal Left and Right Ideal/Lemma 2/Lemma 2.2
:There is no left ideal $\JJ$ of $\CC$ such that $\NN \subsetneq \JJ \subsetneq \CC$
Difference between Two Squares equal to Repdigit
{{begin-eqn}} {{eqn | l = 6^2 - 5^2       | r = 11       | c =  }} {{eqn | l = 56^2 - 45^2       | r = 1111       | c =  }} {{eqn | l = 556^2 - 445^2       | r = 111 \, 111       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 7^2 - 4^2       | r = 33       | c =  }} {{eqn | l = 67^2 - 34^2       | r = 3333       | c =  }} {{eqn | l = 667^2 - 334^2       | r = 333 \, 333       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 8^2 - 3^2       | r = 55       | c =  }} {{eqn | l = 78^2 - 23^2       | r = 5555       | c =  }} {{eqn | l = 778^2 - 223^2       | r = 555 \, 555       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 9^2 - 2^2       | r = 77       | c =  }} {{eqn | l = 89^2 - 12^2       | r = 7777       | c =  }} {{eqn | l = 889^2 - 112^2       | r = 777 \, 777       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}}
Largest Number Not Expressible as Sum of Fewer than 8 Cubes
$8042$ is (probably) the largest positive integer that cannot be expressed as the sum of fewer than $8$ cubes.
Sequence of 5 Consecutive Non-Primable Numbers by Changing 1 Digit
The following sequence of $5$ consecutive positive integers cannot be made into prime numbers by changing just one digit: :$872\,894, 872\,895, 872\,896, 872\,897, 872\,898$ {{OEIS|A192545}}
Smallest 17 Primes in Arithmetic Sequence
The smallest $17$ primes in arithmetic sequence are: :$3\,430\,751\,869 + 87\,297\,210 n$ for $n = 0, 1, \ldots, 16$.
Prime Gap of 654
There exists a prime gap of $654$ between $11\,000\,001\,446\,613\,353$ and $11\,000\,001\,446\,614\,007$.
Pair of Large Twin Primes
The integers defined as: :$1\,159\,142\,985 \times 2^{2304} \pm 1$ are a pair of twin primes each with $703$ digits.
Integers under Subtraction do not form Group
Let $\struct {\Z, -}$ denote the algebraic structure formed by the set of integers under the operation of subtraction. Then $\struct {\Z, -}$ is not a group.
Sequence of Powers of Number less than One/Necessary Condition
Let $x \in \R$ be such that $\size{x} < 1$. Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$. Then $\sequence {x_n}$ is a null sequence.
Sequence of Powers of Number less than One/Sufficient Condition
Let $x \in \R$. Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$. Let $\sequence {x_n}$ be a null sequence. Then $\size x < 1$.
Real Sine Function is neither Injective nor Surjective
The real sine function is neither an injection nor a surjection.
Sequence of Powers of Number less than One/Complex Numbers
Let $z \in \C$. Let $\sequence {z_n}$ be the sequence in $\C$ defined as $z_n = z^n$. Then: :$\size z < 1$  $\sequence {z_n}$ is a null sequence.