Error Correction Capability of Linear Code

Let $C$ be a linear code. Let $C$ have a minimum distance $d$. Then $C$ corrects $e$ transmission errors for all $e$ such that $2 e + 1 \le d$.


Generation of Linear Code from Standard Generator Matrix

Let $G$ be a (standard) generator matrix for a linear code. The following methods can be used to generate a linear code from $G$:


Generation of Linear Code from Standard Generator Matrix/Method 1

A linear code $C$ can be obtained from $G$ by: :considering the rows of $G$ as codewords :forming all possible linear combinations of those codewords, considering them as vectors of a vector space.


Generation of Linear Code from Standard Generator Matrix/Method 2

A linear code $C$ can be obtained from $G$ by: :taking the set $U$ of all sequences of length $k$ over $\Z_p$ and expressing them as $1 \times k$ matrices :forming all possible matrix products $u G$ for all $u \in U$.


Golay Ternary Code has Minimum Distance 5

The Golay ternary code has a minimum distance of $5$.


Golay Ternary Code Corrects 2 Errors

The Golay ternary code corrects $2$ transmission errors.


Decoding Received Word using Coset Decoding Table

Let $C$ be a linear code. Let $v$ be a received word, which may have transmission errors. To find out the transmitted codeword $u$ corresponding to $v$: :$(1): \quad$ Find $v$ in the coset decoding table for $C$. :$(2): \quad$ The corresponding transmitted codeword $u$ will be found at the top of the column where $v$ can be found.


Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 1

:$\norm {\, \cdot \,}_1$ is the trivial norm.


Syndrome is Zero iff Vector is Codeword

Let $C$ be a linear $\tuple {n, k}$-code whose master code is $\map V {n, p}$ Let $G$ be a (standard) generator matrix for $C$. Let $P$ be a standard parity check matrix for $C$. Let $w \in \map V {n, p}$. Then the syndrome of $w$ is zero {{iff}} $w$ is a codeword of $C$.


Condition for Vectors to have Same Syndrome

Let $C$ be a linear $\tuple {n, k}$-code whose master code is $\map V {n, p}$ Let $G$ be a (standard) generator matrix for $C$. Let $P$ be a standard parity check matrix for $C$. Let $u, v \in \map V {n, p}$. Then $u$ and $v$ have the same syndrome {{iff}} they are in the same coset of $C$.


Syndrome Decoding

Let $C$ be a linear $\tuple {n, k}$-code whose master code is $\map V {n, p}$ To decode a given vector $v$ of $\map V {n, p}$, the syndrome of $v$ can be used as follows. Create an array $T$ of $2$ column consisting of the following: :The top row contains: ::in column $1$: the zero of $C$  ::in column $2$: its syndrome. :The $r$th row subsequent contains: ::in column $1$: any element of $\map V {n, p}$ of minimum weight which is not already included in the first $r - 1$ rows ::in column $2$: its syndrome. To decode a given vector $v$ of $\map V {n, p}$: :Calculate its syndrome :Find it in column $2$ of $T$ :See what is in column $1$ of $T$, and call it $u$, say :Subtract $u$ from $v$.


Euler's Equation/Independent of x

Let $y$ be a mapping. Let $J$ be a functional such that: :$\ds J \sqbrk y = \int_a^b \map F {y, y'} \rd x$ Then the corresponding Euler's Equation can be reduced to: :$F - y' F_{y'} = C$  where $C$ is an arbitrary constant.


Euler's Equation/Independent of y

Let $y$ be a mapping Let $J$ be a functional such that :$\ds J \sqbrk y = \int_a^b \map F {x,y'} \rd x$ Then the corresponding Euler's equation can be reduced to: :$F_{y'} = C$  where $C$ is an arbitrary constant.


Subset of Linear Code with Even Weight Codewords

Let $C$ be a linear code. Let $C^+$ be the subset of $C$ consisting of all the codewords of $C$ which have even weight. Then $C^+$ is a subgroup of $C$ such that either $C^+ = C$ or such that $\order {C^+} = \dfrac {\order C} 2$.


Minimal Smooth Surface of Revolution

Let $\map y x$ be a real mapping in 2-dimensional real Euclidean space. Let $y$ pass through the points $\tuple {x_0, y_0}$ and $\tuple {x_1, y_1}$. Consider a surface of revolution constructed by rotating $y$ around the $x$-axis. Suppose this surface is smooth for any $x$ between $x_0$ and $x_1$. Then its surface area is minimized by the following curve, known as a catenoid: :$y = C \map \cosh {\dfrac {x + C_1} C}$ Furthermore, its area is: :$A = \paren {x_1 - x_0} C \pi + \dfrac {\pi C^2} 2 \paren {\map \sinh {\dfrac {2 \paren {x_1 + C_1} } C} - \map \sinh {\dfrac {2 \paren {x_0 + C_1} } C} }$


Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2

:$\forall x \in R: \norm x_1 = \norm x_2^\alpha$


Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.1

Then: :$\forall y \in R: \norm y_1 > 1 \iff \norm y_2 > 1$


Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.2

Then: :$\forall y \in R:\norm y_1 = 1 \iff \norm y_2 = 1$


Homomorphism from Group of Cube Roots of Unity to Itself

Let $\struct {U_3, \times}$ denote the multiplicative group of the complex cube roots of unity. Here, $U_3 = \set {1, \omega, \omega^2}$ where $\omega = e^{2 i \pi / 3}$. Let $\phi: U_3 \to U_3$ be defined as: :$\forall z \in U_3: \map \phi z = \begin{cases} 1 & : z = 1 \\ \omega^2 & : z = \omega \\ \omega & : z = \omega^2 \end{cases}$ Then $\phi$ is a group homomorphism.


Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.3

:$\alpha = \beta$


Max is Associative

The Max operation is associative: : $\max \left({\max \left({x, y}\right), z}\right) = \max \left({x, \max \left({y, z}\right)}\right)$ Thus we are justified in writing $\max \left({x, y, z}\right)$.


Min is Associative

The Min operation is associative: : $\min \left({\min \left({x, y}\right), z}\right) = \min \left({x, \min \left({y, z}\right)}\right)$ Thus we are justified in writing $\min \left({x, y, z}\right)$.


Theoretical Justification for Cycle Notation

Let $\N_k$ be used to denote the initial segment of natural numbers: :$\N_k = \closedint 1 k = \set {1, 2, 3, \ldots, k}$ Let $\rho: \N_n \to \N_n$ be a permutation of $n$ letters. Let $i \in \N_n$. Let $k$ be the smallest (strictly) positive integer for which $\map {\rho^k} i$ is in the set: :$\set {i, \map \rho i, \map {\rho^2} i, \ldots, \map {\rho^{k - 1} } i}$ Then: :$\map {\rho^k} i = i$


Equivalence of Definitions of Equivalent Division Ring Norms/Norm is Power of Other Norm implies Topologically Equivalent

Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy: :$\exists \alpha \in \R_{\gt 0}: \forall x \in R: \norm x_1 = \norm x_2^\alpha$ Then $d_1$ and $d_2$ are topologically equivalent metrics.


Equivalence of Definitions of Equivalent Division Ring Norms/Cauchy Sequence Equivalent implies Open Unit Ball Equivalent

Let $R$ be a division ring. Let $\norm {\, \cdot \,}_1: R \to \R_{\ge 0}$ and $\norm {\, \cdot \,}_2: R \to \R_{\ge 0}$ be norms on $R$. Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy: :for all sequences $\sequence {x_n}$ in $R$: $\sequence {x_n}$ is a Cauchy sequence in $\norm {\, \cdot \,}_1$ {{iff}} $\sequence {x_n}$ is a Cauchy sequence in $\norm {\, \cdot \,}_2$ Then $\forall x \in R$: :$\norm x_1 < 1 \iff \norm x_2 < 1$


Equal Order Elements may not be Conjugate

Let $G$ be a group Let $x, y \in G$ be elements of $G$ such that: :$\order x = \order y$ where $\order x$ denotes the order of $x$. Then it is not necessarily the case that $x$ and $y$ are conjugates.


Group Action of Symmetric Group on Complex Vector Space

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $S_n$ denote the symmetric group on $n$ letters. Let $V$ denote a vector space over the complex numbers $\C$. Let $V$ have a basis: :$\mathcal B := \set {v_1, v_2, \ldots, v_n}$ Let $*: S_n \times V \to V$ be a group action of $S_n$ on $V$ defined as: :$\forall \tuple {\rho, v} \in S_n \times V: \rho * v := \lambda_1 v_{\map \rho 1} + \lambda_2 v_{\map \rho 2} + \dotsb + \lambda_n v_{\map \rho n}$ where: : $v = \lambda_1 v_1 + \lambda_2 v_2 + \dotsb + \lambda_n v_n$ Then $*$ is a group action.


Group Action of Symmetric Group on Complex Vector Space/Orbit

The orbit of an element $v \in V$ is: :$\Orb v = \displaystyle \set {w \in V: \exists \rho \in S_n: w = \sum_{k \mathop = 1}^n \lambda_k v_{\map \rho k} }$


Group Action of Symmetric Group on Complex Vector Space/Stabilizer

The stabilizer of an element $v \in V$ is: :$\Stab v = \displaystyle \set {\rho \in S_n: \sum_{k \mathop = 1}^n \lambda_k v_k = \sum_{k \mathop = 1}^n \lambda_{\map \rho k} v_k}$


Equivalence of Definitions of Equivalent Division Ring Norms/Norm is Power of Other Norm implies Cauchy Sequence Equivalent

Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy: :$\exists \alpha \in \R_{> 0}: \forall x \in R: \norm x_1 = \norm x_2^\alpha$ Then for all sequences $\sequence {x_n}$ in $R$: :$\sequence {x_n}$ is a Cauchy sequence in $\norm {\, \cdot \,}_1$ {{iff}} $\sequence {x_n}$ is a Cauchy sequence in $\norm {\, \cdot \,}_2$


Conjugacy Classes of Symmetric Group

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $S_n$ denote the symmetric group on $n$ letters. The conjugacy classes of $S_n$ are determined entirely by the cycle type. That is, the conjugacy class $\conjclass x$ of an element $x$ of $S_n$ consists of all the elements of $S_n$ whose cycle type is the same as the cycle type of $x$.


Identity of Group is in Center

Let $G$ be a group. Let $e$ be the identity of $G$. Then $e$ is in the center of $G$: :$e \in \map Z G$


Identity of Group is in Singleton Conjugacy Class

Let $G$ be a group. Let $e$ be the identity of $G$. Then $e$ is in its own singleton conjugacy class: :$\conjclass e = \set e$


Finite Group with 2 Conjugacy Classes has 2 Elements

Let $G$ be a finite group. Let $G$ have exactly $2$ conjugacy classes. Then $G$ has exactly $2$ elements.


Group of Order 15 has Cyclic Subgroups of Order 3 and Order 5

Let $G$ be a group whose order is $15$. Then $G$ has :a cyclic subgroup of order $3$ and: :a cyclic subgroup of order $5$.


Number of Sylow p-Subgroups in Group of Order 15

Let $G$ be a group whose order is $15$. Then: :the number of Sylow $3$-subgroups is in the set $\set {1, 4, 7, \ldots}$ :the number of Sylow $5$-subgroups is in the set $\set {1, 6, 11, \ldots}$


Direct Product of Sylow p-Subgroups is Sylow p-Subgroup

Let $G_1$ and $G_2$ be groups. Let $H_1$ and $H_2$ be subgroups of $G_1$ and $G_2$ respectively. Let $H_1$ be a Sylow $p$-subgroup of $G_1$. Let $H_2$ be a Sylow $p$-subgroup of $G_2$. Then $H_1 \times H_2$ is a Sylow $p$-subgroup of $G_1 \times G_2$.


Direct Product of Unique Sylow p-Subgroups is Unique Sylow p-Subgroup

Let $G_1$ and $G_2$ be groups. Let $H_1$ and $H_2$ be subgroups of $G_1$ and $G_2$ respectively. Let $G_1$ be such that $H_1$ is the unique Sylow $p$-subgroup of $G_1$. Let $G_2$ be such that $H_2$ is the unique Sylow $p$-subgroup of $G_2$. Then $H_1 \times H_2$ is the unique Sylow $p$-subgroup of $G_1 \times G_2$.


Intersection of Sylow p-Subgroup with Subgroup not necessarily Sylow p-Subgroup

Let $G$ be a group. Let $P$ be a Sylow $p$-subgroup of $G$. Let $H$ be a subgroup of $G$. Then $P \cap H$ is not necessarily a Sylow $p$-subgroup of $H$.


Sylow p-Subgroups of Group of Order 2p

Let $p$ be an odd prime. Let $G$ be a group of order $2 p$. Then $G$ has exactly one Sylow $p$-subgroup. This Sylow $p$-subgroup is normal.


Groups of Order 2p

Let $p$ be a prime number. Let $G$ be a group. Let the order of $G$ be $2 p$. Then $G$ is either: :the cyclic group $C_{2 p}$ or: :the dihedral group $D_p$.


Group of Order p q has Normal Sylow p-Subgroup

Let $p$ and $q$ be prime numbers such that $p > q$. Let $G$ be a group of order $p q$. Then $G$ has exactly one Sylow $p$-subgroup. This Sylow $p$-subgroup is normal.


Characterisation of Non-Archimedean Division Ring Norms

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$. Then $\norm {\,\cdot\,}$ is non-Archimedean {{iff}}: :$\forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1$ where: :$n \cdot 1_R = \underbrace {1_R + 1_R + \dotsb + 1_R}_{\text {$n$ times} }$


Groups of Order 21

There exist exactly $2$ groups of order $21$, up to isomorphism: :$(1): \quad C_{21}$, the cyclic group of order $21$ :$(2): \quad$ the group whose group presentation is: :::$\gen {x, y: x^7 = e = y^3, y x y^{-1} = x^2}$


Groups of Order 21/Matrix Representation of Non-Abelian Instance

Let $G$ be the group of order $21$ whose group presentation is: :$\gen {x, y: x^7 = e = y^3, y x y^{-1} = x^2}$ Then $G$ can be instantiated by the following pair of matrices over $\Z_7$: :$X = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \qquad Y = \begin{pmatrix} 4 & 0 \\ 0 & 2 \end{pmatrix}$


Normal Sylow p-Subgroups in Group of Order 12

Let $G$ be of order $12$. Then $G$ has either: :a normal Sylow $2$-subgroup or: :a normal Sylow $3$-subgroup.


Group of Order p^2 q has Normal Sylow p-Subgroup

Let $p$ and $q$ be prime numbers such that $p \ne q$. Let $G$ be a group of order $p^2 q$. Then $G$ has a normal Sylow $p$-subgroup.


Group of Order 30 has Normal Cyclic Subgroup of Order 15

Let $G$ be of order $30$. Then $G$ has a normal subgroup of order $15$ which is cyclic.


Groups of Order 30/Lemma

Let $G$ be a group of order $30$. Then $G$ is one of the following: :The cyclic group $C_{30}$ :The dihedral group $D_{15}$ :Isomorphic to one of: ::$\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^4}$ ::$\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^{11} }$


Normal Subgroup of Group of Order 24

Let $G$ be a group of order $24$. Then $G$ has either: :a normal subgroup of order $8$ or: :a normal subgroup of order $4$.


Group of Order 35 is Cyclic Group

Let $G$ be a group whose order is $35$. Then $G$ is cyclic.


Group of Order 105 has Normal Sylow 5-Subgroup or Normal Sylow 7-Subgroup

Let $G$ be a group of order $105$. Then $G$ has either: :exactly one normal Sylow $5$-subgroup or: :exactly one normal Sylow $7$-subgroup.


Group of Order 105 has Normal Cyclic Subgroup of Index 3

Let $G$ be a group of order $105$. Then $G$ has a normal cyclic subgroup $N$ such that: :$\index G N = 3$ where $\index G N$ denotes the index of $N$ in $G$.


Diagonal Relation is Reflexive

The diagonal relation $\Delta_S$ on a set $S$ is a reflexive relation in $S$.


Diagonal Relation is Symmetric

The diagonal relation $\Delta_S$ on a set $S$ is a symmetric relation in $S$.


Diagonal Relation is Transitive

The diagonal relation $\Delta_S$ on a set $S$ is a transitive relation in $S$.


Group of Order 56 has Unique Sylow 2-Subgroup or Unique Sylow 7-Subgroup

Let $G$ be a group of order $56$. Then $G$ has either: :exactly one normal Sylow $2$-subgroup or: :exactly one normal Sylow $7$-subgroup.


Subgroup of Direct Product is not necessarily Direct Product of Subgroups

Let $G$ and $H$ be groups. Let $G \times H$ denote the direct product of $G$ and $H$. Let $K$ be a subgroup of $G \times H$. Then it is not necessarily the case that $K$ is of the form: :$G' \times H'$ where: :$G'$ is a subgroup of $G$ :$H'$ is a subgroup of $H$.


Groups of Order 30

Let $G$ be a group of order $30$. Then $G$ is one of the following: :The cyclic group $C_{30}$ :The dihedral group $D_{15}$ :The group direct product $C_5 \times D_3$ :The group direct product $C_3 \times D_5$


Characterisation of Non-Archimedean Division Ring Norms/Necessary Condition

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$. Then: :$\norm {\,\cdot\,}$ is non-Archimedean $\implies \forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1$. where: $n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} }$


Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$. Then: :$\forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1 \implies \norm {\,\cdot\,}$ is Definition:Non-Archimedean Division Ring Norm where: :$n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} }$


Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 1

Let $y \ne 0_R$ where $0_R$ is the zero of $R$. Then: :$\norm {x + y} \le \max \set {\norm x, \norm y} \iff \norm {x y^{-1} + 1_R} \le \max \set {\norm {x y^{-1} }, 1}$


Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 2

Then for all $i$, $0 \le i \le n$: :$\norm x^i \le \max \set {\norm x^n , 1}$


Group Epimorphism preserves Central Subgroups

Let $G$ and $H$ be groups. Let $\theta: G \to H$ be an epimorphism. Let $Z \le G$ be a central subgroup of $G$. Then $\theta \sqbrk Z$ is a central subgroup of $H$.


Direct Product of Central Subgroups

Let $G$ and $H$ be groups. Let $Z$ and $W$ be central subgroups of $G$ and $H$ respectively. Then $Z \times W$ is a central subgroup of $G \times H$.


Groups of Order 30/C 5 x D 3

Let $G$ be a group of order $30$. Let $G$ have the group presentation: :$\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^{11} }$ Then $G$ is isomorphic to the group direct product of the cyclic group $C_5$ and the dihedral group $D_3$: :$G \cong C_5 \times D_3$


Groups of Order 30/C 3 x D 5

Let $G$ be a group of order $30$. Let $G$ have the group presentation: :$\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^4}$ Then $G$ is isomorphic to the group direct product of the cyclic group $C_3$ and the dihedral group $D_5$: :$G \cong C_3 \times D_5$


Dihedral Group D6 is Internal Direct Product of C2 with D3

The dihedral group $D_6$ is an internal direct product of the cyclic group $C_2$ of order $2$ and the dihedral group $D_3$: :$D_6 = C_2 \times D_3$


Sequence of Integers defining Abelian Group

Let $n \in \Z_{>0}$ be a strictly positive integer. Let $C_n$ be a finite abelian group. Then $C_n$ is of the form: :$C_{n_1} \times C_{n_2} \times \cdots \times C_{n_r}$ such that: :$n = \displaystyle \prod_{k \mathop = 1}^r n_k$ :$\forall k \in \set {2, 3, \ldots, r}: n_k \divides n_{k - 1}$ where $\divides$ denotes divisibility.


Parallelism is Equivalence Relation

Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is parallel to $l_2$. Then $\parallel$ is an equivalence relation on $S$.


Connected Equivalence Relation is Trivial

Let $S$ be a set. Let $\mathcal R$ be a relation on $S$ which is both connected and an equivalence relation. Then $\mathcal R$ is the trivial relation on $S$.


Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 3

Let $\sequence {x_n}$ be the real sequence defined as $x_n = \paren {n + 1}^{1/n}$, using exponentiation. Then $\sequence {x_n}$ converges with a limit of $1$.


Characterisation of Non-Archimedean Division Ring Norms/Corollary 1

$\norm {\,\cdot\,}$ is non-Archimedean {{iff}}: :$\sup \set {\norm {n \cdot 1_R}: n \in \N_{> 0}} = 1$.


Characterisation of Non-Archimedean Division Ring Norms/Corollary 3

$\norm {\,\cdot\,}$ is Archimedean {{iff}}: :$\sup \set {\norm {n \cdot 1_R}: n \in \N_{\gt 0} } = +\infty$


Characterisation of Non-Archimedean Division Ring Norms/Corollary 2

Let $\sup \set {\norm {n \cdot 1_R}: n \in \N_{> 0} } = C < +\infty$.


Norms Equivalent to Absolute Value on Rational Numbers

Let $\alpha \in \R_{\gt 0}$. Let $\norm{\,\cdot\,}:\Q \to \R$ be the mapping defined by: :$\forall x \in \Q: \norm{x} = \size {x}^\alpha$ where $\size {x}$ is the absolute value of $x$ in $\Q$. Then: :$\norm{\,\cdot\,}$ is a norm on $\Q$ {{iff}} $\,\,\alpha \le 1$


Reflexive Relation on Set of Cardinality 2 is Transitive

Let $S$ be a set whose cardinality is equal to $2$: :$\card S = 2$ Let $\odot \subseteq S \times S$ be a reflexive relation on $S$. Then $\odot$ is also transitive.


Relation on Set of Cardinality 2 cannot be Non-Symmetric and Non-Transitive

Let $S$ be a set whose cardinality is equal to $2$: :$\card S = 2$ Let $\odot \subseteq S \times S$ be a relation on $S$. Then it is not possible for $\odot$ to be not symmetric and also not transitive.


Relation on Set of Cardinality 1 is Symmetric and Transitive

Let $S$ be a set whose cardinality is equal to $1$: :$\card S = 1$ Let $\odot \subseteq S \times S$ be a relation on $S$. Then $\odot$ is both symmetric and transitive.


Congruence of Triangles is Equivalence Relation

Let $S$ denote the set of all triangles in the plane. Let $\triangle A \cong \triangle B$ denote the relation that $\triangle A$ is congruent to $\triangle B$. Then $\cong$ is an equivalence relation on $S$.


Even Integer Plus 5 is Odd/Indirect Proof

{{:Even Integer Plus 5 is Odd}}


Even Integer Plus 5 is Odd/Proof by Contradiction

{{:Even Integer Plus 5 is Odd}}


Sum of Sequence of Cubes/Historical Note

The result '''Sum of Sequence of Cubes''' was documented by {{AuthorRef|Aryabhata the Elder}} in his work ''Āryabhaṭīya'' of $499$ CE.


Sum of Sequence of Odd Squares/Formulation 2

:$\displaystyle \forall n \in \Z_{> 0}: \sum_{i \mathop = 1}^n \paren {2 i - 1}^2 = \frac {4 n^3 - n} 3$


Sum of Sequence of Products of Consecutive Odd and Consecutive Even Numbers

{{begin-eqn}} {{eqn | lo= \forall n \in \Z_{>0}:       | l = \sum_{j \mathop = 1}^n j \paren {j + 2}       | r = 1 \times 3 + 2 \times 4 + 3 \times 5 + \dotsb + n \paren {n + 2}       | c =  }} {{eqn | r = \frac {n \paren {n + 1} \paren {2 n + 7} } 6       | c =  }} {{end-eqn}}


Arcsin as an Integral

:$\displaystyle \map \arcsin x = \int_0^\Theta \frac {\d x} {\sqrt {1 - x^2} }$


Sum of Sequence of Products of Consecutive Fibonacci Numbers

=== Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers === {{:Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers}} === Sum of Even Sequence of Products of Consecutive Fibonacci Numbers === {{:Sum of Even Sequence of Products of Consecutive Fibonacci Numbers}}


Lucas Number as Element of Recursive Sequence

Let $L_k$ be the $k$th Lucas number, defined as the sum of two Fibonacci numbers: :$L_n = F_{n - 1} + F_{n + 1}$ Then $L_n$ can be defined as the $n$th element of the recursive sequence: :$L_n = \begin{cases} 2 & : n = 0 \\ 1 & : n = 1 \\ L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$


Product of nth Lucas and Fibonacci Numbers

Let $L_k$ be the $k$th Lucas number. Let $F_k$ be the $k$th Fibonacci number. Then: :$\forall n \in \N_{>0}: F_n L_n = F_{2 n}$


Representation of Integers in Balanced Ternary

Let $n \in \Z$ be an integer. $n$ can be represented uniquely in balanced ternary: :$\displaystyle n = \sum_{j \mathop = 0}^m r_j 3^j$ :$\sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}$ such that: where: :$m \in \Z_{>0}$ is a strictly positive integer such that $3^m < \size {2 n} < 3^{m + 1}$ :all the $r_j$ are such that $r_j \in \set {\underline 1, 0, 1}$, where $\underline 1 := -1$.


Absolute Value is Norm

The absolute value is a norm on the set of real numbers $\R$.


Bounds for Integer Expressed in Base k

Let $n \in \Z$ be an integer. Let $k \in \Z$ such that $k \ge 2$. Let $n$ be expressed in base $k$ notation: :$n = \displaystyle \sum_{j \mathop = 1}^s a_j k^j$ where each of the $a_j$ are such that $a_j \in \set {0, 1, \ldots, k - 1}$. Then: :$0 \le n < k^{s + 1}$


Existence of q for which j - qk is Positive

Let $j, k \in \Z$ be integers such that $k > 0$. Then there exist $q \in \Z$ such that $j - q k > 0$.


Power Function on Base between Zero and One is Strictly Decreasing/Real Number

Let $a \in \R$ be a real number such that $0 \lt a \lt 1$. Let $f: \R \to \R$ be the real function defined as: :$\map f x = a^x$ where $a^x$ denotes $a$ to the power of $x$. Then $f$ is strictly decreasing.


Integral Ideal is Ideal of Ring

Let $J$ be a non-empty subset of the set of integers $\Z$. Then: :$J$ is an integral ideal {{iff}}: :$J$ is an ideal of the ring of integers $\struct {\Z, +, \times}$.


Set of Integer Multiples is Integral Ideal

Let $m \in \Z$ be an integer. Let $m \Z$ denote the set of integer multiples of $m$. Then $m \Z$ is an integral ideal.


Integral Ideal is Set of Integer Multiples

Let $J$ be an integral ideal. Then $J$ is in the form of a set of integer multiples $m \Z$ for some $m \in \Z$.


Difference between Odd Squares is Divisible by 8

Let $a$ and $b$ be odd integers. Then $a^2 - b^2$ is divisible by $8$.


Norms Equivalent to Absolute Value on Rational Numbers/Necessary Condition

Let $\alpha \in \R_{\gt 0}$. Let $\norm{\,\cdot\,}:\Q \to \R$ be the mapping defined by: :$\forall x \in \Q: \norm{x} = \size {x}^\alpha$ where $\size {x}$ is the absolute value of $x$ in $\Q$. Then: :$\norm{\,\cdot\,}$ is a norm on $\Q \implies \,\,\alpha \le 1$


Norms Equivalent to Absolute Value on Rational Numbers/Sufficient Condition

Let $\alpha \in \R_{\gt 0}$. Let $\norm{\,\cdot\,}:\Q \to \R$ be the mapping defined by: :$\forall x \in \Q: \norm{x} = \size {x}^\alpha$ where $\size {x}$ is the absolute value of $x$ in $\Q$. Then: :$\alpha \le 1 \implies \norm{\,\cdot\,}$ is a norm on $\Q$


Intersection of Sets of Integer Multiples

Let $m, n \in \Z$ such that $m n \ne 0$. Let $m \Z$ denote the set of integer multiples of $m$. Then: :$m \Z \cap n \Z = \lcm \set {m, n} \Z$ where $\lcm$ denotes lowest common multiple.


Set of Integer Multiples of GCD

Let $m, n \in \Z$. Let $m \Z$ denote the set of integer multiples of $m$ Then: :$m \Z \cup n \Z \subseteq \gcd \set {m, n} \Z$ where $\gcd$ denotes greatest common divisor.


GCD of Generators of General Fibonacci Sequence is Divisor of All Terms

Let $\mathcal F = \sequence {a_n}$ be a general Fibonacci sequence generated by the parameters $r, s, t, u$: :$a_n = \begin{cases} r & : n = 0 \\ s & : n = 1 \\ t a_{n - 2} + u a_{n - 1} & : n > 1 \end{cases}$ Let: :$d = \gcd \set {r, s}$ where $\gcd$ denotes greatest common divisor. Then: :$\forall n \in \Z_{>0}: d \divides a_n$


GCD of Consecutive Integers of General Fibonacci Sequence

Let $\mathcal F = \sequence {a_n}$ be a general Fibonacci sequence generated by the parameters $r, s, t, u$: :$a_n = \begin{cases} r & : n = 0 \\ s & : n = 1 \\ t a_{n - 2} + u a_{n - 1} & : n > 1 \end{cases}$ Let: :$d = \gcd \set {r, s}$ where $\gcd$ denotes greatest common divisor. Let $f = \gcd \set {a_m, a_{m - 1} }$ for some $m \in \Z$. Let $\gcd \set {f, t} = 1$. Then: :$f \divides d$


Three Points in Ultrametric Space have Two Equal Distances/Corollary 3

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm{\,\cdot\,}$, Let $x, y \in R$ and $\norm x \lt \norm y$. Then: :$\norm {x + y} = \norm {x - y} = \norm {y - x} = \norm y$


Three Points in Ultrametric Space have Two Equal Distances/Corollary 4

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm{\,\cdot\,}$, Let $x, y \in R$. Then: :* $\norm {x + y} \lt \norm y \implies \norm x = \norm y$ :* $\norm {x - y} \lt \norm y \implies \norm x = \norm y$ :* $\norm {y - x} \lt \norm y \implies \norm x = \norm y$


Equivalent Norms are both Non-Archimedean or both Archimedean

Let $R$ be a division ring with unity $1_R$. Let $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ be equivalent norms on $R$. Then $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are either both non-Archimedean or both Archimedean.


Perpendicular Distance from Straight Line in Plane to Origin

Let $L$ be the straight line embedded in the cartesian plane whose equation is given as: :$a x + b y = c$ Then the perpendicular distance $d$ between $L$ and $\tuple {0, 0}$ is given by: :$d = \size {\dfrac c {\sqrt {a^2 + b^2} } }$


Necessary Condition for Integral Functional to have Extremum/Two Variables

Let $D \subset \R^2$. Let $\Gamma$ be the boundary of $D$. Let $S$ be a set of real mappings such that: :$S = \set {\map z {x, y}: \paren {z: S_1 \subseteq \R^2 \to S_2 \subseteq \R}, \paren {\map z {x, y} \in \map {C^2}D}, \paren {\map z \Gamma = 0} }$ Let $J \sqbrk z: S \to S_3 \subseteq \R$ be a functional of the form: :$\ds \iint_D \map F {x, y, z, z_x, z_y} \rd x \rd y$  Then a necessary condition for $J \sqbrk y$ to have an extremum (strong or weak) for a given function $\map z {x, y}$ is that $\map z {x, y}$ satisfy Euler's equation: :$F_z - \dfrac \partial {\partial x} F_{z_x} - \dfrac \partial {\partial y} F_{z_y} = 0$


Line in Plane is Straight iff Gradient is Constant

Let $\mathcal L$ be a curve which can be embedded in the plane. Then $\mathcal L$ is a straight line {{iff}} it is of constant gradient.


Equation of Straight Line in Plane/General Equation

A straight line $\mathcal L$ is the set of all $\tuple {x, y} \in \R^2$, where: :$\alpha_1 x + \alpha_2 y = \beta$ where $\alpha_1, \alpha_2, \beta \in \R$ are given, and not both $\alpha_1, \alpha_2$ are zero.


Slope of Straight Line joining Points in Cartesian Plane

Let $p_1 := \tuple {x_1, y_1}$ and $p_2 := \tuple {x_2, y_2}$ be points in a cartesian plane. Let $\mathcal L$ be the straight line passing through $p_1$ and $p_2$. Then the slope of $\mathcal L$ is given by: :$\tan \theta = \dfrac {y_2 - y_1} {x_2 - x_1}$ where $\theta$ is the angle made by $\mathcal L$ with the $x$-axis.


Equation of Straight Line in Plane/Two-Point Form

Let $p_1 := \tuple {x_1, y_1}$ and $p_2 := \tuple {x_2, y_2}$ be points in a cartesian plane. Let $\LL$ be the straight line passing through $p_1$ and $p_2$. Then $\LL$ can be described by the equation: :$\dfrac {y - y_1} {x - x_1} = \dfrac {y_2 - y_1} {x_2 - x_1}$ or: :$\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$


Equation of Straight Line in Plane/Slope-Intercept Form

Let $\mathcal L$ be the straight line defined by the general equation: :$\alpha_1 x + \alpha_2 y = \beta$ Then $\mathcal L$ can be described by the equation: :$y = m x + c$ where: {{begin-eqn}} {{eqn | l = m       | r = -\dfrac {\alpha_1} {\alpha_2}       | c =  }} {{eqn | l = c       | r = \dfrac {\beta} {\alpha_2}       | c =  }} {{end-eqn}} such that $m$ is the slope of $\mathcal L$ and $c$ is the $y$-intercept.


Equation of Straight Line in Plane/Two-Intercept Form

Let $\mathcal L$ be a straight line which intercepts the $x$-axis and $y$-axis respectively at $\tuple {a, 0}$ and $\tuple {0, b}$, where $a b \ne 0$. Then $\mathcal L$ can be described by the equation: :$\dfrac x a + \dfrac y a = 1$


Equation of Straight Line in Plane/Normal Form

Let $\mathcal L$ be a straight line such that: :the perpendicular distance from $\mathcal L$ to the origin is $p$ :the angle made between that perpendicular and the $x$-axis is $\alpha$. Then $\mathcal L$ can be defined by the equation: :$x \cos \alpha + y \sin \alpha = p$


Perpendicular Distance from Straight Line in Plane to Point

Let $\LL$ be a straight line embedded in a cartesian plane, given by the equation: :$a x + b y = c$ Let $P$ be a point in the cartesian plane whose coordinates are given by: :$P = \tuple {x_0, y_0}$ Then the perpendicular distance $d$ from $P$ to $\LL$ is given by: :$d = \dfrac {\size {a x_0 + b y_0 + c} } {\sqrt {a^2 + b^2} }$


Equation of Straight Line in Plane/Point-Slope Form

Let $\mathcal L$ be a straight line embedded in a cartesian plane, given in slope-intercept form as: :$y = m x + c$ Let $\mathcal L$ pass through the point $\tuple {x_0, y_0}$. Then $\mathcal L$ can be expressed by the equation: :$y - y_0 = m \paren {x - x_0}$


Shortest Possible Distance between Lattice Points on Straight Line in Cartesian Plane

Let $\mathcal L$ be the straight line defined by the equation: :$a x - b y = c$ Let $p_1$ and $p_2$ be lattice points on $\mathcal L$. Then the shortest possible distance $d$ between $p_1$ and $p_2$ is: :$d = \dfrac {\sqrt {a^2 + b^2} } {\gcd \set {a, b} }$ where $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$.


Decomposition into Even-Odd Integers is not always Unique

For every even integer $n$ such that $n > 1$, if $n$ can be expressed as the product of one or more even-times odd integers, it is not necessarily the case that this product is unique.


Decomposition into Product of Power of 2 and Odd Integer is Unique

Let $n \in \Z$ be an integer. Then $n$ can be decomposed into the product of a power of $2$ and an odd integer.


Prime Decomposition of Integer is Unique

Let $n$ be an integer such that $n > 1$. Then the prime decomposition of $n$ is unique.


Expression for Integers as Powers of Same Primes

Let $a, b \in \Z$ be integers. Let their prime decompositions be given by: {{begin-eqn}} {{eqn | l = a       | r = {q_1}^{e_1} {q_2}^{e_2} \cdots {q_r}^{e_r} }} {{eqn | r = \prod_{\substack {q_i \mathop \divides a \\ \text {$q_i$ is prime} } } {q_i}^{e_i} }} {{eqn | l = b       | r = {s_1}^{f_1} {s_2}^{f_2} \cdots {s_u}^{f_u} }} {{eqn | r = \prod_{\substack {s_i \mathop \divides b \\ \text {$s_i$ is prime} } } {s_i}^{f_i} }} {{end-eqn}} Then there exist prime numbers: :$t_1 < t_2 < \dotsb < t_v$ such that: {{begin-eqn}} {{eqn | n = 1       | l = a       | r = {t_1}^{g_1} {t_2}^{g_2} \cdots {t_v}^{g_v} }} {{eqn | n = 2       | l = b       | r = {t_1}^{h_1} {t_2}^{h_2} \cdots {t_v}^{h_v} }} {{end-eqn}}


Expression for Integers as Powers of Same Primes/General Result

Let $a_1, a_2, \dotsc, a_n \in \Z$ be integers. Let their prime decompositions be given by: :$\displaystyle a_i = \prod_{\substack {p_{i j} \mathop \divides a_i \\ \text {$p_{i j}$ is prime} } } {p_{i j} }^{e_{i j} }$ Then there exists a set $T$ of prime numbers: :$T = \set {t_1, t_2, \dotsc, t_v}$ such that: :$t_1 < t_2 < \dotsb < t_v$ :$\displaystyle a_i = \prod_{j \mathop = 1}^v {t_j}^{g_{i j} }$


GCD from Prime Decomposition/General Result

Let $n \in \N$ be a natural number such that $n \ge 2$. Let $\N_n$ be defined as: :$\N_n := \set {1, 2, \dotsc, n}$ Let $A_n = \set {a_1, a_2, \dotsc, a_n} \subseteq \Z$ be a set of $n$ integers. From Expression for Integers as Powers of Same Primes, let: :$\displaystyle \forall i \in \N_n: a_i = \prod_{p_j \mathop \in T} {p_j}^{e_{i j} }$ where: :$T = \set {p_j: j \in \N_r}$ such that: :$\forall j \in \N_{r - 1}: p_j < p_{j - 1}$ :$\forall j \in \N_r: \exists i \in \N_n: p_j \divides a_i$ where $\divides$ denotes divisibility. Then: :$\displaystyle \map \gcd {A_n} = \prod_{j \mathop \in \N_r} {p_j}^{\min \set {e_{i j}: \, i \in \N_n} }$ where $\map \gcd {A_n}$ denotes the greatest common divisor of $a_1, a_2, \dotsc, a_n$.


LCM from Prime Decomposition/General Result

Let $n \in \N$ be a natural number such that $n \ge 2$. Let $\N_n$ be defined as: :$\N_n := \set {1, 2, \dotsc, n}$ Let $A_n = \set {a_1, a_2, \dotsc, a_n} \subseteq \Z$ be a set of $n$ integers. From Expression for Integers as Powers of Same Primes, let: :$\displaystyle \forall i \in \N_n: a_i = \prod_{p_j \mathop \in T} {p_j}^{e_{i j} }$ where: :$T = \set {p_j: j \in \N_r}$ such that: :$\forall j \in \N_{r - 1}: p_j < p_{j - 1}$ :$\forall j \in \N_r: \exists i \in \N_n: p_j \divides a_i$ where $\divides$ denotes divisibility. Then: :$\displaystyle \map \lcm {A_n} = \prod_{j \mathop \in \N_r} {p_j}^{\max \set {e_{i j}: \, i \in \N_n} }$ where $\map \lcm {A_n}$ denotes the greatest common divisor of $a_1, a_2, \dotsc, a_n$.


Alternating Summation of Binomial Coefficient of Summation of Binomial Coefficient of Sequence

Let $\sequence a, \sequence b$ be real sequences which satisfy the condition: :$a_n = \displaystyle \sum_{r \mathop = 0}^n \binom n r b_r$ Then: :$\displaystyle \paren {-1}^n b_n = \sum_{s \mathop = 0}^n \binom n s \paren {-1}^s a_s$


Integer and Fifth Power have same Last Digit

Let $n \in \Z$ be an integer. Then $n^5$ has the same last digit as $n$ when both are expressed in conventional decimal notation.


Sufficient Condition for 5 to divide n^2+1

Let: {{begin-eqn}} {{eqn | l = 5       | o = \nmid       | r = n - 1 }} {{eqn | l = 5       | o = \nmid       | r = n }} {{eqn | l = 5       | o = \nmid       | r = n + 1 }} {{end-eqn}} where $\nmid$ denotes non-divisibility. Then: :$5 \divides n^2 + 1$ where $\divides$ denotes divisibility.


Wilson's Theorem/Necessary Condition

Let $p$ be a prime number. Then: :$\paren {p - 1}! \equiv -1 \pmod p$


Wilson's Theorem/Sufficient Condition

Let $p$ be a (strictly) positive integer such that: :$\paren {p - 1}! \equiv -1 \pmod p$ Then $p$ is a prime number.


Necessary Condition for Integral Functional to have Extremum/Two Variables/Lemma

Let $D \subset \R^2$. Let $\Gamma$ be the boundary of $D$. Let $\alpha : D \to \R$ be a continuous mapping. Let $h : D \to \R$ be a twice differentiable mapping such that $\map h \Gamma = 0$. Suppose for every $h$ we have that: :$\displaystyle \iint_D \map \alpha {x, y} \map h {x,y} \rd x \rd y = 0$. Then: :$\displaystyle \forall x, y \in D : \map \alpha {x, y} = 0$


Number of Different n-gons that can be Inscribed in Circle

Let $C$ be a circle on whose circumference $n$ points are placed which divide $C$ into $n$ equal arcs. The number of different $n$-gons (either stellated or otherwise) that can be described on $C$ whose vertices are those $n$ points is: :$S_n = \dfrac {\paren {n - 1}!} 2$


Number of Regular Stellated Odd n-gons

Let $n \in \Z_{>0}$ be a strictly positive odd integer. Then there are $\dfrac {n - 1} 2$ distinct regular stellated $n$-gons.


Partition of Non-Regular Prime Stellated Cyclic Polygons into Rotation Classes

Let $p$ be an odd prime. Let $C$ be a circle whose center is $O$. Consider the set $P$ of $p$ points on the circumference of $C$ dividing it into $p$ equal arcs. Let $S$ be the set of all non-regular stellated $p$-gons whose vertices are the elements of $P$. Let $\sim$ denote the equivalence relation on $S$ defined as: :$\forall \tuple {a, b} \in S \times S: a \sim b \iff$ there exists a plane rotation about $O$ transforming $a$ to $b$. Then the $\sim$-equivalence classes of $S$ into which $S$ can thereby be partitioned all have cardinality $p$.


Square Modulo n Congruent to Square of Inverse Modulo n

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: :$a^2 \equiv \paren {n - a}^2 \pmod n$ where the notation denotes congruence modulo $n$.


Partition of Integer into Powers of 2 for Consecutive Integers

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\map b n$ denote the number of ways $n$ can be partitioned into (integer) powers of $2$. Then: :$\map b {2 n} = \map b {2 n + 1}$


Number of Partitions with no Multiple of 3 equals Number of Partitions where Parts appear No More than Twice

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\map t n$ denote the number of ways $n$ can be partitioned into parts which are specifically not multiples of $3$. Let $\map v n$ denote the number of ways $n$ can be partitioned such that no part appears twice. Then: :$\forall n \in \Z_{>0}: \map t n = \map v n$


Congruent Numbers are not necessarily Equal

Let $x, y, z \in \R$ be real numbers such that: :$x \equiv y \pmod z$ where $x \equiv y \pmod z$ denotes congruence modulo $z$. Then it is not necessarily the case that $x = y$.


Congruence Modulo Negative Number

Let $a, b, c \in \R$ be real numbers. Then: :$a \equiv b \pmod c \iff a \equiv b \pmod {-c}$


P-adic Norm and Absolute Value are Not Equivalent

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$. Let $\size{\,\cdot\,}$ be the absolute value on the rationals $\Q$. Then $\norm {\,\cdot\,}_p$ and $\size{\,\cdot\,}$ are not equivalent norms. That is, the topology induced by $\norm {\,\cdot\,}_p$ does not equal the topology induced by $\size {\,\cdot\,}$.


P-adic Norm and Absolute Value are Not Equivalent/Proof 1

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$. Let $\size {\,\cdot\,}$ be the absolute value on the rationals $\Q$. Then $\norm {\,\cdot\,}_p$ and $\size {\,\cdot\,}$ are not equivalent norms. That is, the topology induced by $\norm {\,\cdot\,}_p$ does not equal the topology induced by $\size {\,\cdot\,}$.


P-adic Norm and Absolute Value are Not Equivalent/Proof 2


P-adic Norms are Not Equivalent

Let $p_1$ and $p_2$ be prime numbers such that $p_1 \neq p_2$. Let $\norm {\,\cdot\,}_{p_1}$ and $\norm {\,\cdot\,}_{p_2}$ be the $p$-adic norms on the rationals $\Q$. Then $\norm {\,\cdot\,}_{p_1}$ and $\norm {\,\cdot\,}_{p_2}$ are not equivalent norms. That is, the topology induced by $\norm {\,\cdot\,}_{p_1}$ does not equal the topology induced by $\norm {\,\cdot\,}_{p_2}$.


Polynomials of Congruent Integers are Congruent

Let $x, y, m \in \Z$ be integers where $m \ne 0$. Let: :$x \equiv y \pmod m$ where the notation indicates congruence modlo $m$. Let $a_0, a_1, \ldots, a_r$ be integers. Then: :$\displaystyle \sum_{k \mathop = 0}^r a_k x^k \equiv \sum_{k \mathop = 0}^r a_k y^k \pmod m$


Congruent Integers less than Half Modulus are Equal

Let $k \in \Z_{>0}$ be a strictly positive integer. Let $a, b \in \Z$ such that $\size a < \dfrac k 2$ and $\size b < \dfrac k 2$. Then: :$a \equiv b \pmod k \implies a = b$ where $\equiv$ denotes congruence modulo $k$.


Complete Residue System Modulo m has m Elements

Let $m \in \Z_{\ne 0}$ be a non-zero integer. Let $S := \set {r_1, r_2, \dotsb, r_s}$ be a complete residue system modulo $m$. Then $s = m$.


Initial Segment of Natural Numbers forms Complete Residue System

Let $m \in \Z_{\ne 0}$ be a non-zero integer. Let $\N_m = \set {0, 1, 2, \ldots, m - 1}$ denote the initial segment of $\N$ Then $\N_m$ is a complete residue system modulo $m$.


Number of Non-Dividing Primes Less than n is Less than Euler Phi Function of n

Let $n \in \Z_{>0}$ be a strictly positive integer. Let $\map w n$ denote the number of primes strictly less than $n$ which are not divisors of $n$. Let $\map \phi n$ denote the Euler $\phi$ function of $n$. Then: :$\map w n < \map \phi n$


Schatunowsky's Theorem

Let $n \in \Z_{>0}$ be a strictly positive integer. Let $\map w n$ denote the number of primes strictly less than $n$ which are not divisors of $n$. Let $\map \phi n$ denote the Euler $\phi$ function of $n$. Then $30$ is the largest integer $n$ such that: :$\map w n = \map \phi n - 1$


Position of Card after n Modified Perfect Faro Shuffles

Let $D$ be a deck of cards $D$ of size $2 r$. Let $C$ be a card in position $x$ of $D$. Let $n$ modified perfect faro shuffles be performed on $C$. Then $C$ will be in position $w$, where: :$w \equiv 2^n x \pmod {2 r + 1}$


Deck of 52 Cards returns to Original Order after 52 Modified Perfect Faro Shuffles

Let $D$ be a deck of $52$ cards. Let $D$ be given a sequence of modified perfect faro shuffles. Then after $52$ such shuffles, the cards of $D$ will be in the same order they started in.


Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order

Let $D$ be a deck of $2 m$ cards. Let $D$ be given a sequence of modified perfect faro shuffles. Then the cards of $D$ will return to their original order after $n$ such shuffles, where: :$2^n \equiv 1 \pmod {2 m + 1}$


Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 8 Cards

Let $D$ be a deck of $8$ cards. Let $D$ be given a sequence of modified perfect faro shuffles. Then after $6$ such shuffles, the cards of $D$ will be in the same order they started in.


Sum of Two Odd Powers

Let $\F$ be one of the standard number systems, that is $\Z, \Q, \R$ and so on. Let $n \in \Z_{\ge 0}$ be a positive integer. Then: {{begin-eqn}} {{eqn | l = a^{2 n + 1} + b^{2 n + 1}       | r = \paren {a + b} \sum_{j \mathop = 0}^{2 n} \paren {-1}^j a^{2 n - j} b^j       | c =  }} {{eqn | r = \paren {a + b} \paren {a^{2 n} - a^{2 n - 1} b + a^{2 n - 2} b^2 - \dotsb - a b^{2 n - 1} + b^{2 n} }       | c =  }} {{end-eqn}}


Difference of Two Fifth Powers

:$x^5 - y^5 = \paren {x - y} \paren {x^4 + x^3 y + x^2 y^2 + x y^3 + y^4}$


Difference of Two Sixth Powers

:$x^6 - y^6 = \paren {x - y} \paren {x + y} \paren {x^2 + x y + y^2} \paren {x^2 - x y + y^2}$


Difference of Two Odd Powers

Let $\mathbb F$ denote one of the standard number systems, that is $\Z$, $\Q$, $\R$ and $\C$. Let $n \in \Z_{\ge 0}$ be a positive integer. Then for all $a, b \in \mathbb F$: {{begin-eqn}} {{eqn | l = a^{2 n + 1} - b^{2 n + 1}       | r = \paren {a - b} \sum_{j \mathop = 0}^{2 n} a^{2 n - j} b^j       | c =  }} {{eqn | r = \paren {a - b} \paren {a^{2 n} + a^{2 n - 1} b + a^{2 n - 2} b^2 + \dotsb + a b^{2 n - 1} + b^{2 n} }       | c =  }} {{end-eqn}}


Difference of Two Even Powers

Let $\mathbb F$ denote one of the standard number systems, that is $\Z$, $\Q$, $\R$ and $\C$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then for all $a, b \in \mathbb F$: {{begin-eqn}} {{eqn | l = a^{2 n} - b^{2 n}       | r = \paren {a - b} \paren {a + b} \sum_{j \mathop = 0}^{n - 1} a^{2 \paren {n - j - 1} } b^{2 j}       | c =  }} {{eqn | r = \paren {a - b} \paren {a + b} \paren {a^{2 n - 2} + a^{2 n - 4} b^2 + a^{2 n - 6} b^4 + \dotsb + a^2 b^{2 n - 4} + b^{2 n - 2} }       | c =  }} {{end-eqn}}


Factors of Difference of Two Odd Powers

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = x^{2 n + 1} - y^{2 n + 1}       | r = \paren {x - y} \displaystyle \prod_{k \mathop = 1}^n \paren {x^2 - 2 x y \cos \dfrac {2 \pi k} {2 n + 1} + y^2}       | c =  }} {{eqn | r = \paren {x - y} \paren {x^2 - 2 x y \cos \dfrac {2 \pi} {2 n + 1} + y^2} \paren {x^2 - 2 x y \cos \dfrac {4 \pi} {2 n + 1} + y^2} \dotsm \paren {x^2 - 2 x y \cos \dfrac {2 n \pi} {2 n + 1} + y^2}       | c =  }} {{end-eqn}}


Factors of Sum of Two Odd Powers

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = x^{2 n + 1} + y^{2 n + 1}       | r = \paren {x + y} \displaystyle \prod_{k \mathop = 1}^n \paren {x^2 + 2 x y \cos \dfrac {2 \pi k} {2 n + 1} + y^2}       | c =  }} {{eqn | r = \paren {x + y} \paren {x^2 + 2 x y \cos \dfrac {2 \pi} {2 n + 1} + y^2} \paren {x^2 + 2 x y \cos \dfrac {4 \pi} {2 n + 1} + y^2} \dotsm \paren {x^2 + 2 x y \cos \dfrac {2 n \pi} {2 n + 1} + y^2}       | c =  }} {{end-eqn}}


Factors of Difference of Two Even Powers

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: :$x^{2 n} - y^{2 n} = \paren {x - y} \paren {x + y} \displaystyle \prod_{k \mathop = 1}^{n - 1} \paren {x^2 - 2 x y \cos \dfrac {k \pi} n + y^2}$


Factors of Sum of Two Even Powers

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: :$x^{2 n} + y^{2 n} = \displaystyle \prod_{k \mathop = 1}^n \paren {x^2 + 2 x y \cos \dfrac {\paren {2 k - 1} \pi} {2 n} + y^2}$


Topological Properties of Non-Archimedean Division Rings/Spheres are Clopen

:The $r$-sphere of $x$, $\map {S_r} x$, is both open and closed in the metric induced by $\norm {\,\cdot\,}$.


Magnitude of Projection of Complex Number on Another

Let $z_1$ and $z_2$ denote complex numbers in vector form. Let $\map {\pr_1} {z_1, z_2}$ denote the projection of $z_1$ on $z_2$. {{explain|We really need another page to explain the concept of Definition:Projection (Analytic Geometry) in the context of the Definition:Complex Plane}} Then: :$\cmod {\map {\pr_1} {z_1, z_2} } = \dfrac {\cmod {z_1 \circ z_2} } {\cmod {z_2} }$ where: :$z_1 \circ z_2$ denotes complex dot product :$\cmod {z_2}$ denotes complex modulus.


Non-Archimedean Division Ring is Totally Disconnected

Let $\struct {R, \norm{\,\cdot\,} }$ be a non-Archimedean normed division ring. Let $\tau$ be the topology induced by the norm $\norm{\,\cdot\,}$. Then the topological space $\struct {R, \tau}$ is totally disconnected.


Distance between Points in Complex Plane

Let $A$ and $B$ be points in the complex plane such that: :$A = \tuple {x_1, y_1}$ :$B = \tuple {x_2, y_2}$ Then the distance between $A$ and $B$ is given by: {{begin-eqn}} {{eqn | l = \size {AB}       | r = \sqrt {\paren {x_2 - x_1}^2 + \paren {y_2 - y_1}^2}       | c =  }} {{eqn | r = \cmod {z_1 - z_2}       | c =  }} {{end-eqn}} where $z_1$ and $z_2$ are represented by the complex numbers $z_1$ and $z_2$ respectively.


Linear Combination of Non-Parallel Complex Numbers is Zero if Factors are Both Zero

Let $z_1$ and $z_2$ be complex numbers expressed as vectors such taht $z_1$ is not parallel to $z_2$. Let $a, b \in \R$ be real numbers such that: :$a z_1 + b z_2 = 0$ Then $a = 0$ and $b = 0$.


Equation for Line through Two Points in Complex Plane/Formulation 1

$L$ can be expressed by the equation: :$\map \arg {\dfrac {z - z_1} {z_2 - z_1} } = 0$


Equation for Line through Two Points in Complex Plane/Parametric Form 1

$L$ can be expressed by the equation: :$z = z_1 + t \paren {z_2 - z_1}$ or: :$z = \paren {1 - t} z_1 + t z_2$ This form of $L$ is known as the '''parametric form''', where $t$ is the '''parameter'''.


Equation for Line through Two Points in Complex Plane/Symmetric Form

$L$ can be expressed by the equation: :$z = \dfrac {m z_1 + n z_2} {m + n}$ This form of $L$ is known as the '''symmetric form'''.


Valuation Ring of Non-Archimedean Division Ring is Subring

Let $\struct {R, \norm{\,\cdot\,}}$ be a non-Archimedean normed division ring with zero $0_R$ and unity $1_R$. Let $\OO$ be the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}$, that is: :$\OO = \set {x \in R : \norm{x} \le 1}$ Then $\OO$ is a subring of $R$: :with a unity: $1_R$ :in which there are no (proper) zero divisors, that is: :::$\forall x, y \in \OO: x \circ y = 0_R \implies x = 0_R \text{ or } y = 0_R$


Valuation Ideal is Maximal Ideal of Induced Valuation Ring

Let $\struct {R, \norm {\,\cdot\,}}$ be a non-Archimedean normed division ring with zero $0_R$ and unity $1_R$. Let $\OO$ be the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}$, that is: :$\OO = \set{x \in R : \norm x \le 1}$ Let $\PP$ be the valuation ideal induced by the non-Archimedean norm $\norm {\,\cdot\,}$, that is: :$\PP = \set{x \in R : \norm x < 1}$ Then $\PP$ is an ideal of $\OO$: :$(a):\quad \PP$ is a maximal left ideal :$(b):\quad \PP$ is a maximal right ideal :$(c):\quad$ the quotient ring $\OO / \PP$ is a division ring.


Sequence of Imaginary Reciprocals

Consider the subset $S$ of the complex plane defined as: :$S := \set {\dfrac i n : n \in \Z_{>0} }$ That is: :$S := \set {i, \dfrac i 2, \dfrac i 3, \dfrac i 4, \ldots}$ where $i$ is the imaginary unit.


Sequence of Imaginary Reciprocals/Boundedness

The set $S$ is bounded in $\C$.


Sequence of Imaginary Reciprocals/Limit Points

The set $S$ has exactly one limit point, and that is $z = 0$.


Sequence of Imaginary Reciprocals/Closedness

The set $S$ is not closed.


Valuation Ring of P-adic Norm on Rationals

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$. The induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$ is the set: :$\OO = \Z_{\paren p} = \set {\dfrac a b \in \Q : p \nmid b}$


Valuation Ideal of P-adic Norm on Rationals

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$. The induced valuation ideal on $\struct {\Q,\norm {\,\cdot\,}_p}$ is the set: :$\PP = p \Z_{\ideal p} = \set {\dfrac a b \in \Q : p \nmid b, p \divides a}$ where $\Z_{\ideal p}$ is the induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$


Residue Field of P-adic Norm on Rationals

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$. The induced residue field on $\struct {\Q,\norm {\,\cdot\,}_p}$ is isomorphic to the field $\F_p$ of integers modulo $p$.


Sequence of Imaginary Reciprocals/Boundary Points

Every point of $S$, along with the point $z = 0$, is a boundary point of $S$.


Sequence of Imaginary Reciprocals/Interior

No point of $S$ is an interior point.


Sequence of Imaginary Reciprocals/Openness

$S$ is not an open set.


Sequence of Imaginary Reciprocals/Connectedness

$S$ is not connected.


Sequence of Imaginary Reciprocals/Not an Open Region

$S$ is not an open region.


Sequence of Imaginary Reciprocals/Not Compact

The set $S$ is not compact.


Sequence of Imaginary Reciprocals/Closure is Compact

The closure $S^-$ of the set $S$ is compact.


Product of Complex Conjugates/Examples/3 Arguments

Let $z_1, z_2, z_3 \in \C$ be complex numbers. Let $\overline z$ denote the complex conjugate of the complex number $z$. Then: :$\overline {z_1 z_2 z_3} = \overline {z_1} \cdot \overline {z_2} \cdot \overline {z_3}$


Residue Field of P-adic Norm on Rationals/Lemma 1

:$\phi$ is a homomorphism.


Residue Field of P-adic Norm on Rationals/Lemma 2

:$p \Z = \map \ker \phi$


Residue Field of P-adic Norm on Rationals/Lemma 3

:$\phi : \Z \to \Z_{\paren p} / p \Z_{\paren p}$ is a surjection.


Equation for Line through Two Points in Complex Plane/Parametric Form 2

$L$ can be expressed by the equations: {{begin-eqn}} {{eqn | l = x - x_1       | r = t \paren {x_2 - x_1} }} {{eqn | l = y - y_1       | r = t \paren {y_2 - y_1} }} {{end-eqn}} These are the '''parametric equations of $L$''', where $t$ is the parameter.


Equation for Perpendicular Bisector of Two Points in Complex Plane/Parametric Form 1

$L$ can be expressed by the equation: :$z = ...$ or: :$z = ...$ This form of $L$ is known as the '''parametric form''', where $t$ is the '''parameter'''.


Geodesic Equation/2d Surface Embedded in 3d Euclidean Space

Let $\sigma: U \subset \R^2 \to V \subset \R^3$ be a smooth surface specified by a vector-valued function: :$\mathbf r = \map {\mathbf r} {u, v}$ Then a geodesic of $\sigma$ satisfies the following system of differential equations: :$\dfrac {E_u u'^2 + 2 F_u u' v' + G_u v'^2} {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } - \dfrac \d {\d t} \dfrac {2 \paren {E u' + F v'} } {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } = 0$ :$\dfrac {E_v u'^2 + 2 F_v u' v' + G_v v'^2} {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } - \dfrac \d {\d t} \dfrac {2 \paren {F u' + G v'} } {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } = 0$ where $E, F, G$ are the functions of the first fundamental form: :$\displaystyle E = {\mathbf r}_u \cdot {\mathbf r}_u, F = {\mathbf r}_u \cdot {\mathbf r}_v, G = {\mathbf r}_v \cdot {\mathbf r}_v$


Modulus of Exponential of i z where z is on Circle

Let $C$ be the circle embedded in the complex plane given by the equation: :$z = R e^{i \theta}$ Then: :$\cmod {e^{i z} } = e^{-R \sin \theta}$


Sum of Complex Numbers in Exponential Form/General Result

Let $n \in \Z_{>0}$ be a positive integer. For all $k \in \set {1, 2, \dotsc, n}$, let: :$z_k = r_k e^{i \theta_k}$ be non-zero complex numbers in exponential form. Let: :$r e^{i \theta} = \displaystyle \sum_{k \mathop = 1}^n z_k = z_1 + z_2 + \dotsb + z_k$ Then: {{begin-eqn}} {{eqn | l = r       | r = \sqrt {\displaystyle \sum_{k \mathop = 1}^n r_k + \displaystyle \sum_{1 \mathop \le j \mathop < k \mathop \le n} 2 {r_j} {r_k} \, \map \cos {\theta_j - \theta_k} } }} {{eqn | l = \theta       | r = \map \arctan {\dfrac {r_1 \sin \theta_1 + r_2 \sin \theta_2 + \dotsb + r_n \sin \theta_n} {r_1 \cos \theta_1 + r_2 \cos \theta_2 + \dotsb + r_n \cos \theta_n} } }} {{end-eqn}}


Bias of Sample Variance

Let $X_1, X_2, \ldots, X_n$ form a random sample from a population with mean $\mu$ and variance $\sigma^2$. Let:  :$\displaystyle \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$ Then: :$\displaystyle \hat {\sigma^2} = \frac 1 n \sum_{i \mathop = 1}^n \paren {X_i - \bar X}^2$ is a biased estimator of $\sigma^2$, with:  :$\displaystyle \operatorname{bias} \paren {\hat {\sigma ^2}} = -\frac {\sigma^2} n$


Sample Mean is Unbiased Estimator of Population Mean

Let $X_1, X_2, \ldots, X_n$ form a random sample from a population with mean $\mu$ and variance $\sigma^2$. Then:  :$\displaystyle \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$ is an unbiased estimator of $\mu$.


Ostrowski's Theorem/Archimedean Norm

Let $\norm {\, \cdot \,}$ be a non-trivial Archimedean norm on the rational numbers $\Q$. Then $\norm {\, \cdot \,}$ is equivalent to the absolute value $\size {\, \cdot \,}$.


Ostrowski's Theorem/Non-Archimedean Norm

Let $\norm {\, \cdot \,}$ be a non-trivial non-Archimedean norm on the rational numbers $\Q$. Then $\norm {\, \cdot \,}$ is equivalent to the $p$-adic norm $\norm {\, \cdot \,}_p$ for some prime $p$.


Ostrowski's Theorem/Archimedean Norm/Lemma 1.1

:$\forall n \in N: \norm n \le n^\alpha$


Ostrowski's Theorem/Archimedean Norm/Lemma 1.2

:$\forall n \in N: \norm n \ge n^\alpha$


Equivalent Norms on Rational Numbers

Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be norms on the rational numbers $\Q$. Then $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ are equivalent {{iff}}: :$\exists \alpha \in \R_{\gt 0}: \forall n \in \N: \norm n_1 = \norm n_2^\alpha$


Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.1

:$\exists n \in \N: 0 < \norm n < 1$.


Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.2

:$n_0$ is a prime number.


Sum of Cosines of k pi over 5

:$\cos 36 \degrees + \cos 72 \degrees + \cos 108 \degrees + \cos 144 \degrees = 0$


Products of nth Roots of Unity taken up to n-1 at a Time is Zero

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $U_n = \set {z \in \C: z^n = 1}$ be the complex $n$th roots of unity. Then the sum of the products of the elements of $U_n$ taken $2, 3, 4, \dotsc n - 1$ at a time is zero.


Absolute Value of Complex Dot Product is Commutative

Let $z_1$ and $z_2$ be complex numbers. Let $z_1 \circ z_2$ denote the (complex) dot product of $z_1$ and $z_2$. Then: :$\size {z_1 \circ z_2} = \size {z_2 \circ z_1}$ where $\size {\, \cdot \,}$ denotes the absolute value function.


Three Points in Ultrametric Space have Two Equal Distances/Corollary 5

Let $\norm {\, \cdot \,}$ be a non-trivial non-Archimedean norm on the rational numbers $\Q$. Let $a, b \in \Z_{\ne 0}$ be coprime, $a \perp b$ Then: :$\norm a = 1$ or $\norm b = 1$


Exponential of 2 m i Arccotangent of p

:$\map \exp {2 m i \arccot p} \paren {\dfrac {p i + 1} {p i - 1} }^m = 1$


Modulus z - 1 Less than Modulus z + 1 iff Real z Greater than Zero

Let $z \in \C$ be a complex number. Then: :$\cmod {z - 1} < \cmod {z + 1} \iff \map \Re z > 0$


Vertices of Equilateral Triangle in Complex Plane/Sufficient Condition

Let $z_1$, $z_2$ and $z_3$ be complex numbers. Let $z_1$, $z_2$ and $z_3$ represent on the complex plane the vertices of an equilateral triangle. Then: :${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$


Vertices of Equilateral Triangle in Complex Plane/Necessary Condition

Let $z_1$, $z_2$ and $z_3$ be complex numbers. Let $z_1$, $z_2$ and $z_3$ fulfil the condition: :${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$ Then $z_1$, $z_2$ and $z_3$ represent on the complex plane the vertices of an equilateral triangle.


Geodesic Equation/2d Surface Embedded in 3d Euclidean Space/Cylinder

Let $\sigma$ be the surface of a cylinder. Let $\sigma$ be embedded in 3-dimensional Euclidean space. Let $\sigma$ be parameterised by $\tuple {\phi, z}$ as :$\mathbf r = \tuple {a \cos \phi, a \sin \phi, z}$ where :$a > 0$ and :$z, \phi \in \R$ Then geodesics on $\sigma$ are of the following form: :$z = C_1 \phi + C_2$ where $C_1, C_2$ are real arbitrary constants.


Modulus of Sum equals Modulus of Distance implies Quotient is Imaginary

Let $z_1$ and $z_2$ be complex numbers such that: :$\cmod {z_1 + z_2} = \cmod {z_1 - z_2}$ Then $\dfrac {z_2} {z_1}$ is wholly imaginary.


Difference of Even Powers of z + a and z - a

Let $m \in \Z$ be an integer such that $m > 1$. Then for all complex number $z$: :$\paren {z + a}^{2 m} - \paren {z - a}^{2 m} = 4 m a z \displaystyle \prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \dfrac {k \pi} {2 m} }$


Imaginary Part of Complex Product

Let $z_1$ and $z_2$ be complex numbers. Then: :$\map \Im {z_1 z_2} = \map \Re {z_1} \, \map \Im {z_2} + \map \Im {z_1} \, \map \Re {z_2}$


Element of Center in Group whose Order is Power of 2

Let $n \in \Z$ be an integer such that $n \ge 2$. Let $G$ be a group whose order is $2^n$. Let $x \in G$ be of order $2^{n - 1}$ in $G$. Then $x^{2^{n - 2} }$ is an element of the center of $G$.


Intersection of Abelian Subgroups is Normal Subgroup of Subgroup Generated by those Subgroups

Let $G$ be a group. Let $L$ and $M$ be abelian subgroups of $G$. Let $H = \gen {L, M}$ be the subgroup of $G$ generated by $L$ and $M$. Then $L \cap M$ is a normal subgroup of $H$.


Characterisation of Non-Archimedean Division Ring Norms/Corollary 5

If $\norm {\, \cdot \,}$ is non-Archimedean then: :$\sup \set {\norm {n \cdot 1_R}: n \in \Z} = 1$ where $n \cdot 1_R =  \begin{cases} \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} } & : n > 0 \\ 0 & : n = 0 \\ \\ -\underbrace {\paren {1_R + 1_R + \dots + 1_R} }_{\text {$-n$ times} } & : n < 0 \\ \end{cases}$


Subgroup Containing all Squares of Group Elements is Normal

Let $G$ be a group. Let $H$ be a subgroup of $G$ with the property that: :$\forall x \in G: x^2 \in H$ Then $H$ is normal in $G$.


Commutator is Identity iff Elements Commute

Let $G$ be a group whose identity is $e$. Let $x, y \in G$. Let $\sqbrk {x, y}$ denote the commutator of $x$ and $y$. Then $\sqbrk {x, y} = e$ {{iff}} $x$ and $y$ commute.


Commutator of Quotient Group Elements

Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $\sqbrk {x, y}$ denote the commutator of $x, y \in G$: :$\sqbrk {x, y} = x^{-1} y^{-1} x y$ Then: :$\forall x, y \in G: \sqbrk {x N, y N} = \sqbrk {x, y} N$ where $x N$ and $y N$ are left cosets of $N$, and so elements of the quotient group $G / N$ of $G$ by $N$.


Quotient Group is Abelian iff All Commutators in Divisor

Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $G / N$ be the quotient group of $G$ by $N$. Then the quotient group $G / N$ is abelian {{iff}}: :$\forall x, y \in G: \sqbrk {x, y} \in N$ where $\sqbrk {x, y}$ denotes the commutator of $x$ and $y$.


Sufficient Condition for Quotient Group by Intersection to be Abelian

Let $G$ be a group. Let $N$ and $K$ be normal subgroups of $G$. Let the quotient groups $G / N$ and $G / K$ be abelian. Then the quotient group $G / \paren {N \cap K}$ is also abelian.


Quotient Group by Intersection of Normal Subgroups not necessarily Cyclic if Quotient Groups are

Let $G$ be a group. Let $N$ and $K$ be normal subgroups of $G$. Let the quotient groups $G / N$ and $G / K$ be cyclic. Then the quotient group $G / \paren {N \cap K}$ is not necessarily cyclic.


Order of Boolean Group is Power of 2

Let $G$ be a Boolean group. Let $\order G$ denote the order of $G$. Then: :$\order G = 2^n$ where $n \in \Z_{\ge 0}$ is a positive integer.


Subgroup of Index 2 contains all Squares of Group Elements

Let $G$ be a group. Let $H$ be a subgroup of $G$ whose index is $2$. Then: :$\forall x \in G: x^2 \in H$


Center of Non-Abelian Group of Order pq is Trivial

Let $p$ and $q$ be distinct prime numbers. Let $G$ be a non-abelian group of order $p q$ whose identity is $e$. Then the center of $G$ is trivial: :$\map Z G = \set e$


Number of Elements of Order p in Group of Order pq is Multiple of q

Let $p$ and $q$ be distinct prime numbers. Let $G$ be a non-abelian group of order $p q$. Then the number of elements of $G$ of order $p$ is a multiple of $q$.


Intersection of Normal Subgroup with Center in p-Group

Let $p$ be a prime number Let $G$ be a $p$-group. Let $N$ be a non-trivial normal subgroup of $G$. Let $\map Z G$ denote the center of $G$. Then: :$N \cap \map Z G$ is a non-trivial normal subgroup of $G$.


Normal Subgroup of p-Group of Order p is Subset of Center

Let $p$ be a prime number. Let $G$ be a $p$-group. Let $N$ be a normal subgroup of $G$ of order $p$. Then: :$N \subseteq \map Z G$ where $\map Z G$ denotes the center of $G$.


Non-Abelian Group of Order p Cubed has Exactly One Normal Subgroup of Order p

Let $p$ be a prime number. Let $G$ be a non-abelian group of order $p^3$. Then $G$ contains exactly one normal subgroup of order $p$.


Non-Abelian Order 2p Group has Order p Element

Let $p$ be an odd prime. Let $G$ be a non-abelian group of order $2 p$. Then $G$ has at least one element of order $p$.


Fourth Power Modulo 5

Let $n \in \Z$ be an integer. Then: :$n^4 \equiv m \pmod 5$ where $m \in \set {0, 1}$.


Subgroup of Order p in Group of Order 2p is Normal/Corollary

Let $G$ be non-abelian. Every element of $G \setminus K$ is of order $2$, and: :$\forall b \in G \setminus K: b a b^{-1} = a^{-1}$


Subgroup of Order p in Group of Order 2p is Normal

Let $p$ be an odd prime. Let $G$ be a group of order $2 p$. Let $a \in G$ be of order $p$. Let $K = \gen a$ be the subgroup of $G$ generated by $a$. Then $K$ is normal in $G$.


Inner Automorphisms form Subgroup of Automorphism Group

Let $G$ be a group. Then the set $\Inn G$ of all inner automorphisms of $G$ forms a normal subgroup of the automorphism group $\Aut G$ of $G$: :$\Inn G \le \Aut G$


Property of Group Automorphism which Fixes Identity Only

Let $G$ be a finite group whose identity is $e$. Let $\phi: G \to G$ be a group automorphism. Let $\phi$ have the property that: :$\forall g \in G \setminus \set e: \map \phi t \ne t$ That is, the only fixed element of $\phi$ is $e$. Then: :$\forall x, y \in G: x^{-1} \, \map \phi x = y^{-1} \, \map \phi y \implies x = y$


Property of Group Automorphism which Fixes Identity Only/Corollary 1

:$\forall x \in G: \exists g \in G: x = g^{-1} \, \map \phi g$


Square Order 2 Matrices over Real Numbers form Ring with Unity

Let $S$ denote the set of square matrices of order $2$ whose entries are the set of real numbers. Then $S$ forms a ring with unity whose unity is the matrix $\begin {pmatrix} 1 & 0 \\ 0 & 1 \end {pmatrix}$.


Product Formula for Norms on Non-zero Rationals

Let $\Q_{\ne 0}$ be the set of non-zero rational numbers. Let  $\Bbb P$ denote the set of prime numbers. Let $a \in \Q_{\ne 0}$. Then the following infinite product converges: :$\size a \times \displaystyle \prod_{p \mathop \in \Bbb P}^{} \norm a_p = 1$ where: :$\size {\,\cdot\,}$ is the absolute value on $\Q$ :$\norm {\,\cdot\,}_p$ is the $p$-adic norm on $\Q$ for prime number $p$


Null Ring is Subring of Ring

Let $R$ be a ring. Then the null ring is a subring of $R$.


Integers form Subring of Reals

The ring of integers $\struct {\Z, +, \times}$ forms a subring of the field of real numbers.


Unity of Subfield is Unity of Field

The unity of $\struct {K, +, \times}$ is also $1$.


Zero of Subring is Zero of Ring

The zero of $\struct {S, +, \times}$ is also $0$.


Zero of Subfield is Zero of Field/Proof 1

Let $\struct {F, +, \times}$ be a field whose zero is $0$. Let $\struct {K, +, \times}$ be a subfield of $\struct {F, +, \times}$. {{:Zero of Subfield is Zero of Field}}


Zero of Subfield is Zero of Field/Proof 2


P-adic Norm not Complete on Rational Numbers/Proof 1/Case 1

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p > 3$.   Then: :$\struct {\Q, \norm {\,\cdot\,}_p}$ is not a complete normed division ring. That is, there exists a Cauchy sequence in $\struct {\Q, \norm{\,\cdot\,}_p}$ which does not converge to a limit in $\Q$.


P-adic Norm not Complete on Rational Numbers/Proof 1/Case 2

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for $p = 2$ or $3$.   Then: :$\struct {\Q, \norm {\,\cdot\,}_p}$ is not a complete normed division ring. That is, there exists a Cauchy sequence in $\struct {\Q, \norm{\,\cdot\,}_p}$ which does not converge to a limit in $\Q$.


Element in Integral Domain is Divisor iff Principal Ideal is Superset

:$x \divides y \iff \ideal y \subseteq \ideal x$


Residue of Gamma Function

Let $\Gamma$ be the Definition:Gamma Function.  Let $n$ be a non-negative integer. Then:  :$\Res \Gamma {-n} = \dfrac {\paren {-1}^n} {n!}$


Periodic Element is Multiple of Period

Let $f: \R \to \R$ be a real periodic function with period $P$. Let $L$ be a periodic element of $f$. Then $P \divides L$.


Equivalence of Definitions of Associate in Integral Domain

{{TFAE|def = Associate in Integral Domain|context = Integral Domain|view = Associate}} Let $\struct {D, +, \circ}$ be an integral domain. Let $x, y \in D$.


Finite Set of Elements in Principal Ideal Domain has GCD

Let $\struct {D, +, \circ}$ be a principal ideal domain. Let $a_1, a_2, \dotsc, a_n$ be non-zero elements of $D$. Then $a_1, a_2, \dotsc, a_n$ all have a greatest common divisor.


Set of Linear Combinations of Finite Set of Elements of Principal Ideal Domain is Principal Ideal

Let $\struct {D, +, \circ}$ be a principal ideal domain. Let $a_1, a_2, \dotsc, a_n$ be non-zero elements of $D$. Let $J$ be the set of all linear combinations in $D$ of $\set {a_1, a_2, \dotsc, a_n}$ Then for some $x \in D$: :$J = \ideal x$ where $\ideal x$ denotes the principal ideal generated by $x$.


Greatest Common Divisors in Principal Ideal Domain are Associates

Let $\struct {D, +, \circ}$ be a principal ideal domain. Let $S = \set {a_1, a_2, \dotsc, a_n}$ be a set of non-zero elements of $D$. Let $y_1$ and $y_2$ be greatest common divisors of $S$. Then $y_1$ and $y_2$ are associates.


Greatest Common Divisor in Principal Ideal Domain is Expressible as Linear Combination

Let $\struct {D, +, \circ}$ be a principal ideal domain. Let $S = \set {a_1, a_2, \dotsc, a_n}$ be a set of non-zero elements of $D$. Let $y$ be a greatest common divisor of $S$. Then $y$ is expressible in the form: :$y = d_1 a_1 + d_2 a_2 + \dotsb + d_n a_n$ where $d_1, d_2, \dotsc, d_n \in D$.


Complete Factorizations of Proper Element in Principal Ideal Domain are Equivalent

Let $\struct {D, +, \circ}$ be a principal ideal domain. Let $x \in D$ be a proper element of $D$. Let there be two complete factorizations of $x$: :$x = u_y \circ y_1 \circ y_2 \circ \cdots \circ y_m = F_1$ :$x = u_z \circ z_1 \circ z_2 \circ \cdots \circ z_n = F_2$ Then $F_1$ and $F_2$ are equivalent.


Moment Generating Function of Poisson Distribution

Let $X$ be a discrete random variable with a Poisson distribution with parameter $\lambda$ for some $\lambda \in \R_{> 0}$. Then the moment generating function $M_X$ of $X$ is given by:  :$\map {M_X} t = e^{\lambda \paren {e^t - 1} }$


Moment Generating Function of Binomial Distribution

Let $X$ be a discrete random variable with a binomial distribution with parameters $n$ and $p$ for some $n \in \N$ and $0 \le p \le 1$: :$X \sim \Binomial n p$ Then the moment generating function $M_X$ of $X$ is given by:  :$\map {M_X} t = \paren {1 - p + p e^t}^n$


Moment Generating Function of Exponential Distribution

Let $X$ be a continuous random variable with an exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$. Then the moment generating function $M_X$ of $X$ is given by:  :$\displaystyle \map {M_X} t = \frac 1 {1 - \beta t}$ for $t < \dfrac 1 \beta$, and is undefined otherwise.


Moment Generating Function of Geometric Distribution

Let $X$ be a discrete random variable with a geometric distribution with parameter $p$ for some $0 < p \le 1$.  Then the moment generating function $M_X$ of $X$ is given by:  :$\displaystyle \map {M_X} t = \frac p {1 - \paren {1 - p} e^t}$ for $t < -\map \ln {1 - p}$, and is undefined otherwise.


Maximal Ideal iff Quotient Ring is Field/Proof 1/Maximal Ideal implies Quotient Ring is Field

Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $J$ be a maximal ideal. Then the quotient ring $R / J$ is a field.


Maximal Ideal iff Quotient Ring is Field/Proof 1/Quotient Ring is Field implies Ideal is Maximal

Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let the quotient ring $R / J$ be a field. Then $J$ is a maximal ideal.


Ring of Polynomial Forms is not necessarily Isomorphic to Ring of Polynomial Functions

Let $D$ be an integral domain. Let $D \sqbrk X$ be the ring of polynomial forms in $X$ over $D$. Let $\map P D$ be the ring of polynomial functions over $D$. Then it is not necessarily the case that $D \sqbrk X$ is isomorphic with $\map P D$.


Moment Generating Function of Bernoulli Distribution

Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$ for some $0 \le p \le 1$.  Then the moment generating function $M_X$ of $X$ is given by:  :$\map {M_X} t = q + p e^t$ where $q = 1 - p$.


Double of Antiperiodic Element is Periodic

Let $f: X \to X$ be a function, where $X$ is either $\R$ or $\C$. Let $L \in X_{\ne 0}$ be an anti-periodic element of $f$. Then $2L$ is a periodic element of $f$. In other words, every anti-periodic function is also periodic.


Skewness of Gaussian Distribution

Let $X$ be a continuous random variable with a Gaussian distribution with parameters $\mu$ and $\sigma^2$ for some $\mu \in \R$ and $\sigma \in \R_{> 0}$.  Then the skewness $\gamma_1$ of $X$ is equal to $0$.


Inverse of Unit in Centralizer of Ring is in Centralizer

Let $\struct {R, +, \circ}$ be a ring. Let $S$ be a subset of $R$. Let $\map {C_R} S$ denote the centralizer of $S$ in $R$ Let $u \in R$ be a unit of $R$. Then: :$u \in \map {C_R} S \implies u^{-1} \in \map {C_R} S$


Skewness in terms of Non-Central Moments

Let $X$ be a random variable with mean $\mu$ and standard deviation $\sigma$. Then the skewness $\gamma_1$ of $X$ is given by:  :$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$


Intersection of Ring Ideals is Ideal

Let $\struct {R, +, \circ}$ be a ring Let $\mathbb L$ be a non-empty set of ideals of $R$. Then the intersection $\bigcap \mathbb L$ of the members of $\mathbb L$ is itself an ideal of $R$.


Set of Ring Elements forming Zero Product with given Element is Ideal

Let $\struct {R, +, \circ}$ be a commutative ring whose zero is $0_R$. Let $a \in R$ be an arbitrary element of $R$. Let $A$ be the subset of $R$ defined as: :$A = \set {x \in R: x \circ a = 0_R}$ Then $A$ is an ideal of $A$.


Skewness of Bernoulli Distribution

Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$. Then the skewness $\gamma_1$ of $X$ is given by:  :$\gamma_1 = \dfrac {1 - 2 p} {\sqrt {p q} }$ where $q = 1 - p$.


Intersection of Subrings is Subring

Let $\struct {R, +, \circ}$ be a ring. Let $\mathbb L$ be a non-empty set of subrings of $R$. Then the intersection $\displaystyle \bigcap \mathbb L$ of the members of $\mathbb L$ is itself a subring of $R$.


Intersection of Ring Ideals is Largest Ideal Contained in all Ideals

Let $\struct {R, +, \circ}$ be a ring Let $\mathbb L$ be a non-empty set of ideals of $R$. Then the intersection $\bigcap \mathbb L$ of the members of $\mathbb L$ is the largest ideal of $R$ contained in each member of $\mathbb L$.


Intersection of Subrings is Largest Subring Contained in all Subrings

Let $\struct {R, +, \circ}$ be a ring. Let $\mathbb L$ be a non-empty set of subrings of $R$. Then the intersection $\displaystyle \bigcap \mathbb L$ of the members of $\mathbb L$ is the largest subring of $R$ contained in each member of $\mathbb L$.


Median of Continuous Uniform Distribution

Let $X$ be a continuous random variable which is uniformly distributed on a closed real interval $\closedint a b$.  Then the median $M$ of $X$ is given by:  :$M = \dfrac {a + b} 2$


Intersection of Ring Ideals Containing Subset is Smallest

Let $\struct {R, +, \circ}$ be a ring Let $S \subseteq R$ be a subset of $R$. Let $L$ be the intersection of the set of all ideals of $R$ containing $S$. Then $L$ is the smallest ideal of $R$ containing $S$.


Intersection of Subrings Containing Subset is Smallest

Let $\struct {R, +, \circ}$ be a ring. Let $S \subseteq R$ be a subset of $R$. Let $L$ be the intersection of the set of all subrings of $R$ containing $S$. Then $L$ is the smallest subring of $R$ containing $S$.


Intersection of Division Subrings is Division Subring

Let $\struct {D, +, \circ}$ be a division ring. Let $\mathbb K$ be a non-empty set of division subrings of $D$. Then the intersection $\bigcap \mathbb K$ of the members of $\mathbb K$ is itself a division subring of $D$.


Intersection of Subfields is Subfield

Let $\struct {F, +, \circ}$ be a field. Let $\mathbb K$ be a non-empty set of subfields of $F$. Then the intersection $\bigcap \mathbb K$ of the members of $\mathbb K$ is itself a subfield of $F$.


Intersection of Division Subrings is Largest Division Subring Contained in all Division Subrings

Let $\struct {D, +, \circ}$ be a division ring. Let $\mathbb K$ be a non-empty set of division subrings of $D$. Let $\bigcap \mathbb K$ be the intersection of the elements of $\mathbb K$. Then $\bigcap \mathbb K$ is the largest division subring of $D$ contained in each element of $\mathbb K$.


Intersection of Division Subrings Containing Subset is Smallest

Let $\struct {D, +, \circ}$ be a division ring. Let $S \subseteq D$ be a subset of $D$. Let $L$ be the intersection of the set of all division subrings of $D$ containing $S$. Then $L$ is the smallest division subring of $D$ containing $S$.


Intersection of Subfields Containing Subset is Smallest

Let $\struct {F, +, \circ}$ be a field. Let $S \subseteq F$ be a subset of $F$. Let $L$ be the intersection of the set of all subfields of $F$ containing $S$. Then $L$ is the smallest subfield of $F$ containing $S$.


Intersection of Subfields is Largest Subfield Contained in all Subfields

Let $\struct {F, +, \circ}$ be a field. Let $\mathbb K$ be a non-empty set of subfields of $F$. Let $\bigcap \mathbb K$ be the intersection of the elements of $\mathbb K$. Then $\bigcap \mathbb K$ is the largest subfield of $F$ contained in each element of $\mathbb K$.


Equivalent Norms on Rational Numbers/Necessary Condition

Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be norms on the rational numbers $\Q$. Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be equivalent norms. Then: :$\exists \alpha \in \R_{\gt 0}: \forall n \in \N: \norm n_1 = \norm n_2^\alpha$


Equivalent Norms on Rational Numbers/Sufficient Condition

Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be norms on the rational numbers $\Q$. Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy: :$\exists \alpha \in \R_{\gt 0}: \forall n \in \N: \norm n_1 = \norm n_2^\alpha$ Then $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ are equivalent


Commutative and Unitary Ring with 2 Ideals is Field

Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$. Let $\struct {R, +, \circ}$ be such that the only ideals of $\struct {R, +, \circ}$ are: :$\set {0_R}$ and: $\struct {R, +, \circ}$ itself. That is, such that $\struct {R, +, \circ}$ has no non-null proper ideals. Then $\struct {R, +, \circ}$ is a field.


Field has 2 Ideals

Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Then the only ideals of $\struct {F, +, \circ}$ are:$\struct {F, +, \circ}$ and $\set {0_F}$. That is, $\struct {F, +, \circ}$ has no non-null proper ideals.


General Antiperiodicity Property

Let $f: X \to X$ be an antiperiodic function, where $X$ is either $\R$ or $\C$. Let $L$ be an antiperiodic element of $f$. Let $n \in \Z$ be an integer. :If $n$ is even, then $n L$ is a periodic element of $f$. :If $n$ is odd, then $n L$ is an antiperiodic element of $f$.


Antiperiodic Element is Multiple of Antiperiod

Let $f: \R \to \R$ be a real anti-periodic function with anti-period $A$. Let $L$ be an anti-periodic element of $f$. Then $A \divides L$.


Skewness of Poisson Distribution

Let $X$ be a discrete random variable with a Poisson distribution with parameter $\lambda$. Then the skewness $\gamma_1$ of $X$ is given by:  :$\gamma_1 = \dfrac 1 {\sqrt \lambda}$


Periodic Element is Multiple of Antiperiod

Let $f: \R \to \R$ be a real anti-periodic function with anti-period $A$. Let $L$ be a periodic element of $f$. Then $A \divides L$.


Skewness of Continuous Uniform Distribution

Let $X$ be a continuous random variable which is uniformly distributed on a closed real interval $\closedint a b$.  Then the skewness $\gamma_1$ of $X$ is equal to $0$.


Principal Ideal in Integral Domain generated by Power Plus One is Subset of Principal Ideal generated by Power

Let $\struct {D, +, \circ}$ be an integral domain whose unity is $1_D$. Let $a \in D$ be a proper element of $D$. Then: :$\forall n \in \Z_{\ge 0}: \ideal {a^{n + 1} } \subsetneq \ideal {a_n}$ where $\ideal x$ denotes the principal ideal of $D$ generated by $x$.


Non-Field Integral Domain has Infinite Number of Ideals

Let $\struct {D, +, \circ}$ be an integral domain which is not a field. Then $\struct {D, +, \circ}$ has an infinite number of distinct ideals.


Ring Homomorphism from Ring with Unity to Integral Domain Preserves Unity

Let $\struct {R, +_R, \circ_R}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\struct {D, +_D, \circ_D}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $\phi: R \to D$ be a ring homomorphism such that: :$\map \ker \phi \ne R$ where $\map \ker \phi$ denotes the kernel of $\phi$. Then $\map \phi {1_R} = 1_D$.


Unity plus Negative of Nilpotent Ring Element is Unit

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $x \in R$ be nilpotent. Then $1_R - x$ is a unit of $R$.


Quotient of Commutative Ring by Nilradical is Reduced

Let $\struct {R, +, \circ}$ be a commutative ring whose zero is $0_R$ and whose unity is $1_R$. Let $\struct {N, +, \circ}$ denote the nilradical of $R$. The quotient ring $R / N$ is a reduced ring.


Self-Inverse Element of Integral Domain is Unity or its Negative

Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $x \in D$ such that $x^2 = 1_D$. Then either $x = 1_D$ or $x = -1_D$.


Product of Units of Integral Domain with Finite Number of Units

Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $D$ have a finite number of units. Let $U_D$ be the set of units of $\struct {D, +, \circ}$. Then: :$\displaystyle \prod_{x \mathop \in U_D} x = -1_D$


Prime Ideals of Ring of Integers

Let $\struct {\Z, +, \times}$ denote the ring of integers. Let $J$ be a prime ideal of $\Z$. Then either: :$J = \set 0$ or: :$J = \ideal p$ where: :$p$ is a prime number :$\ideal p$ denotes the principal ideal of $\Z$ generated by $p$.


Additive Group and Multiplicative Group of Field are not Isomorphic

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $\struct {F, +}$ denote the additive group of $F$. Let $\struct {F_{\ne 0_F}, \times}$ denote the multiplicative group of $F$. Then $\struct {F, +}$ and $\struct {F_{\ne 0_F}, \times}$ are not isomorphic to each other.


Field Norm of Complex Number is Positive Definite

Let $\C$ denote the set of complex numbers. Let $N: \C \to \R_{\ge 0}$ denote the field norm on complex numbers: :$\forall z \in \C: \map N z = \cmod z^2$ where $\cmod z$ denotes the complex modulus of $z$. Then $N$ is positive definite on $\C$.


Field Norm of Complex Number is Multiplicative Function

Let $\C$ denote the set of complex numbers. Let $N: \C \to \R_{\ge 0}$ denote the field norm on complex numbers: :$\forall z \in \C: \map N z = \cmod z^2$ where $\cmod z$ denotes the complex modulus of $z$. Then $N$ is a multiplicative function on $\C$.


Units of 5th Cyclotomic Ring

Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring. The units of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ are $1$ and $-1$.


Field Norm on 5th Cyclotomic Ring

Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring. Let $\alpha = a + i b \sqrt 5$ be an arbitrary element of $\Z \sqbrk {i \sqrt 5}$. The field norm of $\alpha$ is given by: :$\map N \alpha = a^2 + 5 b^2$


5th Cyclotomic Ring has no Elements with Field Norm of 2 or 3

Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring. There are no elements of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ whose field norm is either $2$ or $3$.


Irreducible Elements of 5th Cyclotomic Ring

Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring. The following elements of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ are irreducible: :$2$ :$3$ :$1 + i \sqrt 5$ :$1 - i \sqrt 5$


Value of Field Norm on 5th Cyclotomic Ring is Integer

Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring. Let $\alpha = a + i b \sqrt 5$ be an arbitrary element of $\Z \sqbrk {i \sqrt 5}$. Let $\map N \alpha$ denoted the field norm of $\alpha$. Then $\map N \alpha$ is an integer.


Elements of 5th Cyclotomic Ring with Field Norm 1

Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring. The only elements of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ whose field norm equals $1$ are the units of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$: $1$ and $-1$.


Polynomials in Integers with Even Constant Term forms Ideal

Let $\Z \sqbrk X$ be the ring of polynomials in $X$ over $\Z$. Let $S \subseteq \Z \sqbrk X$ be the set of polynomials over $\Z$ in $X$ which have a constant term which is even. Then $S$ is an ideal of $\Z \sqbrk X$.


Polynomials in Integers is Unique Factorization Domain

Let $\Z \sqbrk X$ be the ring of polynomials in $X$ over $\Z$. Then $\Z \sqbrk X$ is a unique factorization domain.


Ring of Gaussian Integers is Principal Ideal Domain

The ring of Gaussian integers: :$\struct {\Z \sqbrk i, +, \times}$ forms a principal ideal domain.


Matrix Multiplication is not Commutative

Let $R$ be a ring with unity. Let $n \in \Z_{>0}$ be a (strictly) positive integer such that $n \ne 1$. Let $\map {\MM_R} n$ denote the $n \times n$ matrix space over $R$. Then (conventional) matrix multiplication over $\map {\MM_R} n$ is not commutative: :$\exists \mathbf A, \mathbf B \in \map {\MM_R} n: \mathbf {A B} \ne \mathbf {B A}$ If $R$ is specifically not commutative, then the result holds when $n = 1$ as well.


Matrix Multiplication on Square Matrices over Trivial Ring is Commutative

Let $\struct {R, +, \circ}$ be the trivial ring over an underlying set. Let $\map {\MM_R} n$ denote the $n \times n$ matrix space over $R$. Then (conventional) matrix multiplication is commutative over $\map {\MM_R} n$: :$\forall \mathbf A, \mathbf B \in \map {\MM_R} n: \mathbf {A B} = \mathbf {B A}$


Area between Smooth Curve and Line with Fixed Endpoints is Maximized by Arc of Circle

Let $y$ be a smooth curve, embedded in $2$-dimensional Euclidean space. Let $y$ have a total length of $l$. Let it be contained in the upper halfplane with an exception of endpoints, which are on the x-axis and are given. Suppose, $y$, together with a line segment connecting $y$'s endpoints, maximizes the enclosed area. Then $y$ is an arc of a circle.


Preordering of Products under Operation Compatible with Preordering

Let $\struct {S, \circ}$ be an algebraic structure. Let $\precsim$ be a preordering on $S$. Then $\precsim$ is compatible with $\circ$ {{iff}}: :$\forall x_1, x_2, y_1, y_2 \in S: x_1 \precsim x_2 \land y_1 \precsim y_2 \implies \paren {x_1 \circ y_1} \precsim \paren {x_2 \circ y_2}$


P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 1

:$\exists x \in \Z_{>0}:  p \nmid x, x \ge \dfrac {p + 1} 2$


P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 5

:$\displaystyle \lim_{n \mathop \to \infty} x_n^k = a$ in $\struct {\Q, \norm {\,\cdot\,}_p}$


Characterisation of Cauchy Sequence in Non-Archimedean Norm/Corollary 1

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$. Let $\sequence {x_n}$ be a sequence of integers such that: :$\forall n: x_{n + 1} \equiv x_n \pmod {p^n}$ Then: :$\sequence {x_n}$ is a Cauchy sequence in $\struct {\Q, \norm {\,\cdot\,}_p}$.


P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 2

:$a \in \Z_{> 0}: \nexists \,c \in \Z : c^k = a$


P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 3

:$\map f {x_1} \equiv 0 \pmod p$


P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 4

:$\map {f'} {x_1} \not \equiv 0 \pmod p$


Surjection from Natural Numbers iff Countable/Corollary 1

Let $T$ be a countably infinite set. Let $S$ be a non-empty set. Then $S$ is countable {{iff}} there exists a surjection $f: T \to S$.


Surjection from Natural Numbers iff Countable/Corollary 2

Let $T$ be a countably infinite set. Let $S$ be an uncountable set. Let $f:T \to S$ be a mapping. Then $f$ is not a surjection.


P-adic Numbers are Uncountable

Let $p$ be any prime number. The set of $p$-adic numbers $\Q_p$ is an uncountable set.


Ordered Integral Domain is Totally Ordered Ring

Let $\struct {D, +, \times, \le}$ be an ordered integral domain. Then $\struct {D, +, \times, \le}$ is a totally ordered ring.


Strict Negativity is equivalent to Strictly Preceding Zero

:$\map N a \iff a < 0$


Sum of Strictly Negative Elements is Strictly Negative

:$\map N a, \map N b \implies \map N {a + b}$


Product of Two Strictly Negative Elements is Strictly Positive

:$\map N a, \map N b \implies \map P {a \times b}$


Product of Strictly Negative Element with Strictly Positive Element is Strictly Negative

:$\map N a, \map P b \implies \map N {a \times b}$


Principle of Mathematical Induction/Zero-Based

Let $\map P n$ be a propositional function depending on $n \in \N$. Suppose that: :$(1): \quad \map P 0$ is true :$(2): \quad \forall k \in \N: k \ge 0 : \map P k \implies \map P {k + 1}$ Then: :$\map P n$ is true for all $n \in \N$.


Principle of Mathematical Induction/One-Based

Let $\map P n$ be a propositional function depending on $n \in \N_{>0}$. Suppose that: :$(1): \quad \map P 1$ is true :$(2): \quad \forall k \in \N_{>0}: k \ge 1 : \map P k \implies \map P {k + 1}$ Then: :$\map P n$ is true for all $n \in \N_{>0}$.


Principle of Finite Induction/One-Based

Let $S \subseteq \N_{>0}$ be a subset of the $1$-based natural numbers. Suppose that: :$(1): \quad 1 \in S$ :$(2): \quad \forall n \in \N_{>0} : n \in S \implies n + 1 \in S$ Then: :$S = \N_{>0}$


Second Principle of Finite Induction/One-Based

Let $S \subseteq \N_{>0}$ be a subset of the $1$-based natural numbers. Suppose that: :$(1): \quad 1 \in S$ :$(2): \quad \forall n \in \N_{>0}: \paren {\forall k: 1 \le k \le n \implies k \in S} \implies n + 1 \in S$ Then: :$S = \N_{>0}$


Second Principle of Mathematical Induction/Zero-Based

Let $\map P n$ be a propositional function depending on $n \in \N$. Suppose that: :$(1): \quad \map P 0$ is true :$(2): \quad \forall k \in \N: \map P 0 \land \map P 1 \land \ldots \land \map P {k - 1} \land \map P k \implies \map P {k + 1}$ Then: :$\map P n$ is true for all $n \in \N$.


Second Principle of Mathematical Induction/One-Based

Let $\map P n$ be a propositional function depending on $n \in \N_{>0}$. Suppose that: :$(1): \quad \map P 1$ is true :$(2): \quad \forall k \in \N_{>0}: \map P 1 \land \map P 2 \land \ldots \land \map P {k - 1} \land \map P k \implies \map P {k + 1}$ Then: :$\map P n$ is true for all $n \in \N_{>0}$.


Null Sequences form Maximal Left and Right Ideal/Corollary 1

Then $\NN$ is a maximal ring ideal of $\CC$.


Non-Zero Integer has Finite Number of Divisors

Let $n \in \Z_{\ne 0}$ be a non-zero integer. Then $n$ has a finite number of divisors.


Coprimality Relation is Non-Reflexive

:$\perp$ is non-reflexive.


Ring of Polynomials over Reals is not Field

Let $\R \sqbrk X$ be the ring of polynomials in an indeterminate $X$ over $\R$. Then $\R \sqbrk X$ is not a field.


Ideals of Ring of Integers Modulo m

Let $m \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {\Z_m, +, \times}$ denote the ring of integers modulo $m$. The ideals of $\struct {\Z_m, +, \times}$ are of the form: :$d \Z / m \Z$ where $d$ is a divisor of $m$.


Quotient Ring of Cauchy Sequences is Division Ring/Corollary 1

Then the quotient ring $\CC \,\big / \NN$ is a field.


Quotient Ring of Cauchy Sequences is Normed Division Ring/Corollary 1

Then $\struct {\CC \,\big / \NN, \norm {\, \cdot \,}_1 }$ is a valued field.


Cauchy Sequence Converges Iff Equivalent to Constant Sequence

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. Let $\\CC \,\big / \NN$ be the quotient ring of Cauchy sequences of $\CC$ by the maximal ideal $\NN$. Let $\sequence {x_n} \in \CC$. Then $\sequence {x_n}$ converges in $\struct {R, \norm {\,\cdot\,} }$ {{iff}} :$\exists a \in R: \sequence {x_n} \in \sequence {a, a, a, \dotsc} + \NN$ where $\sequence {a, a, a, \dotsc} + \NN$ is the left coset in $\CC \, \big / \NN$ that contains the constant sequence $\sequence {a, a, a, \dotsc}$.


Homomorphism of Ring Subtraction

Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism. Then: :$\forall a, b \in R_1: \map \phi {a -_1 b} = \map \phi a -_2 \map \phi b$ where $a -_1 b$ denotes subtraction of $b$ from $a$.


Limit of Modulo Operation/Limit 1

Let $x$ and $y$ be real numbers. Let $x \bmod y$ denote the modulo operation. Then $\displaystyle \lim_{y \mathop \to 0} x \bmod y = 0$.


Limit of Modulo Operation/Limit 2

Let $x$ and $y$ be real numbers. Let $x \bmod y$ denote the modulo operation. Then $\displaystyle \lim_{y \mathop \to \infty} x \bmod y = x$ if $x \ge 0$.


Limit of Modulo Operation

Let $x$ and $y$ be real numbers. Let $x \bmod y$ denote the modulo operation. Then holding $x$ fixed gives: :$\displaystyle \lim_{y \mathop \to 0} x \bmod y = 0$ :$\displaystyle \lim_{y \mathop \to \infty} x \bmod y = x$ if $x \ge 0$ {{expand|What about the limits with respect to $x$?}}


Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary

Let $\struct {R, \norm{\,\cdot\,} }$ be a non-Archimedean normed division ring with zero $0_R$ Let $\sequence {x_n}$ be a Cauchy sequence such that $\sequence {x_n}$ does not converge to $0_R$. Then: :$\exists N \in \N: \forall n, m \ge N: \norm {x_n} = \norm {x_m}$


Equivalence of Definitions of Unit of Ring

Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$. {{TFAE|def = Unit of Ring}}


Bézout's Lemma/Euclidean Domain

Let $\struct {D, +, \times}$ be a Euclidean domain whose zero is $0$ and whose unity is $1$. Let $\nu: D \setminus \set 0 \to \N$ be the Euclidean valuation on $D$. Let $a, b \in D$ such that $a$ and $b$ are not both equal to $0$. Let $\gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$. Then: :$\exists x, y \in D: a \times x + b \times y = \gcd \set {a, b}$ such that $\gcd \set {a, b}$ is the element of $D$ such that: :$\forall c = a \times x + b \times y \in D: \map \nu {\gcd \set {a, b} } \le \map \nu c$


Euclidean Valuation of Non-Unit is less than that of Product

Let $\struct {D, +, \times}$ be a Euclidean domain whose zero is $0$, and unity is $1$. Let the valuation function of $D$ be $\nu$. Let $b, c \in D_{\ne 0}$. Then: :If $c$ is not a unit of $D$ then $\map \nu b < \map \nu {b c}$


Element is Unit iff its Euclidean Valuation equals that of 1

Let $\struct {D, +, \times}$ be a Euclidean domain whose zero is $0$, and unity is $1$. Let the valuation function of $D$ be $\nu$. Let $a \in D$. Then: :$a$ is a unit of $D$ {{iff}} $\map \nu a = \map \nu 1$


Gauss's Lemma on Primitive Polynomials over Ring

Let $R$ be a commutative ring with unity. Let $f, g \in R \sqbrk X$ be primitive polynomials. {{explain|Definition:Primitive Polynomial over Ring}} Then $f g$ is primitive.


Rational Polynomial is Content Times Primitive Polynomial/Existence

Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\map f X \in \Q \sqbrk X$. Then: :$\map f X = \cont f \, \map {f^*} X$ where: :$\cont f$ is the content of $\map f X$ :$\map {f^*} X$ is a primitive polynomial.


Rational Polynomial is Content Times Primitive Polynomial/Uniqueness

Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\map f X \in \Q \sqbrk X$ be given. Then there exist unique content $\cont f$ of $\map f X$ and primitive polynomial $\map {f^*} X$ such that: :$\map f X = \cont f \, \map {f^*} X$


Content of Polynomial in Dedekind Domain is Multiplicative

Let $R$ be a Dedekind domain. Let $f, g \in R \sqbrk X$ be polynomials. Let $\cont f$ denote the content of $f$. Then $\cont {f g} = \cont f \cont g$ is the product of $\cont f$ and $\cont g$.


Factors of Polynomial with Integer Coefficients have Integer Coefficients

Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\map h X \in \Q \sqbrk X$ have coefficients all of which are integers. Let it be possible to express $\map h X$ as: :$\map h X = \map f X \, \map g X$ where $\map f X, \map g X \in \Q \sqbrk X$. Then it is also possible to express $\map h X$ as: :$\map h X = \map {f'} X \, \map {g'} X$ where: :$\map {f'} X, \map {g'} X \in \Q \sqbrk X$ :the coefficients of $\map {f'} X$ and $\map {g'} X$ are all integers :$\map {f'} X = a \map f X$ and $\map {g'} X = b \map f X$, for $a, b \in \Q$.


Polynomial which is Irreducible over Integers is Irreducible over Rationals

Let $\Z \sqbrk X$ be the ring of polynomial forms over the integers in the indeterminate $X$. Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\map f X \in \Z \sqbrk X$ be irreducible in $\Z \sqbrk X$. Then $\map f X$ is also irreducible in $\Q \sqbrk X$.


Dirichlet Function is Periodic

Let $D: \R \to \R$ be a Dirichlet function: :$\forall x \in \R: \map D x = \begin{cases} c & : x \in \Q \\ d & : x \notin \Q \end{cases}$ Then $D$ is periodic. Namely, every non-zero rational number is a periodic element of $D$.


Dirichlet Function has no Period

The Dirichlet functions are periodic by Dirichlet Function is Periodic. However, they do not admit a period. That is, there does not exist a smallest value $L \in \R_{> 0}$ such that: :$\forall x \in \R: \map D x = \map D {x + L}$


Existence of Nonconstant Periodic Function with no Period

There exists a real, non-constant function $f$ such that: :$(1): \quad f$ is periodic. :$(2): \quad f$ does '''not''' have a period.


Vector Augend plus Addend equals Augend implies Addend is Zero

Let $V$ be a vector space over a field $F$. Let $\mathbf a, \mathbf b \in V$. Let $\mathbf a + \mathbf b = \mathbf a$. Then: :$\mathbf b = \bszero$ where $\bszero$ is the zero vector of $V$.


Vector Space of All Mappings is Vector Space

Let $\struct {K, +, \circ}$ be a division ring. Let $\struct {G, +_G, \circ}_K$ be a $K$-vector space. Let $S$ be a set. Let $\struct {G^S, +_G', \circ}_R$ be the vector space of all mappings from $S$ to $G$. Then $\struct {G^S, +_G', \circ}_K$ is a $K$-vector space.


Unitary Module of All Mappings is Unitary Module

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module. Let $S$ be a set. Let $\struct {G^S, +_G', \circ}_R$ be the module of all mappings from $S$ to $G$. Then $\struct {G^S, +_G', \circ}_R$ is a unitary module.


Finite Direct Product of Unitary Modules is Unitary Module

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G_1, +_1, \circ_1}_R, \struct {G_2, +_2, \circ_2}_R, \ldots, \struct {G_n, +_n, \circ_n}_R$ be unitary $R$-modules. Let: :$\displaystyle G = \prod_{k \mathop = 1}^n G_k$ be their direct product. Then $G$ is a unitary module.


Nonconstant Periodic Function with no Period is Discontinuous Everywhere

Let $f$ be a real periodic function that does not have a period. Then $f$ is either constant or discontinuous everywhere.