hcju/proofwikips huggingface dataset:
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 : :$J$ 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  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
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 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}}