Numbers in Row of Pascal's Triangle all Odd iff Row number 2^n - 1

Let $n \in \Z_{\ge 0}$ be a positive integer. Then the non-zero elements of the $n$th row of Pascal's triangle are all odd {{iff}}: :$n = 2^m - 1$ for some $m \in \Z_{\ge 0}$. {{:Definition:Pascal's Triangle}} As can be seen, the entries in rows $0, 1, 3, 7$ are all odd.


Sum of Sequence of Fourth Powers

:$\displaystyle \sum_{k \mathop = 0}^n k^4 = \dfrac {\paren {n + 1} n \paren {n + \frac 1 2} \paren {3 n^2 + 3 n - 1} } {15}$


Complete Lattice is Bounded

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Then :$L$ is bounded


Constant Function is Primitive Recursive/General Case

The constant function of $k$ variables: $f_c^k: \N^k \to \N$, defined as: :$\map {f_c^k} {n_1, n_2, \ldots, n_k} = c$ where $c \in \N$ is primitive recursive.


Value of Multiplicative Function at One

Let $f: \N \to \C$ be a multiplicative function. If $f$ is not identically zero, then $\map f 1 = 1$.


Value of Multiplicative Function is Product of Values of Prime Power Factors

Let $f: \N \to \C$ be a multiplicative function. Let $n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$ be the prime decomposition of $n$. Then: :$\map f n = \map f {p_1^{k_1} } \, \map f {p_2^{k_2} } \dotsm \map f {p_r^{k_r} }$


-1^n by -n choose k-1 equals -1^k by -k choose n-1

:$\paren {-1}^n \dbinom {-n} {k - 1} = \paren {-1}^k \dbinom {-k} {n - 1}$


One Choose n

:$\dbinom 1 n = \begin{cases} 1 & : n \in \left\{ {0, 1}\right\} \\ 0 & : \text {otherwise} \end{cases}$


Summations of Products of Binomial Coefficients

This page gathers together some identities concerning summations of products of binomial coefficients. In the following, unless otherwise specified: :$k, m, n \in \Z$ :$r, s, t \in \R$.


Separable Discrete Space is Countable

Let $T = \struct {S, \tau}$ be a discrete topological space. Let $T$ be separable. Then $S$ is countable.


Sum over k of r-tk Choose k by s-t(n-k) Choose n-k by r over r-tk/Proof 1/Basis for the Induction

Let $r, s, t \in \R, n \in \Z$. Consider the equation: :$\displaystyle (1): \quad \sum_{k \mathop \ge 0} \binom {r - t k} k \binom {s - t \paren {n - k} } {n - k} \frac r {r - t k} = \binom {r + s - t n} n$ where $\dbinom {r - t k} k$ etc. are binomial coefficients. Then equation $(1)$ holds for the special case where $s = n - 1 - r + n t$.


Continuous Lattice iff Auxiliary Approximating Relation is Superset of Way Below Relation

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $\mathit{App}\left({L}\right)$ be the set of all auxiliary approximating relation on $S$. Then :$L$ is continuous {{iff}} :$\forall \mathcal R \in \mathit{App}\left({L}\right): \ll \subseteq \mathcal R$ and $\ll$ is an approximating relation


Completely Additive Function is Additive

Let $f: \N \to \C$ be a completely additive function. Then $f$ is also additive.


Real Logarithm is Completely Additive

Let $\log_b: \R_{>0} \to \R$ be the real logarithm to base $b$. Then $\log_b$ is completely additive.


Sum over k of r-tk Choose k by s-t(n-k) Choose n-k by r over r-tk/Proof 1/Lemma

Let this hold for $\tuple {r, s, t, n}$: :$\displaystyle \sum_{k \mathop \ge 0} \binom {r - t k} k \binom {s - t \paren {n - k} } {n - k} \frac r {r - t k} = \binom {r + s - t n} n$ and also for $\tuple {r, s - t, t, n - 1}$. Then it also holds for $\tuple {r, s + 1, t, n}$.


Sum over k of r-kt choose k by r over r-kt by z^k

Let $n \in \Z_{\ge 0}$ be a positive integer. Let $A_n \left({x, t}\right)$ be the polynomial of degree $n$ defined as: :$A_n \left({x, t}\right) := \dbinom {x - n t} n \dfrac x {x - n t}$ for $x \ne n t$. Let $z = x^{t + 1} - x^t$. Then: :$\displaystyle \sum_k A_k \left({r, t}\right) z^k = x^r$ for sufficiently small $z$.


Not Preceding implies Approximating Relation and not Preceding

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $x, y \in S$ such that :$x \npreceq y$ Let $\mathcal R$ be an approximating relation on $S$. Then :$\exists u \in S: \left({u, x}\right) \in \mathcal R \land u \npreceq y$


Auxiliary Approximating Relation has Quasi Interpolation Property

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $x, z \in S$. Let $\mathcal R$ be an auxiliary approximating relation on $S$ such that :$\left({x, z}\right) \in \mathcal R \land x \ne z$ Then :$\exists y \in S: x \preceq y \land \left({y, z}\right) \in \mathcal R \land x \ne y$


Preceding iff Join equals Larger Operand

Let $\left({S, \preceq}\right)$ be a join semilattice. Let $x, y \in S$. Then :$x \preceq y$ {{iff}} $x \vee y = y$


Sum over k of -1^k by n choose k by r-kt choose n by r over r-kt

:$\displaystyle \sum_k \left({-1}\right)^k \dbinom n k \dbinom {r - k t} n \dfrac r {r - k t} = \delta_{n 0}$ where $\delta_{n 0}$ is the Kronecker delta.


Sum over k of r-kt choose k by z^k

Let $n \in \Z_{\ge 0}$ be a non-negative integer. Then: :$\displaystyle \sum_k \dbinom {r - t k} k z^k = \frac {x^{r + 1} } {\left({t + 1}\right)x - t}$ where $\dbinom {r - t k} k$ denotes a binomial coefficient.


Auxiliary Approximating Relation has Interpolation Property

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $x, z \in S$ such that :$x \ll z \land x \ne z$ Let $\mathcal R$ be an auxiliary approximating relation on $S$. Then :$\exists y \in S: \left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \land x \ne y$


Way Below has Strong Interpolation Property

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice. Let $x, z \in S$ such that :$x \ll z \land x \ne z$ Then :$\exists y \in S: x \ll y \land y \ll z \land x \ne y$


Way Below has Interpolation Property

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice. Let $x, z \in S$ such that :$x \ll z$ Then :$\exists y \in S: x \ll y \land y \ll z$


Left Coset Equals Subgroup iff Element in Subgroup

:$x H = H \iff x \in H$


Right Coset Equals Subgroup iff Element in Subgroup

:$H x = H \iff x \in H$


Elements in Same Left Coset iff Product with Inverse in Subgroup

: $x, y$ are in the same left coset of $H$ {{iff}} $x^{-1} y \in H$.


Elements in Same Right Coset iff Product with Inverse in Subgroup

: $x, y$ are in the same right coset of $H$ {{iff}} $x y^{-1} \in H$


Sum over k of m+r+s Choose k by n+r-s Choose n-k by r+k Choose m+n

Let $m, n \in \Z_{\ge 0}$. Then: :$\displaystyle \sum_k \binom {m - r + s} k \binom {n + r - s} {n - k} \binom {r + k} {m - n} = \binom r m \binom s n$


Sum over k of Unsigned Stirling Numbers of First Kind by x^k

:$\displaystyle \sum_k \left[{n \atop k}\right] x^k = x^{\overline n}$ where: :$\displaystyle \left[{n \atop k}\right]$ denotes an unsigned Stirling number of the first kind :$x^{\overline n}$ denotes $x$ to the $n$ rising.


Capelli's Sum

:$\displaystyle \left({x + y}\right)^{\overline n} = \sum_k \binom n k x^{\overline k} y^{\overline {n - k} }$ where: :$\dbinom n k$ denotes a binomial coefficient :$x^{\overline k}$ denotes $x$ to the $k$ rising.


Way Below iff Second Operand Preceding Supremum of Directed Set There Exists Element of Directed Set First Operand Way Below Element

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice. Let $x, y$ be elements of $S$. Then :$x \ll y$ {{iff}} :for every directed subset $D$ of $S$ such that $y \preceq \sup D$ ::there exists an element $d$ of $D$: $x \ll d$


Continuous iff Way Below Closure is Ideal and Element Precedes Supremum

Let $L = \left({S, \wedge, \preceq}\right)$ be an up-complete meet semilattice. Then :$L$ is continuous {{iff}} :$\forall x \in S: x^\ll$ is an ideal in $L$ and $x \preceq \sup \left({x^\ll}\right)$ and ::for every ideal $I$ in $L$: $x \preceq \sup I \implies x^\ll \subseteq I$ where $x^\ll$ denotes the way below closure of $x$.


Characterization of Euclidean Borel Sigma-Algebra/Open equals Closed

Let $\mathcal O^n$, $\mathcal C^n$ and $\mathcal K^n$ be the collections of open and closed subsets of the Euclidean space $\left({\R^n, \tau}\right)$, respectively. Then: :$\sigma \left({\mathcal O^n}\right) = \sigma \left({\mathcal C^n}\right)$ where $\sigma$ denotes generated $\sigma$-algebra.


Characterization of Euclidean Borel Sigma-Algebra/Closed equals Compact

Let $\mathcal C^n$ and $\mathcal K^n$ be the collections of closed and compact subsets of the Euclidean space $\left({\R^n, \tau}\right)$, respectively. Then: :$\sigma \left({\mathcal C^n}\right) = \sigma \left({\mathcal K^n}\right)$ where $\sigma$ denotes generated $\sigma$-algebra.


Characterization of Euclidean Borel Sigma-Algebra/Open equals Rectangle

Let $\mathcal O^n$ be the collection of open subsets of the Euclidean space $\left({\R^n, \tau}\right)$. Let $\mathcal J_{ho}^n$ be the collection of half-open rectangles in $\R^n$. Then: :$\sigma \left({\mathcal O^n}\right) = \sigma \left({\mathcal J_{ho}^n}\right)$ where $\sigma$ denotes generated $\sigma$-algebra.


Characterization of Euclidean Borel Sigma-Algebra/Rectangle equals Rational Rectangle

Let $\mathcal J_{ho}^n$ be the collection of half-open rectangles in $\R^n$. Let $\mathcal J^n_{ho, \text{rat}}$ be the collection of half-open rectangles in $\R^n$ with rational endpoints. Then: :$\sigma \left({\mathcal J_{ho}^n}\right) = \sigma \left({\mathcal J^n_{ho, \text{rat}}}\right)$ where $\sigma$ denotes generated $\sigma$-algebra.


Rising Factorial as Factorial by Binomial Coefficient

Let $x \in \R$ be a real number. Let $n \in \Z_{\ge 0}$ be a positive integer. :$x^{\overline n} = n! \dbinom {x + n - 1} n$ where: :$x^{\overline n}$ denotes $x$ to the $n$ rising :$n!$ denotes the factorial of $n$ :$\dbinom {x + n - 1} n$ denotes a binomial coefficient.


Ramus's Identity

Let $k, m, n \in \Z_{\ge 0}$ be positive integers such that $0 \le k < m$. Then: {{begin-eqn}} {{eqn | l = \sum_{j \mathop \ge 0} \binom n {j m + k}       | r = \dbinom n k + \dbinom n {m + k} +  \dbinom n {2 m + k} + \cdots        | c =  }} {{eqn | r = \dfrac 1 m \sum_{0 \mathop \le j \mathop < m} \paren {2 \cos \dfrac {j \pi} m}^n \cos \dfrac {j \paren {n - 2 k} \pi} m       | c =  }} {{end-eqn}}


Way Below Closure is Lower Set

Let $L = \struct {S, \vee, \preceq}$ be an ordered set. Let $x \in S$. Then :$x^\ll$ is a lower set.


Continuous iff For Every Element There Exists Ideal Element Precedes Supremum

Let $L = \left({S, \wedge, \preceq}\right)$ be an up-complete meet semilattice. Then :$L$ is continuous {{iff}} :for every element $x$ of $S$ there exists ideal $I$ in $L$: ::$x \preceq \sup I$ and for every ideal $J$ in $L: x \preceq \sup J \implies I \subseteq J$


Beta Function of x+1 with y plus Beta Function of x with y+1

:$\Beta \left({x + 1, y}\right) + \Beta \left({x, y + 1}\right) = \Beta \left({x, y}\right)$


Beta Function of x with y+1 by x+y over y

:$\map \Beta {x, y} = \dfrac {x + y} y \map \Beta {x, y + 1}$


Partial Gamma Function expressed as Integral

Let $m \in \Z_{\ge 1}$. Let $\map {\Gamma_m} x$ denote the partial Gamma function, defined as: :$\map {\Gamma_m} x := \dfrac {m^x m!} {x \paren {x + 1} \paren {x + 2} \cdots \paren {x + m} }$ Then: :$\displaystyle \map {\Gamma_m} x = \int_0^m \paren {1 - \frac t m}^m t^{x - 1} \rd t$ for $x > 0$.


Partial Gamma Function expressed as Integral/Lemma

:$(1): \quad \displaystyle \int_0^m \paren {1 - \frac t m}^m t^{x - 1} \rd t = m^x \int_0^1 \paren {1 - t}^m t^{x - 1} \rd t$ for $x > 0$.


Integral Form of Gamma Function equivalent to Euler Form

{{TFAENocat|def = Gamma Function}} === Integral Form === {{:Definition:Integral Form of Gamma Function}} === Euler Form === {{:Definition:Euler Form of Gamma Function}}


Derivative of Composite Function/Second Derivative

:$D_x^2 w = D_u^2 w \paren {D_x^1 u}^2 + D_u^1 w D_x^2 u$


Faà di Bruno's Formula

Let $D_x^k u$ denote the $k$th derivative of a function $u$ {{WRT|Differentiation}} $x$. Then: :$\displaystyle D_x^n w = \sum_{j \mathop = 0}^n D_u^j w \sum_{\substack {\sum_{p \mathop \ge 1} k_p \mathop = j \\ \sum_{p \mathop \ge 1} p k_p \mathop = n \\ \forall p \mathop \ge 1: k_p \mathop \ge 0} } n! \prod_{m \mathop = 1}^n \dfrac {\left({D_x^m u}\right)^{k_m} } {k_m! \left({m!}\right)^{k_m} }$


Cardinality of Reduced Residue System

Let $n \ge 2$. Let $\Z'_n$ be the reduced residue system modulo $n$. Then: :$\card {\Z'_n} = \map \phi n$ where $\map \phi n$ is the Euler phi function.


Equivalence of Definitions of Norm of Linear Functional/Corollary

For all $h \in H$, the following inequality holds: :$\left|{Lh}\right| \le \left\|{L}\right\| \left\|{h}\right\|$


Faà di Bruno's Formula/Example/0/Proof

Consider Faà di Bruno's Formula: {{:Faà di Bruno's Formula}} When $n = 0$ we have: {{:Faà di Bruno's Formula/Example/0}}


Faà di Bruno's Formula/Example/1/Proof

Consider Faà di Bruno's Formula: {{:Faà di Bruno's Formula}} When $n = 1$ we have: {{:Faà di Bruno's Formula/Example/1}}


Faà di Bruno's Formula/Example/2/Proof

Consider Faà di Bruno's Formula: {{:Faà di Bruno's Formula}} When $n = 2$ we have: {{:Faà di Bruno's Formula/Example/2}}


Faà di Bruno's Formula/Lemma 1

:$D_x \left({\left({D_x^m u}\right)^{k_m} }\right) = k_m \left({D_x^m u}\right)^{k_m - 1} D_x^{m + 1} u$


Faà di Bruno's Formula/Lemma 2

:$\displaystyle D_x \left({\prod_{m \mathop = 1}^r \left({\dfrac {\left({D_x^m u}\right)^{k_m} } {k_m! \left({m!}\right)^{k_m} } }\right) }\right) = \prod_{m \mathop = 1}^r \left({\dfrac {\left({D_x^m u}\right)^{k_m} } {k_m! \left({m!}\right)^{k_m} } }\right) \sum_{m \mathop = 1}^r k_m \dfrac {D_x^{m + 1} u} {D_x^m u}$


Supremum of Ideals is Increasing

Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $\mathit{Ids}\left({L}\right)$ be the set of all ideals in $L$. Let $P = \left({\mathit{Ids}\left({L}\right), \precsim}\right)$ be an ordered set where $\mathord \precsim = \subseteq\restriction_{\mathit{Ids}\left({L}\right)\times \mathit{Ids}\left({L}\right)}$ Let $f: \mathit{Ids}\left({L}\right) \to S$ be a mapping such that :$\forall I \in \mathit{Ids}\left({L}\right): f\left({I}\right) = \sup I$ Then $f$ is an increasing mapping.


Derivative of Composite Function/Third Derivative

:$D_x^3 w = D_u^3 w \paren {D_x^1 u}^3 + 3 D_u^2 w D_x^2 u D_x^1 u + D_u^1 w D_x^3 u$


Number of Set Partitions by Number of Components

Let $S$ be a (finite) set whose cardinality is $n$. Let $\map f {n, k}$ denote the number of different ways $S$ can be partitioned into $k$ components. Then: :$\displaystyle \map f {n, k} = {n \brace k}$ where $\displaystyle {n \brace k}$ denotes a Stirling number of the second kind.


Floor of Half of n+m plus Floor of Half of n-m+1

Let $n, m \in \Z$ be integers. :$\floor {\dfrac {n + m} 2} + \floor {\dfrac {n - m + 1} 2} = n$ where $\floor x$ denotes the floor of $x$.


Ceiling of Half of n+m plus Ceiling of Half of n-m+1

Let $n, m \in \Z$ be integers. :$\left\lceil{\dfrac {n + m} 2}\right\rceil + \left\lceil{\dfrac {n - m + 1} 2}\right\rceil = n + 1$ where $\left\lceil{x}\right\rceil$ denotes the ceiling of $x$.


Floor of Non-Integer

Let $x \in \R$ be a real number. Let $x \notin \Z$. Then: :$\left\lfloor{x}\right\rfloor < x$ where $\left\lfloor{x}\right\rfloor$ denotes the floor of $x$.


Logarithm of Base

Let $b \in \R_{>0}$ such that $b \ne 1$. Then: :$\log_b b = 1$ where $\log_b$ denotes the logarithm to base $b$.


Floor of x+m over n

Let $m, n \in \Z$ such that $n > 0$. Let $x \in \R$. Then: :$\floor {\dfrac {x + m} n} = \floor {\dfrac {\floor x + m} n}$ where $\floor x$ denotes the floor of $x$.


McEliece's Theorem (Integer Functions)

Let $f: \R \to \R$ be a continuous, strictly increasing real function defined on an interval $A$. Let: :$\forall x \in A: \left \lfloor{x}\right \rfloor \in A, \left \lceil{x}\right \rceil \in A$ where: :$\left \lfloor{x}\right \rfloor$ denotes the floor of $x$ :$\left \lceil{x}\right \rceil$ denotes the ceiling of $x$ Then: :$\forall x \in A: \left \lfloor{f \left({x}\right)}\right \rfloor = \left \lfloor{f \left({\left \lfloor{x}\right \rfloor}\right)}\right \rfloor \iff \left \lceil{f \left({x}\right)}\right \rceil = \left \lceil{f \left({\left \lceil{x}\right \rceil}\right)}\right \rceil \iff \left(f \left({x}\right) \in \Z \implies x \in \Z)\right)$


Ceiling of Non-Integer

Let $x \in \R$ be a real number. Let $x \notin \Z$. Then: :$\left \lceil{x}\right \rceil > x$ where $\left \lceil{x}\right \rceil$ denotes the ceiling  of $x$.


Ceiling of x+m over n

Let $m, n \in \Z$ such that $n > 0$. Let $x \in \R$. Then: :$\ceiling {\dfrac {x + m} n} = \ceiling {\dfrac {\ceiling x + m} n}$ where $\ceiling x$ denotes the ceiling of $x$.


Summation over k of Floor of mk+x over n

Let $m, n \in \Z$ such that $n > 0$. Let $x \in \R$. Then: :$\displaystyle \sum_{0 \mathop \le k \mathop < n} \left \lfloor{\dfrac {m k + x} n}\right \rfloor = \dfrac {\left({m - 1}\right) \left({n - 1}\right)} 2 + \dfrac {d - 1} 2 + d \left \lfloor{\dfrac x d}\right \rfloor$ where: :$\left \lfloor{x}\right \rfloor$ denotes the floor of $x$ :$d$ is the greatest common divisor of $m$ and $n$.


Summation over k of Floor of x plus k over y

Let $x, y \in \R$ such that $y > 0$. Then: :$\displaystyle \sum_{0 \mathop \le k \mathop < y} \left \lfloor{x + \dfrac k y}\right \rfloor = \left \lfloor{x y + \left \lfloor{x + 1}\right \rfloor \left({\left \lceil{y}\right \rceil - y}\right)}\right \rfloor$


Semantic Consequence preserved in Supersignature

Let $\mathcal L, \mathcal L'$ be signatures for the language of predicate logic. Let $\mathcal L'$ be a supersignature of $\mathcal L$. Let $\mathbf A$ be an $\mathcal L$-sentence. Let $\Sigma$ be a set of $\mathcal L$-sentences. Then the following are equivalent: :$\mathcal A \models_{\mathrm{PL}} \mathbf A$ for all $\mathcal L$-structure $\mathcal A$ for which $\mathcal A \models_{\mathrm{PL}} \Sigma$ :$\mathcal A' \models_{\mathrm{PL}} \mathbf A$ for all $\mathcal L'$-structure $\mathcal A'$ for which $\mathcal A' \models_{\mathrm{PL}} \Sigma$ where $\models_{\mathrm{PL}}$ denotes the models relation. That is to say, the notion of semantic consequence is preserved in passing to a supersignature.


Satisfiability preserved in Supersignature

Let $\mathcal L, \mathcal L'$ be signatures for the language of predicate logic. Let $\mathcal L'$ be a supersignature of $\mathcal L$. Let $\Sigma$ be a set of $\mathcal L$-sentences. Then the following are equivalent: :$\mathcal A \models_{\mathrm{PL}} \Sigma$ for some $\mathcal L$-structure $\mathcal A$ :$\mathcal A' \models_{\mathrm{PL}} \Sigma$ for some $\mathcal L'$-structure $\mathcal A'$ where $\models_{\mathrm{PL}}$ is the models relation. That is to say, the notion of satisfiability is preserved in passing to a supersignature.


Subtract Half is Replicative Function

Let $f: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map f x = x - \dfrac 1 2$ Then $f$ is a replicative function.


Membership of Set of Integers is Replicative Function

Let $f: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map f x = \sqbrk {x \in \Z}$ where $\sqbrk {\cdots}$ is Iverson's convention. Then $f$ is a replicative function.


Logarithm of Absolute Value of 2 times Sine of pi x is Replicative Function

Let $f: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map f x = \log \, \size {2 \sin \pi x}$ Then $f$ is a replicative function.


Constant Multiple of Replicative Function is Replicative

Let $f: \R \to \R$ be a real function. Let $f$ be a replicative function. Let $c \in \R$ be a constant. Let $g: \R \to \R$ be the real function defined as: :$g \left({x}\right) = c f \left({x}\right)$ Then $g$ is also a replicative function.


Replicative Function of x minus Floor of x is Replicative

Let $f: \R \to \R$ be a real function. Let $f$ be a replicative function. Let $g: \R \to \R$ be the real function defined as: :$g \left({x}\right) = f \left({x - \left \lfloor{x}\right \rfloor}\right)$ Then $g$ is also a replicative function.


Closed Form for Sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ...

Let $a_1, a_2, a_3, \ldots$ be the sequence: :$\left \langle{a_n}\right \rangle = 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots$ Then: :$a_n = \left \lceil{\dfrac {\sqrt {1 + 8 n} - 1} 2}\right \rceil$


Sum of Sequence as Summation of Difference of Adjacent Terms

Let $n \in \Z_{> 0}$ be a strictly positive integer. Then: :$\displaystyle \sum_{k \mathop = 1}^n a_k = n a_n - \sum_{k \mathop = 1}^{n - 1} k \left({a_{k + 1} - a_k}\right)$


Continuous Lattice and Way Below implies Preceding implies Preceding

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a continuous complete lattice. Let $a, b \in S$. Let :$\forall c \in S: c \ll a \implies c \preceq b$ Then $a \preceq b$


Sum over k of Floor of Log base b of k

Let $n \in \Z_{> 0}$ be a strictly positive integer. Let $b \in \Z$ such that $b \ge 2$. Then: :$\displaystyle \sum_{k \mathop = 1}^n \left \lfloor{\log_b k}\right \rfloor = \left({n + 1}\right) \left \lfloor{\log_b n}\right \rfloor - \dfrac {b^{\left \lfloor{\log_b n}\right \rfloor + 1} - b} {b - 1}$


Sum over k of Floor of Root k

Let $n \in \Z_{> 0}$ be a strictly positive integer. Let $b \in \Z$ such that $b \ge 2$. Then: :$\displaystyle \sum_{k \mathop = 1}^n \left \lfloor{\sqrt k}\right \rfloor = \left \lfloor{\sqrt n}\right \rfloor \left({n - \dfrac {\left({2 \left \lfloor{\sqrt n}\right \rfloor + 5}\right) \left({\left \lfloor{\sqrt n}\right \rfloor - 1}\right)} 6 }\right)$


Top is Meet Irreducible

Let $\left({S, \wedge, \preceq}\right)$ be a bounded above meet semilattice. Then $\top$ is meet irreducible where $\top$ denotes the greatest element in $S$.


Sum over k of Sum over j of Floor of n + jb^k over b^k+1

Let $n, b \in \Z$ such that $n \ge 0$ and $b \ge 2$. Then: :$\displaystyle \sum_{k \mathop \ge 0} \sum_{1 \mathop \le j \mathop < b} \left \lfloor{\dfrac {n + j b^k} {b^{k + 1} } }\right \rfloor = n$


Equivalence of Definitions of Legendre Symbol

Let $p$ be an odd prime. Let $a \in \Z$. {{TFAE|def = Legendre Symbol}}


Upper Closure of Element without Element is Filter implies Element is Meet Irreducible

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Let $x \in S$. Let : $x^\succeq \setminus \left\{ {x}\right\}$ be a filter in $L$. Then $x$ is meet irreducible.


Maximal Element of Complement of Filter is Meet Irreducible

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Let $F$ be a filter in $L$. Let $p \in S$. Let $p = \max \complement_S\left({F}\right)$. Then $p$ is meet irreducible.


De Polignac's Formula/Technique

When calculating $\mu$, the easiest way to calculate the next term is simply to divide the previous term by $p$ and discard the remainder: :$\floor {\dfrac n {p^{k + 1} } } = \floor {\floor {\dfrac n {p^k} } / p}$


Not Preceding implies There Exists Meet Irreducible Element Not Preceding

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice. Let $x, y \in S$ such that :$y \npreceq x$ Then :$\exists p \in S: p$ is meet irreducible and $x \preceq p$ and $y \npreceq p$


Characteristics of Floor and Ceiling Function

Let $f: \R \to \Z$ be an integer-valued function which satisfies both of the following: :$(1): \quad \map f {x + 1} = \map f x + 1$ :$(2): \quad \forall n \in \Z_{> 0}: \map f x = \map f {\dfrac {\map f {n x} } n}$ Then either: :$\forall x \in \Q: \map f x = \floor x$ or: :$\forall x \in \Q: \map f x = \ceiling x$


Continuous Replicative Function

Let $f: \R \to \R$ be a real function. Let $f$ be continuous on $\R$. Let $f$ also be a replicative function. Then $f$ is of the form: :$f \left({x}\right) = \left({x - \dfrac 1 2}\right) a$ where $a \in \R$.


Sum over k to p over 2 of Floor of 2kq over p

Let $p \in \Z$ be an odd prime. Let $q \in \Z$ be an odd integer. Then: :$\displaystyle \sum_{0 \mathop \le k \mathop < p / 2} \floor {\dfrac {2 k q} p}r \equiv  \sum_{0 \mathop \le k \mathop < p / 2} \floor {\dfrac {k q} p} \pmod 2$


Way Below implies There Exists Way Below Open Filter Subset of Way Above Closure

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice. Let $x, y \in S$ such that :$x \ll y$ where $\ll$ denotes the way below relation. Then there exists a way below open filter in $L$: $y \in F \land F \subseteq x^\gg$ where $x^\gg$ denotes the way above closure of $x$.


Powers of Group Elements/Negative Index/Additive Notation

:$\forall n \in \Z: -\paren {n g} = \paren {-n} g = n \paren {-g}$


Powers of Group Elements/Product of Indices/Additive Notation

:$\forall m, n \in \Z: n \paren {m g} = \paren {m n} g = m \paren {n g}$


Power of Product in Abelian Group/Additive Notation

: $k \left({x + y}\right) = k x + k y$


Power of Idempotent Element

Let $\struct {S, \circ}$ be an algebraic structure. Let $s \in S$ be an idempotent element with respect to $\circ$. Then: :$\forall n \in \Z_{> 0}: s^n = s$ where $s^n$ is defined as: :$s^n = \begin{cases} s & : n = 1 \\ s^{n - 1} \circ s & : n > 1 \end{cases}$


Powers of Group Element Commute

Let $\struct {G, \circ}$ be a group. Let $g \in G$. Let $m, n \in \N_{>0}$. Then: :$\forall m, n \in \N_{>0}: g^n \circ g^m = g^m \circ g^n$


Strictly Positive Real Numbers under Multiplication form Subgroup of Non-Zero Real Numbers

Let $\R_{>0}$ be the set of strictly positive real numbers, that is: :$\R_{>0} = \set {x \in \R: x > 0}$ The structure $\struct {\R_{>0}, \times}$ forms a subgroup of $\struct {\R_{\ne 0}, \times}$, where $\R_{\ne 0}$ is the set of real numbers without zero, that is: :$\R_{\ne 0} = \R \setminus \set 0$


Strictly Positive Rational Numbers under Multiplication form Subgroup of Non-Zero Rational Numbers

Let $\Q_{> 0}$ be the set of strictly positive rational numbers, that is $\Q_{> 0} = \set { x \in \Q: x > 0}$. The structure $\struct {\Q_{> 0}, \times}$ is a subgroup of $\struct {\Q_{\ne 0}, \times}$, where $\Q_{\ne 0}$ is the set of rational numbers without zero: $\Q_{\ne 0} = \Q \setminus \set 0$.


Union of Upper Sets is Upper

Let $\left({S, \preceq}\right)$ be a preordered set. Let $A$ be a set of subsets of $S$ such that :$\forall X \in A: X$ is an upper set. Then: :$\bigcup A$ is also an upper set.


Set of Rotations in Space about Fixed Point forms Infinite Group

Let $\SS$ be a rigid body in space. Let $O$ be a fixed point in space. The set of all rotations of $\SS$ through some line through $O$ forms an infinite group.


Group of Rotations about Fixed Point is not Abelian

Let $\SS$ be a rigid body in space. Let $O$ be a fixed point in space. Let $\GG$ be the group of all rotations of $\SS$ around $O$. Then $\GG$ is not an abelian group.


Union of Filtered Sets is Filtered

Let $\left({S, \preceq}\right)$ be a preordered set. Let $A$ be a set of subsets of $S$ such that :$\forall X \in A: X$ is filtered and :$\forall X, Y \in A: \exists Z \in A: X \cup Y \subseteq Z$ Then: :$\bigcup A$ is also filtered.


Special Orthogonal Group is Group

Let $k$ be a field. The $n$th orthogonal group on $k$ is a group.


Unit Matrix is its own Inverse

The inverse of the unit matrix $\mathbf I_n$ of order $n$ is $\mathbf I_n$. That is, a unit matrix it its own inverse.


Way Above Closure is Subset of Upper Closure of Element

Let $\left({S, \preceq}\right)$ be an ordered set. Let $x \in S$. Then $x^\gg \subseteq x^\succeq$ where :$x^\gg$ denotes the way above closure of $x$, :$x^\succeq$ denotes the upper closure of $x$.


Diagonal Matrix is Symmetric

Let $D$ be a diagonal matrix. Then $D$ is symmetric.


Upper Way Below Open Subset Complement is Non Empty implies There Exists Maximal Element of Complement

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Let $X$ be upper way below open subset of $S$. Let $x \in S$ such that :$x \in \relcomp S X$ Then :$\exists m \in S: x \preceq m \land m = \max \relcomp S X$


Unit Matrix is Orthogonal

The unit matrix $\mathbf I_n$ of order $n$ is orthogonal.


Unit Matrix is Proper Orthogonal

The unit matrix $\mathbf I_n$ of order $n$ is proper orthogonal.


Inverse of Orthogonal Matrix is Orthogonal

Let $\mathbf A$ be an orthogonal matrix. Then its inverse $\mathbf A^{-1}$ is also orthogonal.


Inverse of Inverse of Matrix

Let $\mathbf A$ be an invertible matrix. Then: :$\paren {\mathbf A^{-1} }^{-1} = \mathbf A$ That is, an invertible matrix equals the inverse of its inverse.


Inverse of Proper Orthogonal Matrix is Proper Orthogonal

Let $\mathbf A$ be a proper orthogonal matrix. Then its inverse $\mathbf A^{-1}$ is also proper orthogonal.


Special Orthogonal Group is Subgroup of Orthogonal Group

Let $k$ be a field. Let $\map {\operatorname O} {n, k}$ be the $n$th orthogonal group on $k$. Let $\map {\operatorname {SO} } {n, k}$ be the $n$th special orthogonal group on $k$. Then $\map {\operatorname {SO} } {n, k}$ is a subgroup of $\map {\operatorname O} {n, k}$.


Negative Matrix is Inverse for Hadamard Product

Let $\struct {G, \cdot}$ be a group whose identity is $e$. Let $\map {\MM_G} {m, n}$ be a $m \times n$ matrix space over $\struct {G, \cdot}$. Let $\mathbf A$ be an element of $\map {\MM_G} {m, n}$. Let $-\mathbf A$ be the negative of $\mathbf A$. Then $-\mathbf A$ is the inverse for the operation $\circ$, where $\circ$ is the Hadamard product.


Left Regular Representation wrt Left Cancellable Element on Finite Semigroup is Bijection

Let $\struct {S, \circ}$ be a finite semigroup. Let $a \in S$ be left cancellable. Then the left regular representation $\lambda_a$ of $\struct {S, \circ}$ with respect to $a$ is a bijection.


Singleton is Chain

Let $\left({S, \preceq}\right)$ be an ordered set. Let $x \in S$. Then $\left\{ {x}\right\}$ is a chain of $\left({S, \preceq}\right)$.


Identity Elements occupy Diagonal of Cayley Table in Inverse Row Form

Let $\struct {G, \circ}$ be a finite group. Let $\CC$ be a Cayley table for $\struct {G, \circ}$ presented in inverse row form. Then all the entries in the main diagonal of $\CC$ are instances of the identity element.


Scaling preserves Modulo Addition

Let $m \in \Z_{> 0}$. Let $x, y, c \in \Z$. Let $x \equiv y \pmod m$. Then: :$c x \equiv c y \pmod m$


Modulo Addition is Linear

Let $m \in \Z_{> 0}$. Let $x_1, x_2, y_1, y_2, c_1, c_2 \in \Z$. Let: :$x_1 \equiv y_1 \pmod m$ :$x_2 \equiv y_2 \pmod m$ Then: :$c_1 x_1 + c_2 x_2 \equiv c_1 y_1 + c_2 y_2 \pmod m$


Euler Phi Function of Prime

Let $p$ be a prime number $p > 1$. Then: :$\map \phi p = p - 1$ where $\phi: \Z_{>0} \to \Z_{>0}$ is the Euler $\phi$ function.


Chain is Directed

Let $\struct {S, \preceq}$ be an ordered set. Let $C$ be a non-empty chain of $S$. Then $C$ is directed.


Reduced Residue System Modulo Prime

Let $p$ be a prime number. The reduced residue system modulo $p$ contains $p - 1$ elements: :$Z'_p = \set {\eqclass 1 m, \eqclass 2 m, \ldots, \eqclass {p - 1} m}$ and so can be defined as: :$Z'_p = Z_p \setminus \set {\eqclass 0 m}$


Modulo Multiplication on Reduced Residue System is Closed

Let $m \in \Z_{> 0}$ be a (strictly) positive integer. Let $\Z'_m$ be the reduced residue system modulo $m$: :$\Z'_m = \set {\eqclass k m \in \Z_m: k \perp m}$ Let $S = \struct {\Z'_m, \times_m}$ be the algebraic structure consisting of $\Z'_m$ under the modulo multiplication. Then $S$ is closed, in the sense that: :$\forall a, b \in \Z'_m: a \times_m b \in \Z'_m$


Modulo Multiplication on Reduced Residue System is Cancellable

Let $m \in \Z_{> 0}$ be a (strictly) positive integer. Let $\Z'_m$ be the reduced residue system modulo $m$: :$\Z'_m = \set {\eqclass k m \in \Z_m: k \perp m}$ Let $S = \struct {\Z'_m, \times_m}$ be the algebraic structure consisting of $\Z'_m$ under modulo multiplication. Then $\times_m$ is cancellable, in the sense that: :$\forall a, b, c \in \Z'_m: a \times_m c = b \times_m c \implies a = b$ and: :$\forall a, b, c \in \Z'_m: c \times_m a = c \times_m b \implies a = b$


Order Generating iff Every Element is Infimum

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Let $X$ be a subset of $S$. Then :$X$ is order generating {{iff}} :$\forall x \in S: \exists Y \subseteq X: x = \inf Y$


Weierstrass Factorization Theorem

Let $f$ be an entire function. Let $0$ be a zero of $f$ of multiplicity $m \ge 0$. Let the sequence $\sequence {a_n}$ consist of the nonzero zeroes of $f$, repeated according to multiplicity.


Equivalence of Axiom Schemata for Finite Group

The following axiom schemata for the definition of a finite group are logically equivalent:


Order Generating iff Every Superset Closed on Infima is Whole Space

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $X$ be a subset of $S$. Then :$X$ is order generating {{iff}} :$\forall Y \subseteq S: Y \supseteq X \land \left({\forall Z \subseteq Y: \inf Z \in Y}\right) \implies Y = S$


Group Isomorphism Preserves Inverses

Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group isomorphism. Let: :$e_G$ be the identity of $\struct {G, \circ}$ :$e_H$ be the identity of $\struct {H, *}$. Then: : $\forall g \in G: \map \phi {g^{-1} } = \paren {\map \phi g}^{-1}$


Preimage of Composite Relation

Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be relations. Let $\RR_2 \circ \RR_1 \subseteq S_1 \times T_2$ be the composition of $\RR_1$ and $\RR_2$. Then the preimage of $\RR_2 \circ \RR_1$ is given by: :$\Preimg {\RR_2 \circ \RR_1} = \Preimg {\Img {\RR_1} \cap \Preimg {\RR_2} }$


Image of Composite Relation

Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be relations. Let $\RR_2 \circ \RR_1 \subseteq S_1 \times T_2$ be the composition of $\RR_1$ and $\RR_2$. Then the image of $\RR_2 \circ \RR_1$ is given by: :$\Img {\RR_2 \circ \RR_1} = \Img {\Img {\RR_1} \cap \Preimg {\RR_2} }$


Weierstrass Product Theorem

Let $\sequence {a_k}$ be a sequence of non-zero complex numbers such that: :$\cmod {a_n} \to \infty$ as $n \to \infty$ Let $\sequence {p_n}$ be a sequence of non-negative integers for which the series: :$\displaystyle \sum_{n \mathop = 1}^\infty \size {\dfrac r {a_n} }^{1 + p_n}$ converges for every $r \in \R_{> 0}$. Let: :$\displaystyle \map f z = \prod_{n \mathop = 1}^\infty \map {E_{p_n} } {\frac z {a_n} }$ where $E_{p_n}$ are Weierstrass elementary factors. Then $f$ is entire and its zeroes are the points $a_n$, counted with multiplicity.


Isomorphism between Gaussian Integer Units and Rotation Matrices Order 4

Let $\struct {U_\C, \times}$ be the group of Gaussian integer units under complex multiplication. Let $\struct {R_4, \times}$ be the group of rotation matrices of order $4$ under modulo addition. Then $\struct {U_\C, \times}$ and $\struct {R_4, \times}$ are isomorphic algebraic structures.


Isomorphism between Gaussian Integer Units and Reduced Residue System Modulo 5 under Multiplication

Let $\struct {U_\C, \times}$ be the group of Gaussian integer units under complex multiplication. Let $\struct {\Z'_5, \times_5}$ be the multiplicative group of reduced residues modulo $5$. Then $\struct {U_\C, \times}$ and $\struct {\Z'_5, \times_5}$ are isomorphic algebraic structures.


Group Generated by Reciprocal of z and 1 minus z

Let $\struct {S, \circ}$ denote the '''group generated by $\dfrac 1 z$ and $1 - z$'''. Then $\struct {S, \circ}$ is a finite group of order $6$.


Group Generated by Reciprocal of z and Minus z is Klein Four-Group

Let $K_4$ denote the Klein $4$-group. Let $S$ be the group generated by $1 / z$ and $-z$. Then $K_4$ and $S$ are isomorphic algebraic structures.


Group of Reflection Matrices Order 4 is Klein Four-Group

Let $K_4$ denote the Klein $4$-group. Let $R_4$ be the Group of Reflection Matrices Order $4$. Then $K_4$ and $R_4$ are isomorphic algebraic structures.


Klein Four-Group and Group of Cyclic Group of Order 4 are not Isomorphic

The Klein $4$-group $K_4$ and the cyclic group of order $4$ $C_4$ are not isomorphic.


Order Generating iff Not Preceding implies There Exists Element Preceding and Not Preceding

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $X$ be a subset of $S$. Then :$X$ is order generating {{iff}} :$\forall x, y \in S: \left({ y \npreceq x \implies \exists p \in X: x \preceq p \land y \npreceq p}\right)$


Permutation Induces Equivalence Relation/Corollary

:$i \mathrel {\mathcal R_\pi} j$ {{iff}} $i$ and $j$ are in the same cycle of $\pi$.


Order of Group Element not less than Order of Power

Let $\left({G, \circ}\right)$ be a group whose identity is $e$. Let $g \in G$ be an element of $g$. Let $\left\lvert{g}\right\rvert$ denote the order of $g$ in $G$. Then: :$\forall m \in \Z: \left\lvert{g^m}\right\rvert \le \left\lvert{g}\right\rvert$ where $g^m$ denotes the $m$th power of $g$ in $G$.


Order of Group Element equals Order of Coprime Power

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $g \in G$ be an element of $g$. Let $\order g$ denote the order of $g$ in $G$. Then: :$\forall m \in \Z: \order {g^m} = \order g \iff m \perp \order g$ where: :$g^m$ denotes the $m$th power of $g$ in $G$ :$\perp$ denotes coprimality.


Unique Composition of Group Element whose Order is Product of Coprime Integers

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $g \in G$ be an element of $g$. Let: :$\order g = m n$ where: :$\order g$ denotes the order of $g$ in $G$ :$m$ and $n$ are coprime integers. Then $g$ can be expressed uniquely as the product of two commuting elements $a$ and $b$ of order $m$ and $n$ respectively.


Order of Cyclic Group equals Order of Generator

Let $G$ be a finite cyclic group which is generated by $a \in G$. Then: :$\order a = \order G$ where: :$\order a$ denotes the order of $a$ in $G$ :$\order G$ denotes the order of $G$.


Group whose Order equals Order of Element is Cyclic

Let $G$ be a finite group of order $n$. Let $g \in G$ have order $n$. Then $G$ is a cyclic group which is generated by $g$.


Integer Powers of 2 under Multiplication form Infinite Abelian Group

Let $S$ be the set of integers defined as: :$S = \set {2^k: k \in \Z}$ Then $\struct {S, \times}$ is an infinite abelian group.


Numbers of form 1 + 2m over 1 + 2n form Infinite Abelian Group under Multiplication

Let $S$ be the set of integers defined as: :$S = \set {\dfrac {1 + 2 m} {1 + 2 n}: m, n \in \Z}$ Then $\struct {S, \times}$ is an infinite abelian group.


Superset of Order Generating is Order Generating

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $X, Y$ be subsets of $S$ such that :$X$ is order generating and :$X \subseteq Y$ Then $Y$ is order generating.


Modulus 1 Rational Argument Complex Numbers under Multiplication form Infinite Abelian Group

Let $S$ be the set defined as: :$S = \set {\cos \theta + i \sin \theta: \theta \in \Q}$ Then the algebraic structure $\struct {S, \times}$ is an infinite abelian group.


Inversion Mapping is Automorphism iff Group is Abelian

Let $\struct {G, \circ}$ be a group. Let $\iota: G \to G$ be the inversion mapping on $G$, defined as: :$\forall g \in G: \map \iota g = g^{-1}$ Then $\iota$ is an automorphism {{iff}} $G$ is abelian.


Even Order Group has Odd Number of Order 2 Elements

Let $G$ be a group whose identity is $e$. Let $G$ be of even order. Then $G$ has an odd number of elements of order $2$.


Roots of Resolvent of Cubic

Let $P$ be the cubic equation: : $a x^3 + b x^2 + c x + d = 0$ with $a \ne 0$ Let $\map r P$ be the resolvent equation of $P$, given by: : $u^6 - 2 R u^3 - Q^3$ Let the roots of $P$ be $\alpha_1, \alpha_2, \alpha_3$. Then the roots of $\map r P$ can be expressed as: {{begin-eqn}} {{eqn | l = v       | r = \frac 1 3 \paren {\alpha_1 + \omega \alpha_2 + \omega^2 \alpha_3} }} {{eqn | l = \omega v       | r = \frac 1 3 \paren {\alpha_3 + \omega \alpha_1 + \omega^2 \alpha_2}       | c =  }} {{eqn | l = \omega^2 v       | r = \frac 1 3 \paren {\alpha_2 + \omega \alpha_3 + \omega^2 \alpha_1}       | c =  }} {{eqn | l = u       | r = \frac 1 3 \paren {\alpha_1 + \omega \alpha_3 + \omega^2 \alpha_2} }} {{eqn | l = \omega u       | r = \frac 1 3 \paren {\alpha_2 + \omega \alpha_1 + \omega^2 \alpha_3}       | c =  }} {{eqn | l = \omega^2 u       | r = \frac 1 3 \paren {\alpha_3 + \omega \alpha_2 + \omega^2 \alpha_1}       | c =  }} {{end-eqn}} where $\omega = -\dfrac {-1 + \sqrt {-3} } 2$ is one of the primitive (complex) cube roots of $1$.


Roots of Complex Number/Examples/Cube Roots

Let $z := \polar {r, \theta}$ be a complex number expressed in polar form, such that $z \ne 0$. Then the complex cube roots of $z$ are given by: :$z^{1 / 3} = \set {r^{1 / 3} \paren {\map \cos {\dfrac {\theta + 2 \pi k} 3} + i \, \map \sin {\dfrac {\theta + 2 \pi k} 3} }: k \in \set {0, 1, 2} }$ There are $3$ distinct such complex cube roots. These can also be expressed as: :$z^{1 / 3} = \set {r^{1 / 3} e^{i \paren {\theta + 2 \pi k} / 3}: k \in \set {0, 1, 2} }$ or: :$z^{1 / 3} = \set {r^{1 / 3} e^{i \theta / 3} \omega^k: k \in \set {0, 1, 2} }$ where $\omega = e^{2 i \pi / 3} = -\dfrac 1 2 + \dfrac {i \sqrt 3} 2$ is the first cube root of unity.


Roots of Complex Number/Corollary

Let $z := \polar {r, \theta}$ be a complex number expressed in polar form, such that $z \ne 0$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $w$ be one of the complex $n$th roots of $z$. Then the $n$th roots of $z$ are given by: :$z^{1 / n} = \set {w \epsilon^k: k \in \set {1, 2, \ldots, n - 1} }$ where $\epsilon$ is a primitive $n$th root of unity.


Top is Prime Element

Let $L = \left({S, \wedge, \preceq}\right)$ be a bounded above meet semilattice. Then $\top$ is a prime element where $\top$ denotes the greatest element of $L$.


Characterization of Prime Element in Meet Semilattice

Let $L = \struct {S, \wedge, \preceq}$ be a meet semilattice. Let $p \in S$, Then: :$p$ is prime element {{iff}}: :for all non-empty finite subsets $A$ of $S$: ::if $\inf A \preceq p$, then there exists element $x$ of $A$ such that $x \preceq p$.


Prime Element is Meet Irreducible

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Let $p \in S$. Let $p$ be a prime element of $L$. Then $p$ is meet irreducible in $L$.


Prime Element iff Meet Irreducible in Distributive Lattice

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a distributive lattice. Let $p \in S$. Then $p$ is a prime element {{iff}} $p$ is meet irreducible.


Multiple Equilibrium Points all have Equal Payoffs

Let $G$ be a two-person zero-sum game. Let $G$ have more than one equilibrium point. Then every equilibrium point of $G$ has the same payoff.


Definition:Information of Game

The '''information''' of a game is the knowledge each player has of the history of all the moves of all the players of the game.


Multigraph with Finite Vertex Set may not be Finite

Let $G$ be a multigraph. Let the vertex set of $G$ be finite. Then it is not necessarily the case that $G$ is also finite.


Modulus Larger than Real Part

:$\cmod z \ge \size {\map \Re z}$


Modulus Larger than Imaginary Part

:$\cmod z \ge \size {\map \Im z}$


Product Space is T0 iff Factor Spaces are T0/General Result

Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle T = \struct{S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$. Then $T$ is a $T_0$ (Kolmogorov) space {{iff}} each of $\struct{S_\alpha, \tau_\alpha}$ is a $T_0$ (Kolmogorov) space.


Separation Properties Preserved under Topological Product/Corollary

$T = \struct {S, \tau}$ has one of the following properties {{iff}} each of $\struct {S_i, \tau_i}$ has the same property: :Regular Property :Tychonoff (Completely Regular) Property If $T = \struct {S, \tau}$ has one of the following properties then each of $\struct {S_i, \tau_i}$ has the same property: :Normal Property :Completely Normal Property but the converse does not necessarily hold.


Quotient and Remainder to Number Base/General Result

:$\left \lfloor {\dfrac n {b^s}} \right \rfloor = \left[{r_m r_{m-1} \ldots r_{s+1} r_s}\right]_b$ :$\displaystyle n \,\bmod\, {b^s} = \sum_{j \mathop = 0}^{s-1} {r_j b^j} = \left[{r_{s-1} r_{s-2} \ldots r_1 r_0}\right]_b$


Remainder is Primitive Recursive

Let $m, n \in \N$ be natural numbers. Let us define the function $\operatorname{rem}: \N^2 \to \N$: :$\map \rem {n, m} = \begin{cases} \text{the remainder when } n \text{ is divided by } m & : m \ne 0 \\ 0 & : m = 0 \end{cases}$ where the $\text{remainder}$ is as defined in the Division Theorem: :If $n = m q + r$, where $0 \le r < m$, then $r$ is the remainder. Then $\rem$ is primitive recursive.


Quotient is Primitive Recursive

Let $m, n \in \N$ be natural numbers. Let us define the function $\operatorname {quot}: \N^2 \to \N$: :$\map {\operatorname {quot} } {n, m} = \begin{cases} \text{the quotient when } n \text{ is divided by } m & : m \ne 0 \\ 0 & : m = 0 \end{cases}$ where the $\text {quotient}$ and $\text {remainder}$ are as defined in the Division Theorem: :If $n = m q + r$, where $0 \le r < m$, then $q$ is the quotient. Then $\operatorname {quot}$ is primitive recursive.


Not Every Two-Person Zero-Sum Game has Saddle Point

Not every two-person zero-sum game has a saddle point.


Two-Person Zero-Sum Game with Multiple Solutions

There exists a two-person zero-sum game with more than one solution.


Two-Person Zero-Sum Game with Finite Strategies has Solution

Let $G$ be a two-person zero-sum game. Let each player of $G$ have a finite set of strategies available. Then $G$ has at least one solution.


Matching Pennies is Completely Mixed Game

The game of matching pennies is a completely mixed game.


Value of Skew-Symmetric Game is Zero

Let $G$ be a two-person zero-sum game. Let $G$ be represented by a payoff table that is skew-symmetric. Then the value of $G$ is zero.


Prime Element iff Complement of Lower Closure is Filter

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded above lattice. Let $p \in S$ such that :$p \ne \top$ where $\top$ denotes the top of $L$. Then: :$p$ is a prime element {{iff}} :$\relcomp S {p^\preceq}$ is filter in $L$ where :$p^\preceq$ denotes the lower closure of $p$ :$\relcomp S {p^\preceq}$ denotes the relative complement of $p^\preceq$ relative to $S$.


Oscillation at Point (Infimum) equals Oscillation at Point (Epsilon-Neighborhood)

Let $f: D \to \R$ be a real function where $D \subseteq \R$. Let $x$ be a point in $D$. Let $N_x$ be the set of open subset neighborhoods of $x$. Let $E_x$ be the set of $\epsilon$-neighborhoods of $x$. Let $\map {\omega_f} x$ be the oscillation of $f$ at $x$ based on $N_x$: :$\map {\omega_f} x = \inf \set {\map {\omega_f} I: I \in N_x}$ where $\map {\omega_f} I$ is the oscillation of $f$ on $I$: :$\map {\omega_f} I = \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$ Let $\map {\omega^E_f} x$ be the oscillation of $f$ at $x$ based on $E_x$: :$\map {\omega^E_f} x = \inf \set {\map {\omega_f} I: I \in E_x}$ Then: :$\map {\omega_f} x \in \R$ {{iff}} $\map {\omega^E_f} x \in \R$ and, if $\map {\omega_f} x$ and $\map {\omega^E_f} x$ exist as real numbers: :$\map {\omega_f} x = \map {\omega^E_f} x$


Prime Element iff There Exists Way Below Open Filter which Complement has Maximum

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous distributive lattice. Let $p \in S$ such that :$p \ne \top$ where $\top$ denotes the top of $L$. Then :$p$ is a prime element {{iff}} :there exists a way below open filter $F$ in $L$: $p = \max \left({ \complement_S\left({F}\right) }\right)$


Characterization of Prime Ideal

Let $L = \struct {S, \wedge, \preceq}$ be a meet semilattice. Let $I$ be an proper ideal in $L$. Then :$I$ is a prime ideal {{iff}} :$\forall x, y \in S: \paren {x \wedge y \in I \implies x \in I \lor y \in I}$


Matching Pennies is Strictly Competitive

The game of Matching Pennies is strictly competitive.


Second Price Auction has Inefficient Equilibria

A second price auction has Nash equilibria which are inefficient.


Expected-Utility Maximization Theorem

Let $P$ be a rational decision-maker. Let $A$ be a set of possible moves available to $P$. Then there exists a way of assigning values to a payoff function $u: A \to \R$ so that $P$ will always choose the move such that $u$ is a maximum.


Epic Equalizer is Isomorphism

Let $\mathbf C$ be a metacategory. Let $e: E \to C$ be the equalizer of two morphisms $f, g: C \to D$. Let $e$ be an epimorphism. Then $e$ is an isomorphism.


Infimum of Subset of Extended Real Numbers is Arbitrarily Close

Let $A \subseteq \overline \R$ be a subset of the extended real numbers. Let $b$ be an infimum (in $\R$) of $A$. Let $\epsilon \in \R_{>0}$. Then: :$\exists x \in \paren {A \cap \R}: x - b < \epsilon$


Oscillation on Set is an Extended Real Number

Let $f: D \to \R$ be a real function where $D \subseteq \R$. Let $x$ be a point in $D$. Let $I$ be a real set that contains (as an element) $x$. Let $\omega_f \left({I}\right)$ be the oscillation of $f$ on $I$: :$\omega_f \left({I}\right) = \sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ Then: :$\omega_f \left({I}\right) \in \overline \R_{\ge 0}$ and: :$\omega_f \left({I}\right) = \begin{cases} \text{a positive real number} & \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\} \text{is bounded above} \\ \infty & \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\} \text{is not bounded above} \end{cases}$


Oscillation on Subset

Let $f: D \to \R$ be a real function where $D \subseteq \R$. Let $x$ be a point in $D$. Let $S_x$ be a set of real sets that contain (as an element) $x$. Let $\map {\omega_f} I$ be the oscillation of $f$ on a set $I$ in $S_x$: :$\map {\omega_f} I = \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$ Let $I \in S_x$. Let $\map {\omega_f} I \in \R$. Let $J \in S_x$ be a subset of $I$. Then: :$\map {\omega_f} J \in \R$ :$\map {\omega_f} J \le \map {\omega_f} I$


Generating Fraction for Lucas Numbers

The fraction: :$\dfrac {199} {9899}$ has a decimal expansion which contains within it the start of the Lucas sequence: :$0 \cdotp 02010 \, 30407 \, 11 \ldots$


Transcendental Numbers are Uncountable

The set of transcendental real numbers is uncountable.


Almost All Real Numbers are Transcendental

Almost all real numbers are transcendental.


Champernowne Constant is Transcendental

The Champernowne constant: :$0 \cdotp 12345 \, 67891 \, 01112 \, 13141 \, 51617 \, 18192 \, 02122 \ldots$ is transcendental.


Champernowne Constant is Normal

The Champernowne constant: :$0 \cdotp 12345 \, 67891 \, 01112 \, 13141 \, 51617 \, 18192 \, 02122 \ldots$ is normal with respect to base $10$.


Anomalous Cancellation/Examples/3544 over 7531

:$\dfrac {344} {731} = \dfrac {3544} {7531} = \dfrac {35544} {75531} = \cdots$


Anomalous Cancellation/Examples/143 185 over 17 018 560

:$\dfrac {1435} {170 \, 560} = \dfrac {143 \, 185} {17 \, 018 \, 560} = \dfrac {14 \, 318 \, 185} {1 \, 701 \, 818 \, 560} = \cdots$


Number of Digits in Power of 2

Let $n$ be a positive integer. Expressed in conventional decimal notation, the number of digits in the $n$th power of $2$: :$2^n$ is equal to: :$\ceiling {n \log_{10} 2}$ where $\ceiling x$ denotes the ceiling of $x$.


One Third as Quotient of Sequences of Odd Numbers

:$\dfrac 1 3 = \dfrac {1 + 3} {5 + 7} = \dfrac {1 + 3 + 5} {7 + 9 + 11} = \dfrac {1 + 3 + 5 + 7} {9 + 11 + 13 + 15} = \cdots$


Euler's Number as Sum of Egyptian Fractions

The reciprocal of Euler's number $e$ can be approximated by the following sequence of Egyptian fractions: :$\dfrac 1 e = \dfrac 1 3 + \dfrac 1 {29} + \dfrac 1 {15 \, 786} + \dfrac 1 {513 \, 429 \, 610} + \cdots$ {{OEIS|A006526}}


One Half as Pandigital Fraction

There are $12$ ways $\dfrac 1 2$ can be expressed as a pandigital fraction: :$\dfrac 1 2 = \dfrac {6729} {13 \, 458}$ :$\dfrac 1 2 = \dfrac {6792} {13 \, 584}$ :$\dfrac 1 2 = \dfrac {6927} {13 \, 854}$ :$\dfrac 1 2 = \dfrac {7269} {14 \, 538}$ :$\dfrac 1 2 = \dfrac {7293} {14 \, 586}$ :$\dfrac 1 2 = \dfrac {7329} {14 \, 658}$ :$\dfrac 1 2 = \dfrac {7692} {15 \, 384}$ :$\dfrac 1 2 = \dfrac {7923} {15 \, 846}$ :$\dfrac 1 2 = \dfrac {7932} {15 \, 864}$ :$\dfrac 1 2 = \dfrac {9267} {18 \, 534}$ :$\dfrac 1 2 = \dfrac {9273} {18 \, 546}$ :$\dfrac 1 2 = \dfrac {9327} {18 \, 654}$


One Seventh as Pandigital Fraction

There are $7$ ways $\dfrac 1 7$ can be expressed as a pandigital fraction: :$\dfrac 1 2 = \dfrac {2394} {16 \, 758}$ :$\dfrac 1 2 = \dfrac {2637} {18 \, 459}$ :$\dfrac 1 2 = \dfrac {4527} {31 \, 689}$ :$\dfrac 1 2 = \dfrac {5274} {36 \, 918}$ :$\dfrac 1 2 = \dfrac {5418} {37 \, 926}$ :$\dfrac 1 2 = \dfrac {5976} {41 \, 832}$ :$\dfrac 1 2 = \dfrac {7614} {53 \, 298}$


One Third as Pandigital Fraction

There are $2$ ways $\dfrac 1 3$ can be expressed as a pandigital fraction: :$\dfrac 1 3 = \dfrac {5823} {17469}$ :$\dfrac 1 3 = \dfrac {5832} {17496}$


One Quarter as Pandigital Fraction

There are $4$ ways $\dfrac 1 4$ can be expressed as a pandigital fraction: :$\dfrac 1 4 = \dfrac {3942} {15768}$ :$\dfrac 1 4 = \dfrac {4392} {17568}$ :$\dfrac 1 4 = \dfrac {5796} {23184}$ :$\dfrac 1 4 = \dfrac {7956} {31824}$


One Fifth as Pandigital Fraction

There are $12$ ways $\dfrac 1 5$ can be expressed as a pandigital fraction: :$\dfrac 1 5 = \dfrac {2697} {13485}$ :$\dfrac 1 5 = \dfrac {2769} {13845}$ :$\dfrac 1 5 = \dfrac {2937} {14685}$ :$\dfrac 1 5 = \dfrac {2967} {14835}$ :$\dfrac 1 5 = \dfrac {2973} {14865}$ :$\dfrac 1 5 = \dfrac {3297} {16485}$ :$\dfrac 1 5 = \dfrac {3729} {18645}$ :$\dfrac 1 5 = \dfrac {6297} {31485}$ :$\dfrac 1 5 = \dfrac {7629} {38145}$ :$\dfrac 1 5 = \dfrac {9237} {46185}$ :$\dfrac 1 5 = \dfrac {9627} {48135}$ :$\dfrac 1 5 = \dfrac {9723} {48615}$


One Sixth as Pandigital Fraction

There are $3$ ways $\dfrac 1 6$ can be expressed as a pandigital fraction: :$\dfrac 1 6 = \dfrac {2943} {17658}$ :$\dfrac 1 6 = \dfrac {4653} {27918}$ :$\dfrac 1 6 = \dfrac {5697} {34182}$


One Eighth as Pandigital Fraction

There are $46$ ways $\dfrac 1 8$ can be expressed as a pandigital fraction: :$\dfrac 1 8 = \dfrac {3187} {25496}$ :$\dfrac 1 8 = \dfrac {4589} {36712}$ :$\dfrac 1 8 = \dfrac {4591} {36728}$ :$\dfrac 1 8 = \dfrac {4689} {37512}$ :$\dfrac 1 8 = \dfrac {4691} {37528}$ :$\dfrac 1 8 = \dfrac {4769} {38152}$ :$\dfrac 1 8 = \dfrac {5237} {41896}$ :$\dfrac 1 8 = \dfrac {5371} {42968}$ :$\dfrac 1 8 = \dfrac {5789} {46312}$ :$\dfrac 1 8 = \dfrac {5791} {46328}$ :$\dfrac 1 8 = \dfrac {5839} {46712}$ :$\dfrac 1 8 = \dfrac {5892} {47136}$ :$\dfrac 1 8 = \dfrac {5916} {47328}$ :$\dfrac 1 8 = \dfrac {5921} {47368}$ :$\dfrac 1 8 = \dfrac {6479} {51832}$ :$\dfrac 1 8 = \dfrac {6741} {53928}$ :$\dfrac 1 8 = \dfrac {6789} {54312}$ :$\dfrac 1 8 = \dfrac {6791} {54328}$ :$\dfrac 1 8 = \dfrac {6839} {54712}$ :$\dfrac 1 8 = \dfrac {7123} {56984}$ :$\dfrac 1 8 = \dfrac {7312} {58496}$ :$\dfrac 1 8 = \dfrac {7364} {58912}$ :$\dfrac 1 8 = \dfrac {7416} {59328}$ :$\dfrac 1 8 = \dfrac {7421} {59368}$ :$\dfrac 1 8 = \dfrac {7894} {63152}$ :$\dfrac 1 8 = \dfrac {7941} {63528}$ :$\dfrac 1 8 = \dfrac {8174} {65392}$ :$\dfrac 1 8 = \dfrac {8179} {65432}$ :$\dfrac 1 8 = \dfrac {8394} {67152}$ :$\dfrac 1 8 = \dfrac {8419} {67352}$ :$\dfrac 1 8 = \dfrac {8439} {67512}$ :$\dfrac 1 8 = \dfrac {8932} {71456}$ :$\dfrac 1 8 = \dfrac {8942} {71536}$ :$\dfrac 1 8 = \dfrac {8953} {71624}$ :$\dfrac 1 8 = \dfrac {8954} {71632}$ :$\dfrac 1 8 = \dfrac {9156} {73248}$ :$\dfrac 1 8 = \dfrac {9158} {73264}$ :$\dfrac 1 8 = \dfrac {9182} {73456}$ :$\dfrac 1 8 = \dfrac {9316} {74528}$ :$\dfrac 1 8 = \dfrac {9321} {74568}$ :$\dfrac 1 8 = \dfrac {9352} {74816}$ :$\dfrac 1 8 = \dfrac {9416} {75328}$ :$\dfrac 1 8 = \dfrac {9421} {75368}$ :$\dfrac 1 8 = \dfrac {9523} {76184}$ :$\dfrac 1 8 = \dfrac {9531} {76248}$ :$\dfrac 1 8 = \dfrac {9541} {76328}$


One Ninth as Pandigital Fraction

There are $3$ ways $\dfrac 1 9$ can be expressed as a pandigital fraction: :$\dfrac 1 9 = \dfrac {6381} {57429}$ :$\dfrac 1 9 = \dfrac {6471} {58239}$ :$\dfrac 1 9 = \dfrac {8361} {75249}$


Four Fifths as Pandigital Fraction

The fraction $\dfrac 4 5$ can be expressed as a pandigital fraction in the following interesting way: :$\dfrac 4 5 = \dfrac {9876} {12 \, 345}$


Probability of Two Random Integers having no Common Divisor

Let $a$ and $b$ be integers chosen at random. The probability that $a$ and $b$ are coprime is given by: :$\map \Pr {a \perp b} = \dfrac 1 {\map \zeta 2} = \dfrac 6 {\pi^2}$ where $\zeta$ denotes the zeta function. The decimal expansion of $\dfrac 1 {\map \zeta 2}$ starts: :$\dfrac 1 {\map \zeta 2} = 0 \cdotp 60792 \, 71018 \, 54026 \, 6 \ldots$ {{OEIS|A059956}}


Probability of Random Integer being Square-Free

Let $a$ be an integer chosen at random. The probability that $a$ is square-free is given by: :$\map \Pr {\neg \exists b \in \Z: b^2 \divides a} = \dfrac 1 {\map \zeta 2} = \dfrac 6 {\pi^2}$ where $\zeta$ denotes the zeta function. The decimal expansion of $\dfrac 1 {\map \zeta 2}$ starts: :$\dfrac 1 {\map \zeta 2} = 0 \cdotp 60792 \, 71018 \, 54026 \, 6 \ldots$ {{OEIS|A059956}}


Kepler's Conjecture

The densest packing of identical spheres in space is obtained when the spheres are arranged with their centers at the points of a face-centered cubic lattice. This obtains a density of $\dfrac \pi {3 \sqrt 2} = \dfrac \pi {\sqrt {18} }$: :$\dfrac \pi {\sqrt {18} } = 0 \cdotp 74048 \ldots$


Prime Ideal is Prime Element

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a lattice. Let $I$ be an ideal in $L$. Then: :$I$ is a prime ideal {{iff}}: :$I$ is a prime element in $\struct {\map {\mathit {Ids} } L, \precsim}$ where: :$\map {\mathit {Ids} } L$ denotes the set of all ideals in $L$ :$\mathord \precsim := \mathord \subseteq \restriction_{\map {\mathit {Ids} } L \times \map {\mathit {Ids} } L}$


Probability of Three Random Integers having no Common Divisor

Let $a, b$ and $c$ be integers chosen at random. The probability that $a, b$ and $c$ have no common divisor: :$\map \Pr {\map \perp {a, b, c} } = \dfrac 1 {\map \zeta 3}$ where $\zeta$ denotes the zeta function: :$\map \zeta 3 = \dfrac 1 {1^3} + \dfrac 1 {2^3} + \dfrac 1 {3^3} + \dfrac 1 {4^3} + \cdots$ The decimal expansion of $\dfrac 1 {\map \zeta 3}$ starts: :$\dfrac 1 {\map \zeta 3} = 0 \cdotp 83190 \, 73725 \, 80707 \ldots$ {{OEIS|A088453}}


Densest Packing of Identical Circles

The densest packing of identical circles in the plane obtains a density of $\dfrac \pi {2 \sqrt 3} = \dfrac \pi {\sqrt {12} }$: :$\dfrac \pi {2 \sqrt 3} = 0 \cdotp 90689 \, 96821 \ldots$ {{OEIS|A093766}} This happens when they are packed together in a hexagonal array, with each circle touching $6$ others.


Meet is Intersection in Set of Ideals

Let $\mathscr S = \struct {S, \wedge, \preceq}$ be a meet semilattice. Let $\map {\mathit {Ids} } {\mathscr S}$ be the set of all ideals in $\mathscr S$. Let $P = \struct {\map {\mathit {Ids} } {\mathscr S}, \precsim}$ be an ordered set where $\mathord \precsim = \mathord \subseteq \restriction_{\map {\mathit {Ids} } {\mathscr S} \times \map {\mathit {Ids} } {\mathscr S} }$ Let $I_1, I_2$ be ideals in $\mathscr S$. Then :$I_1 \wedge_P I_2 = I_1 \cap I_2$


Characterization of Prime Ideal by Finite Infima

Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $I$ be an ideal in $L$. Then :$I$ is a prime ideal {{iff}} :for all non-empty finite subset $A$ of $S: \left({ \inf A \in I \implies \exists a \in A: a \in I}\right)$


Doubling the Cube/Archytas Curve

The Archytas curve can be used for Doubling the Cube.


Doubling the Cube/Intersection of Parabolas

The problem of Doubling the Cube can be solved by finding the intersection of two parabolas.


Doubling the Cube/Intersection of Parabola and Hyperbola

The problem of Doubling the Cube can be solved by finding the intersection of a parabola and a hyperbola.


Unordered Pair is Finite

Let $x, y$ be arbitrary. Then $\left\{ {x, y}\right\}$ is finite.


Pair is Union of Singletons

Let $x, y$ be arbitrary. {{questionable|Need to state that $x \ne y$.}} Then $\left\{ {x, y}\right\} = \left\{ {x}\right\} \cup \left\{ {y}\right\}$


Doubling the Cube/Conchoid of Nicomedes

The problem of Doubling the Cube can be solved by using a conchoid of Nicomedes.


Doubling the Cube/Cissoid of Diocles

The problem of Doubling the Cube can be solved by using a cissoid of Diocles.


Length of Diagonal of Unit Square

The length of a diagonal of a square of side length $1$ is $\sqrt 2$) (the square root of $2$).


Sequence of Best Rational Approximations to Square Root of 2

A sequence of best rational approximations to the square root of $2$ starts: :$\dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$ where: :the numerators are half of the Pell-Lucas numbers, $\dfrac 1 2 Q_n$ :the denominators are the Pell numbers $P_n$ starting from $\dfrac {\tfrac12 Q_1} {P_1}$. {{OEIS-Numerators|A001333}} {{OEIS-Denominators|A000129}}


Difference between Adjacent Convergents of Simple Continued Fraction

Then for $k \ge 1$: :$p_k q_{k - 1} - p_{k - 1} q_k = \paren {-1}^{k + 1}$ That is: :$C_k - C_{k - 1} = \dfrac {p_k} {q_k} - \dfrac {p_{k - 1} } {q_{k - 1} } = \dfrac {\paren {-1}^{k + 1} } {q_k q_{k - 1} }$


Difference between Adjacent Convergents But One of Simple Continued Fraction

For $k \ge 2$: :$p_k q_{k - 2} - p_{k - 2} q_k = \paren {-1}^k a_k$ That is: :$C_k - C_{k-2} = \dfrac {p_k} {q_k} - \dfrac {p_{k - 2} } {q_{k - 2} } = \dfrac {\paren {-1}^k a_k} {q_k q_{k - 2} }$


Convergents of Simple Continued Fraction are Rationals in Canonical Form

For all $k \ge 1$, $\dfrac {p_k} {q_k}$ is in canonical form: :$p_k$ and $q_k$ are coprime :$q_k > 0$.


Dual of Preordered Set is Preordered Set

Let $P = \left({S, \preceq}\right)$ be a preordered set. Then dual of $P$, $P^{-1} = \left({S, \succeq}\right)$ is also a preordered set. {{finish|Dual Ordered Set $\ne$ Dual Preordered Set}}


Dual Ordered Set is Ordered Set

Let $P = \left({S, \preceq}\right)$ be an ordered set. Then its dual, $P^{-1} = \left({S, \succeq}\right)$, is also an ordered set.


Value of Finite Continued Fraction equals Numerator Divided by Denominator

Let $F$ be a field. Let $\tuple {a_0, a_1, \ldots, a_n}$ be a finite continued fraction of length $n \ge 0$. Let $p_n$ and $q_n$ be its $n$th numerator and denominator. Then the value $\sqbrk {a_0, a_1, \ldots, a_n}$ equals $\dfrac {p_n} {q_n}$.


Simple Infinite Continued Fraction Converges

Let $C = (a_0, a_1, \ldots)$ be a simple infinite continued fraction in $\R$. Then $C$ converges.


Continued Fraction Identities/First/Infinite

Let $\left[{a_0, a_1, a_2, \ldots}\right]$ be a simple infinite continued fraction. Then: :$\left[{a_1, a_2, a_3, \ldots}\right] = a_1 + \dfrac 1 {\left[{a_2, a_3, \ldots}\right]}$


Relation between Adjacent Best Rational Approximations to Root 2

Consider the Sequence of Best Rational Approximations to Square Root of 2: :$\sequence S := \dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$ Let $\dfrac {p_n} {q_n}$ and $\dfrac {p_{n + 1} } {q_{n + 1} }$ be adjacent terms of $\sequence S$. Then: :$\dfrac {p_{n + 1} } {q_{n + 1} } = \dfrac {p_n + 2 q_n} {p_n + q_n}$


Parity of Pell Numbers

Consider the Pell numbers $P_0, P_1, P_2, \ldots$ :$0, 1, 2, 5, 12, 29, \ldots$ $P_n$ has the same parity as $n$. That is: :if $n$ is odd then $P_n$ is odd :if $n$ is even then $P_n$ is even.


Parity of Best Rational Approximations to Root 2

Consider the Sequence of Best Rational Approximations to Square Root of 2: :$\sequence S := \dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$ where $S_1 := \dfrac 1 1$. The numerators of the terms of $\sequence S$ are all odd. For all $n$, the parity of the denominator of term $S_n$ is the same as the parity of $n$.


Prime Ideal is Prime Filter in Dual Lattice

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a lattice. Let $X$ be a subset of $S$. Then :$X$ is a prime ideal in $L$ {{iff}}: :$X$ is a prime filter in $L^{-1}$ where $L^{-1} = \struct {S, \succeq}$ denotes the dual of $L$.


Ideal is Filter in Dual Ordered Set

Let $P = \left({S, \preceq}\right)$ be an ordered set. Let $X$ be a subset of $S$. Then :$X$ is ideal in $P$ {{iff}} :$X$ is filter in $P^{-1}$ where $P^{-1} = \left({S, \succeq}\right)$ denotes the dual of $P$.


Square Root of 2 as Sum of Egyptian Fractions

The square root of $2$ can be approximated by the following sequence of Egyptian fractions: :$\sqrt 2 = 1 + \dfrac 1 3 + \dfrac 1 {253} + \dfrac 1 {218 \, 201} + \dfrac 1 {61 \, 323 \, 543 \, 802} + \cdots$ {{OEIS|A006487}}


Sprague's Property of Root 2

Let $S = \sequence {s_n}$ be the sequence of fractions defined as follows: Let the numerator of $s_n$ be: :$\floor {n \sqrt 2}$ where $\floor x$ denotes the floor of $x$. Let the denominators of the terms of $S$ be the (strictly) positive integers missing from the numerators of $S$: :$S := \dfrac 1 3, \dfrac 2 6, \dfrac 4 {10}, \dfrac 5 {13}, \dfrac 7 {17}, \dfrac 8 {20}, \ldots$ Then the difference between the numerator and denominator of $s_n$ is equal to $2 n$.


Steiner's Calculus Problem

Let $f: \R_{>0} \to \R$ be the real function defined as: :$\forall x \in \R_{>0}: \map f x = x^{1/x}$ Then $\map f x$ reaches its maximum at $x = e$ where $e$ is Euler's number .


Ratio of Lengths of Arms of Pentagram

Consider a pentagram. :400px Let $AC$ be the length of one of the lines which span the pentagram and define it. Let $B$ be one of the points where $AC$ intersects one of the other such lines such that $AB > AC$. Then: :$\dfrac {AC} {AB} = \phi$ where $\phi$ denotes the golden mean.


Filter is Ideal in Dual Ordered Set

Let $P = \left({S, \preceq}\right)$ be an ordered set. Let $X$ be a subset of $S$. Then :$X$ is filter in $P$ {{iff}} :$X$ is ideal in $P^{-1}$ where $P^{-1} = \left({S, \succeq}\right)$ denotes the dual of $P$.


Prime Filter is Prime Ideal in Dual Lattice

Let $L = \struct {S, \preceq}$ be a lattice. Let $X$ be a subset of $S$. Then :$X$ is a prime filter in $L$ {{iff}}: :$X$ is a prime ideal in $L^{-1}$ where $L^{-1} = \struct {S, \succeq}$ denotes the dual of $L$.


Trefoil Knot in Paper forms Pentagon

Take a strip of paper with parallel edges. Tie a simple trefoil knot in it. Flatten the knot and pull the paper tight. The knot will be in the shape of a regular pentagon. The edges of the paper strip trace out a pentagram. :500px


Sequence of Golden Rectangles

A golden rectangle can be divided into a square and another golden rectangle. That golden rectangle can in turn be divided into a square and another golden rectangle. The sequence can be continued indefinitely. :700px


Sequence of Golden Rectangles/Equiangular Spiral

The points where the vertices of successive squares of this sequence meet can be joined together by an equiangular spiral. This equiangular spiral can be approximated by quarter circles constructed as shown. The equiangular spiral tends towards the point of intersection of the diagonals of the golden rectangles.


Continued Fraction Expansion of Golden Mean/Rate of Convergence

This continued fraction expansion has the slowest rate of convergence of all simple infinite continued fractions.


Continued Fraction Expansion of Golden Mean/Successive Convergents

The $n$th convergent is given by: :$C_n = \dfrac {F_{n + 1} } {F_n}$ where $F_n$ denotes the $n$th Fibonacci number.


Continued Fraction Expansion of Pi

The constant $\pi$ (pi) has the continued fraction expansion: :$\pi = \sqbrk {3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, \ldots}$


Continued Fraction Expansion of Pi/Convergents

The convergents of the continued fraction expansion to $\pi$ (pi) are: :$3, \dfrac {22} 7, \dfrac {333} {106}, \dfrac {355} {113}, \dfrac {103993} {33102}, \dfrac {104348} {33215}$ {{OEIS-Numerators|A002485}} {{OEIS-Denominators|A002486}} These best rational approximations are accurate to $0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, \ldots$ decimals. {{OEIS|A114526}} === Zu Chongzhi Fraction === {{:Definition:Zu Chongzhi Fraction}}


Filter is Prime iff For Every Element Element either Negation Belongs to Filter in Boolean Lattice

Let $B = \struct {S, \vee, \wedge, \neg, \preceq}$ be a Boolean lattice. Let $F$ be a filter in $B$. Then :$F$ is prime {{iff}} :$\forall x \in S: x \in F \lor \left({\neg x}\right) \in F$


Beatty's Theorem

Let $r, s \in \R \setminus \Q$ be an irrational number such that $r > 1$ and $s > 1$. Let $\BB_r$ and $\BB_s$ be the Beatty sequences on $r$ and $s$ respectively. Then $\BB_r$ and $\BB_s$ are complementary Beatty sequences {{iff}}: :$\dfrac 1 r + \dfrac 1 s = 1$


Proper and Prime iff Ultrafilter in Boolean Lattice

Let $B = \left({S, \vee, \wedge, \neg, \preceq}\right)$ be a Boolean lattice. Let $F$ be a filter in $B$. Then :$F$ is a proper subset of $S$ and $F$ is a prime filter in $B$ {{iff}}: :$F$ is ultrafilter on $B$


Difference between Terms of Wythoff Pair

Let $\tuple {\floor {k \phi}, \floor {k \phi^2} }$ be a Wythoff pair. The difference between the coordinates of this Wythoff pair is $k$. That is: :$\floor {k \phi^2} - \floor {k \phi} = k$


Finite Infima Set and Upper Closure is Smallest Filter

Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $X$ be a non-empty subset of $S$. Then :$X \subseteq {\operatorname{fininfs}\left({X}\right)}^\succeq$ and :for every a filter $F$ in $L$: $\left({X \subseteq F \implies {\operatorname{fininfs}\left({X}\right)}^\succeq \subseteq F}\right)$ where : $\operatorname{fininfs}\left({X}\right)$ denotes the finite infima set of $X$, : $X^\succeq$ denotes the upper closure of $X$.


Smaller Number of Wythoff Pair is Smallest Number not yet in Sequence

Consider the sequence of Wythoff pairs arranged in sequential order: :$\tuple {0, 0}, \tuple {1, 2}, \tuple {3, 5}, \tuple {4, 7}, \tuple {6, 10}, \tuple {8, 13}, \ldots$ The first coordinate of each Wythoff pair is the smallest positive integer which has not yet appeared in the sequence.


Sequence of Wythoff Pairs contains all Positive Integers exactly Once Each

Consider the sequence of Wythoff pairs arranged in sequential order: :$\tuple {0, 0}, \tuple {1, 2}, \tuple {3, 5}, \tuple {4, 7}, \tuple {6, 10}, \tuple {8, 13}, \ldots$ Apart from the first Wythoff pair $\tuple {0, 0}$, every positive integer appears in this sequence exactly once.


Wythoff Pairs are Winning Positions in Wythoff's Game

The Wythoff pairs are the winning positions in Wythoff's game.


Zeta of 2 as Product of Fractions with Prime Numerators

{{begin-eqn}} {{eqn | l = \map \zeta 2       | r = \prod_p \paren {\frac p {p - 1} } \paren {\frac p {p + 1} }       | c =  }} {{eqn | r = \dfrac 2 1 \times \dfrac 2 3 \times \dfrac 3 2 \times \dfrac 3 4 \times \dfrac 5 4 \times \dfrac 5 6 \times \dfrac 7 6 \times \dfrac 7 8 \times \dfrac {11} {10} \times \dfrac {11} {12} \times \dfrac {13} {12} \times \dfrac {13} {14} \times \cdots       | c =  }} {{end-eqn}} where: : $\zeta$ denotes the Riemann zeta function : $\displaystyle \prod_p$ denotes the product over all prime numbers.


Snub Cube Inscribed in Octahedron

A snub cube inscribed in an octahedron involves the Tribonacci constant. {{explain|a bit more detail could help here}}


Snub Cube Inscribed in Cube

A snub cube inscribed in a cube involves the Tribonacci constant. {{explain|a bit more detail could help here}}


Icosahedron Inscribed in Octahedron

An icosahedron inscribed in an octahedron involves the Golden Ratio. {{stub|a bit more detail could help here}}


Top in Filter

Let $\left({S, \preceq}\right)$ be a bounded above ordered set. Let $F$ be a filter on $S$. Then $\top \in F$ where $\top$ denotes the greatest element of $S$.


Finite Infima Set and Upper Closure is Filter

Let $P = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $X$ be a non-empty subset of $S$. Then :${\operatorname{fininfs}\left({X}\right)}^\succeq$ is filter in $P$. where :$\operatorname{fininfs}\left({X}\right)$ denotes the finite infima set of $X$, :$X^\succeq$ denotes the upper closure of $X$.


Brun's Theorem

The sum of the reciprocals of all the twin primes converges.


Divisibility by 2

An integer $N$ expressed in decimal notation is divisible by $2$ {{iff}} the least significant digit of $N$ is divisible by $2$. That is: :$N = [a_n \ldots a_2 a_1 a_0]_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $2$  {{iff}}: :$a_0$ is divisible by $2$.


Divisibility by Power of 2

Let $r \in \Z_{\ge 1}$ be a strictly positive integer. An integer $N$ expressed in decimal notation is divisible by $2^r$ {{iff}} the last $r$ digits of $N$ form an integer divisible by $2^r$. That is: :$N = [a_n \ldots a_2 a_1 a_0]_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $2^r$  {{iff}}: :$a_0 + a_1 10 + a_2 10^2 + \cdots + a_r 10^r$ is divisible by $2^r$.


Equivalence of Definitions of Deficient Number

The following definitions of a deficient number are equivalent:


Power of Prime is Deficient

Let $n \in \Z_{>0}$ be a power of a prime number $p$: :$n = p^k$ for some $k \in \Z_{>0}$. Then $n$ is deficient.


Set is Subset of Finite Infima Set

Let $\left({S, \preceq}\right)$ be an ordered set. Let $X$ be a subset of $S$. Then $X \subseteq \operatorname{fininfs}\left({X}\right)$ where $\operatorname{fininfs}\left({X}\right)$ denotes the finite infima set of $X$.


Set is Subset of Upper Closure

Let $\left({S, \preceq}\right)$ be an ordered set. Let $X$ be a subset of $S$. Then $X \subseteq X^\succeq$ where $X^\succeq$ denotes the upper closure of $X$.


Tamref's Last Theorem

The Diophantine equation: :$n^x + n^y = n^z$ has exactly one form of solutions in integers: :$2^x + 2^x = 2^{x + 1}$ for all $x \in \Z$.


3^x + 4^y equals 5^z has Unique Solution

The Diophantine equation: :$3^x + 4^y = 5^z$ has exactly one solution in integers: :$3^2 + 4^2 = 5^2$


5^x + 12^y equals 13^z has Unique Solution

The Diophantine equation: :$5^x + 12^y = 13^z$ has exactly one solution in integers: :$5^2 + 12^2 = 13^2$


Upper Closure of Subset is Subset of Upper Closure

Let $\left({S, \preceq}\right)$ be an ordered set. Let $X, Y$ be subsets of $S$. Then :$X \subseteq Y \implies X^\succeq \subseteq Y^\succeq$ where $X^\succeq$ denotes the upper closure of $X$.


Euler's Number as Limit of n over nth Root of n Factorial

:$\displaystyle e = \lim_{n \mathop \to \infty} \dfrac n {\sqrt [n] {n!} }$ where: :$e$ denotes Euler's number :$n!$ denotes $n$ factorial.


Continued Fraction Expansion of Euler's Number

The constant Euler's number $e$ has the continued fraction expansion: {{begin-eqn}} {{eqn | l = e       | r = \sqbrk {2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \ldots }       | c = }} {{eqn | l =        | r = \sqbrk {1, 0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \ldots }       | c = }} {{end-eqn}}


Continued Fraction Expansion of Euler's Number/Convergents

The convergents of the continued fraction expansion to Euler's number $e$ are: :$2, 3, \dfrac 8 3, \dfrac {11} 4, \dfrac {19} 7, \dfrac {87} {32}, \dfrac {106} {39}, \dfrac {193} {71}, \dfrac {1264} {465}, \dfrac {1457} {536}, \dfrac {2721} {1001}, \ldots$ {{OEIS-Numerators|A007676}} {{OEIS-Denominators|A007677}} These best rational approximations are accurate to $0, 0, 1, 1, 2, 3, 3, 4, 5, 5, \ldots$ decimals. {{OEIS|A114539}} The fraction $\dfrac {878} {323}$ is exceptionally easy to remember: :$\dfrac {878} {323} = 2 \cdotp 71826 \, 625 \ldots$ although this does not occur in the above continued fraction expansion.


Number of Random Fractional Reals whose Total Exceeds 1

Let real numbers be selected at random following a continuous uniform distribution from the interval $\closedint 0 1$ until their total sum is greater than $1$. The expectation of the number of selections is Euler's number $e$.


Sine of X over X as Infinite Product

Let $z$ be a non-zero Complex Number. Then: :$\displaystyle \frac {\sin z} z = \cos \frac z 2 \cos \frac z 4 \cos \frac z 8 \cdots = \prod_{i \mathop = 1}^{\infty} \cos \frac z {2^i}$ where $\sin$ denotes the sine function and $\cos$ denotes the cosine function.


Maximum of Three Mutually Perpendicular Lines in Ordinary Space

In ordinary space, there can be no more than $3$ straight lines which are pairwise perpendicular. Thus, in a configuration of $4$ straight lines in space, at least one pair will not be perpendicular to each other.


Complex Logarithm of 1

:$\ln 1 = \set {2 k \pi i: k \in \Z}$


Sum Rule for Complex Derivatives

Let $\map f z, \map j z, \map k z$ be single-valued continuous complex functions in a domain $D \subseteq \C$, where $D$ is open. Let $f$, $j$, and $k$ be complex-differentiable at all points in $D$. Let $\map f z = \map j z + \map k z$. Then: : $\forall z \in D: \map {f'} z = \map {j'} z + \map {k'} z$


Product Rule for Complex Derivatives

Let $\map f z, \map j z, \map k z$ be single-valued continuous complex functions in a domain $D \subseteq \C$, where $D$ is open. Let $f$, $j$, and $k$ be complex-differentiable at all points in $D$. Let $\map f z = \map j z \, \map k z$. Then: :$\forall z \in D: \map {f'} z = \map j z \, \map {k'} z + \map {j'} z \, \map k z$


Derivative of Constant/Complex

Let $f_c \left({z}\right)$ be the constant function on an open domain $D \in \C$, where $c \in \C$. Then: :$\forall z \in D : f_c' \left({z}\right) = 0$


Derivative of Identity Function/Real

Let $I_\R: \R \to \R$ be the identity mapping on the real numbers $\R$. Then: :$\map {I_\R'} x = 1$


Derivative of Identity Function/Complex

Let $I_\C: \C \to \C$ be the identity function. Then: :$\map {I_\C'} z = 1$


Chain Rule for Real-Valued Functions/Corollary

Let $\Psi$ represent a differentiable function of $x$ and $y$. Let $y$ represent a differentiable function of $x$. Then: {{begin-eqn}} {{eqn | l = \frac {\d \Psi} {\d x}       | r = \frac {\partial \Psi} {\partial x} + \frac {\partial \Psi} {\partial y} \frac {\d y} {\d x} }} {{end-eqn}}


Trisecting the Angle/Hyperbola

Let $\alpha$ be an angle which is to be trisected. This can be achieved by means of a hyperbola.


Trisecting the Angle/Quadratrix of Hippias

Let $\alpha$ be an angle which is to be trisected. This can be achieved by means of a quadratrix of Hippias.


Trisecting the Angle/Cissoid of Diocles

Let $\alpha$ be an angle which is to be trisected. This can be achieved by means of a cissoid of Diocles.


Trisecting the Angle/Conchoid of Nicomedes

Let $\alpha$ be an angle which is to be trisected. This can be achieved by means of a conchoid of Nicomedes.


Laplace Transform of Periodic Function

Let $f$ be periodic, that is: :$\exists T \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + T}$ Then: :$\laptrans {\map f t} = \dfrac 1 {1 - e^{-s T} } \displaystyle \int_0^T e^{-s t} \map f t \rd t$ where $\laptrans {\map f t}$ denotes the Laplace transform.


Triangle is Medial Triangle of Larger Triangle

Let $\triangle ABC$ be a triangle. $\triangle ABC$ is the medial triangle of a larger triangle.


Altitudes of Triangle Meet at Point

Let $\triangle ABC$ be a triangle. The altitudes of $\triangle ABC$ all intersect at the same point.


Perpendicular Bisectors of Triangle Meet at Point

Let $\triangle ABC$ be a triangle. The perpendicular bisectors of $AB$, $BC$ and $AC$ all intersect at the same point.


Centroid of Triangle is Centroid of Medial

Let $\triangle ABC$ be a triangle. Let $\triangle DEF$ be the medial triangle of $\triangle ABC$. Let $G$ be the centroid of $\triangle ABC$. Then $G$ is also the centroid of $\triangle DEF$.


Diameters of Parallelogram Bisect each other

Let $\Box ABCD$ be a parallelogram with diameters $AC$ and $BD$. Let $AC$ and $BD$ intersect at $E$. Then $E$ is the midpoint of both $AC$ and $BD$.


Position of Centroid on Euler Line

Let $\triangle ABC$ be a triangle which is not equilateral. Let $O$ be the circumcenter of $\triangle ABC$. Let $G$ be the centroid of $\triangle ABC$. Let $H$ be the orthocenter of $\triangle ABC$. Then $G$ lies on the straight line connecting $O$ and $H$ such that: :$OG : GH = 1 : 2$ The line $OGH$ is the '''Euler line''' of $\triangle ABC$.


Perpendicular Bisector of Triangle is Altitude of Medial Triangle

Let $\triangle ABC$ be a triangle. Let $\triangle DEF$ be the medial triangle of $\triangle ABC$. Let a perpendicular bisector be constructed on $AC$ at $F$ to intersect $DE$ at $P$. Then $FP$ is an altitude of $\triangle DEF$.


Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral

Let $\triangle ABC$ be a triangle. Let $O$ be the circumcenter of $\triangle ABC$. Let $G$ be the centroid of $\triangle ABC$. Let $H$ be the orthocenter of $\triangle ABC$. Then $O$, $G$ and $H$ are the same points {{iff}} $\triangle ABC$ is equilateral. If $\triangle ABC$ is not equilateral, then $O$, $G$ and $H$ are all distinct.


Altitude, Median and Perpendicular Bisector Coincide iff Triangle is Isosceles

Let $\triangle ABC$ be a triangle. Then: ::the altitude from $AB$ to $C$ ::the median from $AB$ to $C$ ::the perpendicular bisector of $AB$ :are all the same straight line {{iff}}: :$\triangle ABC$ is isosceles where $AB$ is the base.


Three Points Describe a Circle

Let $A$, $B$ and $C$ be points which are not collinear. Then there exists exactly one circle whose circumference passes through all $3$ points $A$, $B$ and $C$.


Three Regular Tessellations

There exist exactly $3$ regular tessellations of the plane.


Integer is Sum of Three Triangular Numbers

Let $n$ be a positive integer. Then $n$ is the sum of $3$ triangular numbers.


Fermat's Last Theorem/Cubic

The Diophantine equation $a^3 + b^3 = c^3$ has no solutions in strictly positive integers.


Smallest Magic Square is of Order 3

Apart from the trivial order $1$ magic square: {{:Magic Square/Examples/Order 1}} the smallest magic square is the order $3$ magic square: {{:Magic Square/Examples/Order 3}}


Euler Phi Function of n equal to Euler Phi Function of n+3

Let $\phi$ denote the Euler $\phi$ function. The only solutions to the equation: :$\phi \left({n}\right) = \phi \left({n + 3}\right)$ less than $1 \, 000 \, 000$ are: :$\phi \left({3}\right) = \phi \left({6}\right) = 2$ :$\phi \left({5}\right) = \phi \left({8}\right) = 4$


Friendship Theorem

Let there be a group of $6$ people. The traditional setting is that these $6$ people are at a party. Then (at least) one of the following $2$ statements is true: :$(1): \quad$ At least $3$ of these $6$ people have all met each other before :$(2): \quad$ At least $3$ of these $6$ people have never met each other before. That is, either there exists a set of $3$ mutual acquaintances, or there exists a set of $3$ mutual strangers.


Volume of Smallest Tetrahedron with Integer Edges and Integer Volume

The volume of the smallest tetrahedron with integer edges and integer volume is $3$. There are $2$ possible sets of edges: :$32, 33, 35, 40, 70, 76$ :$21, 32, 47, 56, 58, 76$


Barbier's Theorem

Let $K$ be a (closed) curve of constant diameter. {{explain|"constant diameter"}} Let the circumference of $K$ be $c$. Let the diameter of $K$ be $d$. Then: :$\dfrac c d = \pi$


Machin's Formula for Pi

:$\dfrac \pi 4 = 4 \arctan \dfrac 1 5 - \arctan \dfrac 1 {239} \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$


Ordinals under Addition form Ordered Semigroup

$\left({\operatorname{On}, +, \le}\right)$ forms an ordered semigroup, where: : $\operatorname{On}$ denotes the ordinal class, and : $+$ denotes ordinal addition.


Subset is Left Compatible with Ordinal Addition

Let $x, y, z$ be ordinals. Then: :$x \le y \implies \paren {z + x} \le \paren {z + y}$


Subset is Compatible with Ordinal Addition

Let $x, y, z$ be ordinals. Then: : $(1): x \le y \implies \left({z + x}\right) \le \left({z + y}\right)$ : $(2): x \le y \implies \left({x + z}\right) \le \left({y + z}\right)$


Ordinals under Addition form Semigroup

$\left({\operatorname{On}, +}\right)$ forms an semigroup, where: : $\operatorname{On}$ denotes the ordinal class, and : $+$ denotes ordinal addition.


Ordinals under Multiplication form Semigroup

$\struct {\On, \times}$ forms an semigroup, where: : $\On$ denotes the ordinal class, and : $\times$ denotes ordinal multiplication.


Subset is Left Compatible with Ordinal Multiplication

Let $x, y, z$ be ordinals. Then: :$x \le y \implies \left({z \cdot x}\right) \le \left({z \cdot y}\right)$


Subset is Compatible with Ordinal Multiplication

Let $x, y, z$ be ordinals. Then: : $(1): x \le y \implies \left({z \cdot x}\right) \le \left({z \cdot y}\right)$ : $(2): x \le y \implies \left({x \cdot z}\right) \le \left({y \cdot z}\right)$


Ordinals under Multiplication form Ordered Semigroup

$\struct {\On, \times, \le}$ forms an ordered semigroup, where: : $\On$ denotes the ordinal class, and : $\times$ denotes ordinal multiplication.


Ordinals under Addition form Monoid

$\struct {\On, +}$ forms an monoid, where: :$\On$ denotes the ordinal class :$+$ denotes ordinal addition.


Ordinals under Addition form Ordered Monoid

$\left({\operatorname{On}, +, \le}\right)$ forms an ordered monoid, where: : $\operatorname{On}$ denotes the ordinal class, and : $+$ denotes ordinal addition.


Ordinals under Multiplication form Monoid

$\left({\operatorname{On}, \cdot}\right)$ forms an monoid, where: : $\operatorname{On}$ denotes the ordinal class, and : $\cdot$ denotes ordinal multiplication.


Ordinals under Multiplication form Ordered Monoid

$\struct {\On, \cdot, \le}$ forms an ordered monoid, where: :$\On$ denotes the ordinal class, and :$\cdot$ denotes ordinal multiplication.


Buffon's Needle

Let a horizontal plane be divided into strips by a series of parallel lines a fixed distance apart, like floorboards. Let a needle whose length equals the distance between the parallel lines be dropped onto the plane randomly from a random height. Then the probability that the needle falls across one of the parallel lines is $\dfrac 2 \pi$.


Pi as Sum of Odd Reciprocals Alternating in Sign in Pairs

:$\dfrac {\pi \sqrt 2} 4 = 1 + \dfrac 1 3 - \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} - \dfrac 1 {13} - \dfrac 1 {15} \cdots$


Sum of Reciprocals of Squares of Odd Integers

{{begin-eqn}} {{eqn | l = \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^2}       | r = 1 + \dfrac 1 {3^2} + \dfrac 1 {5^2} + \dfrac 1 {7^2} + \dfrac 1 {9^2} + \cdots }} {{eqn | r = \dfrac {\pi^2} 8 }} {{end-eqn}}


Basel Problem as Infinite Product

:$\displaystyle \dfrac {\pi^2} 6 = \prod_{p \mathop \in \mathbb P} \dfrac {p^2} {p^2 - 1}$


Convergence of Taylor Series of Function Analytic on Disk

Let $F$ be a complex function. Let $x_0$ be a point in $\R$. Let $R$ be an extended real number greater than zero. Let $F$ be analytic at every point $z \in \C$ satisfying $\cmod {z - x_0} < R$. Let $f = F {\restriction_\R}$ be a real function. Then: :the Taylor series of $f$ about $x_0$ converges to $f$ at every point $x \in \R$ satisfying $\size {x - x_0} < R$


Taylor Series of Analytic Function has infinite Radius of Convergence

Let $F$ be a complex function. Let $F$ be analytic everywhere. Let $f = F {\restriction_{\R}}$ be a real function. Let $x_0$ be a point in $\R$. Then the Taylor series of $f$ about $x_0$ converges to $f$ at every point in $\R$.


Number of Binary Digits in Power of 10

Let $n$ be a positive integer. Expressed in binary notation, the number of digits in the $n$th power of $10$: :$10^n$ is equal to: :$\ceiling {n \log_2 10}$ where $\ceiling x$ denotes the ceiling of $x$.


Regular Octahedron is Dual of Cube

The regular octahedron is the dual of the cube.


Plane Figure with Bilateral Symmetry about Two Lines has 4 Congruent Parts

Let $F$ be a plane figure. Let $F$ have two different axes of bilateral symmetry. Then those two axes divide $F$ into $4$ congruent parts. :500px


Hyperbola can be Drawn through Four Non-Collinear Points

Let $A, B, C, D$ be points in the plane of which no $3$ are collinear. Then a hyperbola can be drawn so that it passes through all points $A, B, C, D$.


Classification of Finite Simple Groups

The finite simple groups can be classified as follows:


Divisibility of n-1 Factorial by Composite n

Let $n \in \Z$ be composite. Then: :$n \divides \paren {n - 1}! \iff n \ne 4$ where: :$\divides$ denotes divisibility :$n!$ denotes the factorial of $n$.


Divisibility by 4

An integer $N$ expressed in decimal notation is divisible by $4$ {{iff}} the $2$ least significant digits of $N$ form a $2$-digit integer divisible by $4$. That is: :$N = \sqbrk {a_n \ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $4$  {{iff}}: :$10 a_1 + a_0$ is divisible by $4$.


Ratio of Number to Reversal which is Multiple

Take a (strictly) positive integer $n$, written in conventional decimal notation. Let $m$ be the reversal of $n$. Let $m = k n$ where $k$ is an integer. Then $k$ is either $4$ or $9$.


Smallest Pythagorean Triangle is 3-4-5

The smallest Pythagorean triangle has sides of length $3$, $4$ and $5$. :300px


Pythagorean Triangle with Sides in Arithmetic Sequence

The $3-4-5$ triangle is the only Pythagorean triangle such that: :the lengths of whose sides are in arithmetic sequence and: :the lengths of whose sides form a primitive Pythagorean triple.


Injectivity of Laplace Transform

Let $f$, $g$ be functions from $\left [{0 \,.\,.\, \to} \right ) \to \mathbb F$ of a real variable $t$, where $\mathbb F \in \left\{ {\R, \C}\right\}$. Further let $f$ and $g$ be continuous everywhere on their domains. Let $f$ and $g$ both admit Laplace transforms. Suppose that the Laplace transforms $\mathcal L \left\{{f}\right\}$ and $\mathcal L \left\{{g}\right\}$ satisfy: :$\forall t \ge 0: \mathcal L \left\{{f\left({t}\right)}\right\} = \mathcal L \left\{{g\left({t}\right)}\right\}$ Then $f = g$ everywhere on $\left[{0 \,.\,.\, \to}\right)$.


Pythagorean Triangle whose Area is Half Perimeter

The $3-4-5$ triangle is the only Pythagorean triangle whose area is half its perimeter.


Square Modulo 5/Corollary

When written in conventional base $10$ notation, no square number ends in one of $2, 3, 7, 8$.


Prime equals Plus or Minus One modulo 6

Let $p$ be a prime number greater than $3$. Then $p$ is either of the form: :$p = 6 n + 1$ or: :$p = 6 n - 1$ That is: :$p = \pm 1 \pmod 6$


Conic Section through Five Points

Let $A, B, C, D, E$ be distinct points in the plane such that no $3$ of them are collinear. Then it is possible to draw a conic section that passes through all $5$ points.


Lamé's Theorem

Let $a, b \in \Z_{>0}$ be (strictly) positive integers. Let $c$ and $d$ be the number of digits in $a$ and $b$ respectively when expressed in decimal notation. Let the Euclidean Algorithm be employed to find the GCD of $a$ and $b$. Then it will take no more than $5 \times \min \set {c, d}$ cycles around the Euclidean Algorithm to find $\gcd \set {a, b}$.


Volume of Unit Hypersphere

The volume of the unit sphere in $n$-dimensional space increases as $n$ goes up to $5$, but decreases thereafter.


Finite Subset Bounds Element of Finite Infima Set and Upper Closure

Let $L = \left({S, \wedge, \preceq}\right)$ be meet semilattice. Let $F$ be filter in $L$. Let $X$ be non empty finite subset of $S$. Let $x \in S$ such that : $x \in \left({\operatorname{fininfs}\left({F \cup X}\right)}\right)^\succeq$ where :$\operatorname{fininfs}$ denotes the finite infima set, :$X^\succeq$ denotes the upper closure of $X$. Then there exists $a \in S$: $a \in F \land x \succeq a \wedge \inf X$


Length of Reciprocal of Product of Powers of 2 and 5

Let $n \in \Z$ be an integer. Let $\dfrac 1 n$, when expressed as a decimal expansion, terminate after $m$ digits. Then $n$ is of the form $2^p 5^q$, where $m$ is the greater of $p$ and $q$.


Structure of Recurring Decimal

Let $\dfrac 1 m$, when expressed as a decimal expansion, recur with a period of $p$ digits. Let $\dfrac 1 n$, when expressed as a decimal expansion, terminate after $q$ digits. Then $\dfrac 1 {m n}$ has a nonperiodic part of $q$ digits, and a recurring part of $p$ digits.


Characteristics of Pentatope

A pentatope has $5$ cells, $10$ faces, $10$ edges and $5$ vertices.


Pentatope is Self-Dual

A pentatope is self-dual.


Riemann Hypothesis implies Odd Number is Sum of at most 5 Primes

Let the truth of the Riemann Hypothesis be assumed. Let $n$ be an odd integer. Then $n$ is the sum of at most $5$ primes.


Numbers of Zeroes that Factorial does not end with

Let $n \in \Z_{\ge 0}$ be a positive integer. Let $n!$ denote the factorial of $n$. Let $n!$ be expressed in decimal notation. Then $n!$ cannot end in the following numbers of zeroes: :$5, 11, 17, 23, 29, 30, 36, 42, \ldots$ {{OEIS|A000966}}


Euler-Binet Formula/Historical Note

The Euler-Binet Formula, derived by {{AuthorRef|Jacques Philippe Marie Binet|Binet}} in $1843$, was already known to {{AuthorRef|Leonhard Paul Euler|Euler}}, {{AuthorRef|Abraham de Moivre|de Moivre}} and {{AuthorRef|Daniel Bernoulli}} over a century earlier. However, it was {{AuthorRef|Jacques Philippe Marie Binet|Binet}} who derived the more general Binet Form of which this is an elementary application.


Cosine is of Exponential Order Zero

Let $\cos t$ be the cosine of $t$, where $t \in \R$. Then $\cos t$ is of exponential order $0$.


Sine is of Exponential Order Zero

Let $\sin t$ be the sine of $t$, where $t \in \R$. Then $\sin t$ is of exponential order $0$.


Scalar Multiple of Function of Exponential Order

Let $f: \R \to \F$ be a function, where $\F \in \set {\R, \C}$. Let $\lambda$ be a complex constant. Suppose $f$ is of exponential order $a$. Then $\lambda f$ is also of exponential order $a$.


Function of Exponential Order of Scalar Multiple

Let $f: \R \to \F$ be a function, where $\F \in \set {\R, \C}$. Let $\lambda$ be a real constant. Let $\map f t$ be of exponential order $a$. Then the function defined by $t \mapsto \map f {\lambda t}$ is of exponential order $a\lambda$.


Identity is of Exponential Order Epsilon

Let $I_\R: t \mapsto t$ be the identity mapping on $\R_{\ge 0}$. Then $I_\R$ is of exponential order $\epsilon$ for any  $\epsilon > 0$ arbitrarily small in magnitude.


Product of Functions of Exponential Order

Let $f, g: \R \to \F$ be functions, where $\F \in \set {\R, \C}$. Let $f$ be of exponential order $a$ and $g$ be of exponential order $b$. Then $f g: t \mapsto \map f t \, \map g t$ is of exponential order $a+b$.


Sum of Functions of Exponential Order

Let $f, g: \R \to \F$ be functions, where $\F \in \set {\R, \C}$. Suppose $f$ is of exponential order $a$ and $g$ is of exponential order $b$. Then $f + g: t \mapsto \map f t + \map g t$ is of exponential order $\max \set {a, b}$.


Linear Combination of Functions of Exponential Order

Let $f, g: \R \to \F$ be functions, where $\F \in \set {\R, \C}$. Let $\lambda, \mu$ be complex numbers. Suppose $f$ is of exponential order $a$ and $g$ is of exponential order $b$. Then $\map {\paren {\lambda f + \mu g} } t = \lambda \, \map f t + \mu \, \map g t$ is of exponential order $\max \set {a, b}$.


Constant Function is of Exponential Order Zero

Let $f_C: \R \to \mathbb F: t \mapsto C$ be a constant function, where $\mathbb F \in \set {\R, \C}$. Then $f_C$ is of exponential order $0$.


Polynomial is of Exponential Order Epsilon

Let $P: \R \to \mathbb F$ be a polynomial, where $\mathbb F \in \set {\R, \C}$. Then $P$ is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude.


Fibonacci Number by Power of 2

{{begin-eqn}} {{eqn | ll= \forall n \in \Z_{\ge 0}:       | l = 2^{n - 1} F_n       | r = \sum_k 5^k \dbinom n {2 k + 1}       | c =  }} {{eqn | r = \dbinom n 1 + 5 \dbinom n 3 + 5^2 \dbinom n 5 + \cdots       | c =  }} {{end-eqn}} where: :$F_n$ denotes the $n$th Fibonacci number :$\dbinom n {2 k + 1} \ $ denotes a binomial coefficient.


Approximation to Golden Rectangle using Fibonacci Squares

An approximation to a golden rectangle can be obtained by placing adjacent to one another squares with side lengths corresponding to consecutive Fibonacci numbers in the following manner: :800px It can also be noted, as from Sequence of Golden Rectangles, that an equiangular spiral can be approximated by constructing quarter circles as indicated.


Fibonacci Number of Index 2n as Sum of Squares of Fibonacci Numbers

Let $F_n$ denote the $n$th Fibonacci number. Then: :$F_{2 n} = {F_{n + 1} }^2 - {F_{n - 1} }^2$


Euler-Binet Formula/Also known as

The '''Euler-Binet Formula''' is also known as '''{{AuthorRef|Jacques Philippe Marie Binet|Binet}}'s formula'''.


If Ideal and Filter are Disjoint then There Exists Prime Ideal Including Ideal and Disjoint from Filter

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a distributive lattice. Let $I$ be an ideal in $L$. Let $F$ be a filter on $L$ such that :$I \cap F = \O$ Then there exists a prime ideal $P$ in $L$: $I \subseteq P$ and $P \cap F = \O$


No 4 Fibonacci Numbers can be in Arithmetic Sequence

Let $a, b, c, d$ be distinct Fibonacci numbers. Then, except for the trivial case: :$a = 0, b = 1, c = 2, d = 3$ it is not possible that $a, b, c, d$ are in arithmetic sequence.


Sum of Alternating Sign Reciprocals of Sequence of Pairs of Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared

{{begin-eqn}} {{eqn | l = \sum_{k \mathop \ge 2} \left({-1}\right)^k \dfrac 1 {F_k F_{k + 1} }       | r = \dfrac 1 {1 \times 2} - \dfrac 1 {2 \times 3} + \dfrac 1 {3 \times 5} - \dfrac 1 {5 \times 8} + \cdots       | c =  }} {{eqn | r = \phi^{-2}       | c =  }} {{end-eqn}} where: :$F_k$ denotes the $k$th Fibonacci number :$\phi$ denotes the golden mean.


Sum of Reciprocals of Sequence of Pairs of Even Index Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared

{{begin-eqn}} {{eqn | l = \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k} F_{2 k + 2} }       | r = \dfrac 1 {1 \times 3} + \dfrac 1 {3 \times 8} + \dfrac 1 {8 \times 21} + \dfrac 1 {21 \times 55} + \cdots       | c =  }} {{eqn | r = \phi^{-2}       | c =  }} {{end-eqn}} where: :$F_k$ denotes the $k$th Fibonacci number :$\phi$ denotes the golden mean.


Number of Fibonacci Numbers between n and 2n

Let $n \in \Z_{> 0}$ be a (strictly) positive integer. Then there exists either one or two Fibonacci numbers between $n$ and $2 n$ inclusive.


Square Root is of Exponential Order Epsilon

The positive square root function: :$t \mapsto \sqrt t$ is of exponential order $\epsilon$ for any  $\epsilon > 0$ arbitrarily small in magnitude.


Number of Fibonacci Numbers with Same Number of Decimal Digits

Let $n$ be an integer such that $n > 1$. When expressed in decimal notation, there are either $4$ or $5$ Fibonacci numbers with $n$ digits.


Prime Number divides Infinite Number of Fibonacci Numbers

Let $p$ be a prime number. Then there exist an infinite number of Fibonacci numbers which are divisible by $p$.


Prime Number divides Fibonacci Number

For $n \in \Z$, let $F_n$ denote the $n$th Fibonacci number. Let $p$ be a prime number. Then: :$p \equiv \pm 1 \pmod 5 \implies p \divides F_{p - 1}$ :$p \equiv \pm 2 \pmod 5 \implies p \divides F_{p + 1}$ where $\divides$ denotes divisibility. Thus in all cases, except where $p = 5$ itself: :$p \divides F_{p \pm 1}$


Existence of Fibonacci Number Divisible by Number

Let $m \in \Z$ be an integer. Then in the first $m^2$ Fibonacci numbers there exists at least one Fibonacci number which is divisible by $m$.