Complement of Element is Irreducible implies Element is Meet Irreducible

Let $T = \left({S, \tau}\right)$ be a topological space. Let $P = \left({\tau, \preceq}\right)$ be an ordered set where $\mathord\preceq = \mathord\subseteq \cap \left({\tau \times \tau}\right)$ Let $A \in \tau$. Then $\complement_S\left({A}\right)$ is irreducible implies $A$ is meet irreducible in $P$ where $\complement_S\left({A}\right)$ denotes the relative complement of $A$ relative to $S$.


Pentagonal Number as Sum of Triangular Numbers

Let $P_n$ be the $n$th pentagonal number. Then: :$P_n = T_n + 2 T_{n - 1}$ where $T_n$ is the $n$th triangular number.


Conditions for Integral Functionals to have same Euler's Equations

Let $\mathbf y$ be a real $n$-dimensional vector-valued function. Let $\map F {x, \mathbf y, \mathbf y'}$, $\map \Phi {x, \mathbf y}$  be real functions. Let $\Phi$ be twice differentiable. Let: {{begin-eqn}} {{eqn | l = \Psi       | r = \frac {\d \Phi} {\d x} }} {{eqn | r = \frac {\partial \Phi} {\partial x} + \sum_{i \mathop = 1}^n \frac {\partial \Phi} {\partial y_i} y_i' }} {{end-eqn}} Let $J_1$, $J_2$ be functionals such that: :$\ds J_1 \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ :$\ds J_2 \sqbrk {\mathbf y} = \int_a^b \paren {\map F {x, \mathbf y, \mathbf y'} + \map \Psi {x, \mathbf y, \mathbf y'} } \rd x$  Then $J_1$ and $J_2$ have same Euler's Equations.


Square of Odd Number as Difference between Triangular Numbers

Let $n \in \Z_{\ge 0}$ be a positive integer. Then: :$\exists a, b \in \Z_{\ge 0}: \paren {2 n + 1}^2 = T_a - T_b$ where: :$T_a$ and $T_b$ are triangular numbers :$T_a$ and $T_b$ are coprime. That is, the square of every odd number is the difference between two coprime triangular numbers.


Meet in Inclusion Ordered Set

Let $P = \left({X, \subseteq}\right)$ be an inclusion ordered set. Let $A, B \in X$ such that :$A \cap B \in X$ Then $A \wedge B = A \cap B$


Cube Number as Difference between Squares of Triangular Numbers

Let $n \in \Z_{> 0}$ be a positive integer. Then: :$n^3 = {T_n}^2 - {T_{n - 1} }^2$ where $T_n$ denotes the $n$th triangular number.


Square of Triangular Number equals Sum of Sequence of Cubes

:$\displaystyle \sum_{i \mathop = 1}^n i^3 = {T_n}^2$ where $T_n$ denotes the $n$th triangular number.


Sum of Sequence of Fifth Powers

:$\displaystyle \sum_{i \mathop = 1}^n i^5 = \dfrac { {T_n}^2 \paren {4 T_n - 1} } 3$ where $T_n$ denotes the $n$th triangular number.


Sum of Adjacent Sequences of Triangular Numbers

{{begin-eqn}} {{eqn | l = T_1 + T_2 + T_3       | r = T_4       | c =  }} {{eqn | l = T_5 + T_6 + T_7 + T_8       | r = T_9 + T_{10}       | c =  }} {{eqn | l = T_{11} + T_{12} + T_{13} + T_{14} + T_{15}       | r = T_{16} + T_{17} + T_{18}       | c =  }} {{end-eqn}} and so on. The $n$th line of the pattern can be written as: :$\displaystyle \sum_{k \mathop = n^2 + n - 1}^{n^2 + 2 n} T_n = \sum_{k \mathop = n^2 + 2 n + 1}^{n^2 + 3 n} T_n$


Conditions for Transformation to be Canonical

Let: :$\ds J_1 \sqbrk {\sequence {y_i}_{1 \mathop \le i \mathop \le n}, \sequence {p_i}_{1 \mathop \le i \mathop \le n} } = \int_a^b \paren {\sum_{i \mathop = 1}^n p_i y_i'-H} \rd x$ :$\ds J_2 \sqbrk {\sequence {Y_i}_{1 \mathop \le i \mathop \le n}, \sequence {P_i}_{1 \mathop \le i \mathop \le n} } = \int_a^b \paren {\sum_{i \mathop = 1}^n P_i Y_i'-H^*} \rd x$ be functionals. Then $\paren {\sequence {y_i}_{1 \mathop \le i \mathop \le n}, \sequence{p_i}_{1 \mathop \le i \mathop \le n}, H} \to \paren {\sequence{Y_i}_{1 \mathop \le i \mathop \le n}, \sequence {P_i}_{1 \mathop \le i \mathop \le n}, H^*}$ is a canonical transformation if: :$\ds \sum_{i \mathop = 1}^n p_i y_i' - H = \sum_{i \mathop = 1}^n P_i Y_i' - H^* \pm \dfrac {\d \Phi} {\d x}$   and: :$\ds p_i = \mp \frac {\partial \Phi} {\d y_i}, \quad P_i = \pm \frac {\partial \Phi} {\d Y_i}, \quad H = H^* \mp \frac {\partial \Phi} {\partial x}$


Square of Triangular Numbers as Sum of Triangular Numbers

:${T_n}^2 = T_n + T_{n - 1} T_{n + 1}$ where $T_n$ denotes the $n$th triangular number.


Sum of Sequence of Reciprocals of Triangular Numbers

:$\displaystyle \sum_{k \mathop \ge 1} \dfrac 1 {T_k} = 2$ where $T_k$ denotes the $k$th triangular number.


Triangular Number Pairs with Triangular Sum and Difference

The sequence of pairs of triangular numbers whose sum and difference are also both triangular begins: :$\tuple {15, 21}, \tuple {105, 171}, \tuple {378, 703}, \tuple {780, 990}, \tuple {1485, 4186}, \tuple {2145, 3741}, \tuple {5460, 6786}, \tuple {7875, 8778}$ {{OEIS|A185129|order = first}} {{OEIS|A185128|order = second}}


Triangular Number whose Square is Triangular

The only triangular number with less than $660$ digits, whose square is also triangular, is $6$.


Factors of Integer Congruent to 5 modulo 6

Let $m$ be an positive integer. Let $m \equiv 5 \pmod 6$. Then $m$ has two divisors whose sum is divisible by $6$.


Top in Ordered Set of Topology

Let $T = \left({S, \tau}\right)$ be a topological space. Let $P = \left({\tau, \subseteq}\right)$ be an inclusion ordered set of $\tau$. Then $P$ is bounded above and $\top_P = S$ where $\top_P$ denotes the greatest element in $P$.


Triangular Numbers which are also Square

Let $A_n$ be the $n$th non-negative integer whose square is also a triangular number. Then: :$A_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ 6 A_{n - 1} - A_{n - 2} & : n > 1 \end{cases}$


Index of Square Triangular Number from Preceding

Let $T_n$ be the $n$th triangular number. Let $T_n$ be square. Then $T_{4 n \paren {n + 1} }$ is also square.


Triangular Number cannot be Fourth Power

Let $T_n$ be the $n$th triangular number such that $n > 1$. Then $T_n$ cannot be the $4$th power of an integer.


Triangular Number cannot be Fifth Power

Let $T_n$ be the $n$th triangular number such that $n > 1$. Then $T_n$ cannot be the $5$th power of an integer.


Hilbert-Waring Theorem/Variant Form

For each $k \in \Z: k \ge 2$, there exists a positive integer $G \left({k}\right)$ such that every sufficiently large positive integer can be expressed as a sum of at most $G \left({k}\right)$ $k$th powers.


Equality of Integers to the Power of Each Other

$2$ and $4$ are the only pair of positive integers $m, n$ such that $m \ne n$ such that: :$m^n = n^m$


Square whose Perimeter equals its Area

The $4 \times 4$ square is the only square whose area in square units equals its perimeter in units. The area and perimeter of this square are $16$.


Magic Constant of Order 4 Magic Square

The magic constant of the order $4$ magic square is $34$.


Sum of Cubes on Diagonals of Moessner's Order 4 Magic Square

The sums of the cubes of the entries on the diagonals of Moessner's order $4$ magic square are equal.


Sum of Squares on Pairs of Rows and Columns of Moessner's Order 4 Magic Square

The sums of the squares of the entries are equal on the following pairs of rows and columns of Moessner's order $4$ magic square: :Rows $1$ and $4$ :Rows $2$ and $3$ :Columns $1$ and $4$ :Columns $2$ and $3$.


Equivalence of Definitions of Quasiperfect Number

The following definitions of a quasiperfect number are equivalent:


Quasiperfect Number is Square of Odd Integer

Let $n$ be a quasiperfect number. Then: :$n = \paren {2 k + 1}^2$ for some $k \in \Z_{>0}$. That is, a quasiperfect number is the square of an odd integer.


Bottom in Ordered Set of Topology

Let $T = \left({S, \tau}\right)$ be a topological space. Let $P = \left({\tau, \subseteq}\right)$ be an inclusion ordered set of $\tau$. Then $P$ is bounded below and $\bot_P = \varnothing$


Period of Reciprocal of 17 is of Maximal Length

The decimal expansion of the reciprocal of $17$ has the maximum period, that is: $16$: :$\dfrac 1 {17} = 0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$ {{OEIS|A007450}}


Prime Dudeney Number

The only prime Dudeney number is $17$:


Sequence of Dudeney Numbers

The only Dudeney numbers are: :$0, 1, 8, 17, 18, 26, 27$ two of which are themselves cubes, and one of which is prime.


Derivation of Hamilton-Jacobi Equation

Let $\map S {x_0, x_1, \mathbf y} = \map S {x, \mathbf y}$ be the geodetic distance, where $x_0$ is fixed and $x_1=x$. Let $H$ be Hamiltonian. Then the following equation holds: :$\dfrac {\partial S} {\partial x} + \map H {x, \mathbf y, \nabla_{\mathbf y} S} = 0$ and is known as the Hamilton-Jacobi Equation.


Partial Derivatives of Solution of Hamilton-Jacobi Equation are First Integrals of Euler's Equations

Let $\mathbf y = \sequence {y_i}_{1 \mathop \le i \mathop \le n}$, $\boldsymbol \alpha = \sequence {\alpha_i}_{1 \mathop \le i \mathop \le m}$ be vectors, where $m \le n$. Let $S = \map S {x, \mathbf y, \boldsymbol \alpha}$ be a solution of the Hamilton-Jacobi equation, where $\boldsymbol \alpha$ are parameters. Then each partial derivative: :$\dfrac {\partial S} {\partial \alpha_i}$ is a first integral of canonical Euler's equations.


Jacobi's Theorem

Let $\mathbf y = \sequence {y_i}_{1 \le i \le n}$, $\boldsymbol \alpha = \sequence {\alpha_i}_{1 \le i \le n}$, $\boldsymbol \beta = \sequence {\beta_i}_{1 \le i \le n}$ be vectors, where $\alpha_i$ and $ \beta_i$ are parameters. Let $S = \map S {x, \mathbf y, \boldsymbol \alpha}$ be a a complete solution of the Hamilton-Jacobi equation. Let: :$\begin {vmatrix} \dfrac {\partial^2 S} {\partial \alpha_i \partial y_k} \end{vmatrix} \ne 0$ where $\begin {vmatrix} \cdot \end{vmatrix}$ is a determinant. Let: :$\dfrac {\partial S} {\partial \alpha_i} = \beta_i$ Then: :$p_i = \map {\dfrac {\partial S} {\partial y_i} } {x, \mathbf y, \boldsymbol \alpha}$ :$y_i = \map {y_i} {x, \boldsymbol \alpha, \boldsymbol \beta}$ constitute a general solution of the canonical Euler's equations.


Equivalence of Definitions of Principal Ideal

Let $\left({S, \preceq}\right)$ be a preordered set. Let $I$ be an ideal in $S$. Then :the definitions of principal ideal are equivalent, That means that :$\exists x \in I: x$ is upper bound for $I$ {{iff}} :$\exists x \in S: I = x^\preceq$ where $x^\preceq$ denotes the lower closure of $x$.


Divisions of Numbers in Unit Interval with Numbers in Different Intervals

Let $I = \openint 0 1$ be the open unit interval. Let $a_1, a_2, a_3, \ldots, a_n$ be real numbers chosen in $I$ such that: :$a_1$ and $a_2$ are in different halves of $I$ :$a_1, a_2$ and $a_3$ are in different thirds of $I$ :$a_1, a_2, a_3$ and $a_4$ are in different quarters of $I$ and so on. Then $n \le 17$. That is, for the conditions to be fulfilled, no more than $17$ numbers can be chosen.


Seventeen Different Wallpaper Patterns

There are $17$ different symmetry groups for a wallpaper pattern.


Compact Element iff Principal Ideal

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $P = \left({\mathit{Ids}\left({L}\right), \precsim}\right)$ be an inclusion ordered set where :$\mathit{Ids}\left({L}\right)$ denotes the set of all ideals in $L$, :$\mathord\precsim = \mathord\subseteq \cap \left({\mathit{Ids}\left({L}\right) \times \mathit{Ids}\left({L}\right)}\right)$ Let $x \in \mathit{Ids}\left({L}\right)$ Then $x$ is compact element {{iff}} $x$ is principal ideal in $L$


Integers as Sum of Three Pairwise Coprime Integers

Let $n$ be an integer greater than $17$. Then $n$ is the sum of $3$ integers greater than $1$ which are pairwise coprime.


Primes for which Powers to Themselves minus 1 have Common Factors

Let $p$ and $q$ be prime numbers such that $p^p - 1$ and $q^q - 1$ have a common divisor $d$. The only known $p$ and $q$ such that $d < 400 \, 000$ are $p = 17, q = 3313$.


Reversal formed by Repdigits of Base minus 1 by Addition and Multiplication

Let $b \in \Z_{>1}$ be an integer greater than $1$. Let $n = b^k - 1$ for some integer $k$ such that $k \ge 1$. Then: : $n + n$ is the reversal of $\left({b - 1}\right) n$ when both are expressed in base $b$ representation.


Power of Base minus 1 is Repdigit Base minus 1

Let $b \in \Z_{>1}$ be an integer greater than $1$. Let $n = b^k - 1$ for some integer $k$ such that $k \ge 1$. Let $n$ be expressed in base $b$ representation. Then $n$ is a repdigit number consisting of $k$ instances of digit $b - 1$.


Multiple of Repdigit Base minus 1

Let $b \in \Z_{>1}$ be an integer greater than $1$. Let $n$ be a repdigit number of $k$ instances of the digit $b - 1$ for some integer $k$ such that $k \ge 1$. Let $m \in \Z_{>1}$ be an integer such that $1 < m < b$. Then $m \times n$, when expressed in base $b$, is of the form: :$n = \left[{r d d \cdots d s}\right]_b$ where: :$d = b - 1$ :$r = m - 1$ :$s = b - m$ :there are $k - 1$ occurrences of $d$.


Only Number Twice Sum of Digits is 18

There exists only one (strictly) positive integer that is exactly twice the sum of its digits.


Period of Reciprocal of 19 is of Maximal Length

The decimal expansion of the reciprocal of $19$ has the maximum period, that is: $18$: :$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$ {{OEIS|A021023}}


Divisibility by 19

Let $n$ be an integer expressed in the form: :$n = 100 a + b$ Then $n$ is divisible by $19$ {{iff}} $a + 4 b$ is divisible by $19$.


Diameter of Closure of Subset is Diameter of Subset

Let $M = \struct {A, d}$ be a metric space. Let $S \subseteq A$ be bounded in $M$. Then: :$\map {\operatorname {diam} } S = \map {\operatorname {diam} } {S^-}$ where $\map {\operatorname {diam} } S$ denotes the diameter of $S$, and $S^-$ denotes the closure of $S$ in $M$.


Magic Constant of Order 3 Magic Hexagon

The magic constant of the order 3 magic hexagon is $38$. It is noted that the central cell is $5$, the same as that of the order 3 magic square.


Sum of Sequence of Alternating Positive and Negative Factorials being Prime

Let $n \in \Z_{\ge 0}$ be a positive integer. Let: {{begin-eqn}} {{eqn | l = m       | r = \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!       | c =  }} {{eqn | r = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1       | c =  }} {{end-eqn}} The sequence of $n$ such that $m$ is prime begins: :$3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, \ldots$ {{OEIS|A001272}} The sequence of those values of $m$ begins: :$5, 19, 101, 619, 4421, 35 \, 899, 3 \, 301 \, 819, 1 \, 226 \, 280 \, 710 \, 981, \ldots$ {{OEIS|A071828}}


Ideals are Continuous Lattice Subframe of Power Set

Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice. Let $I = \left({\mathit{Ids}\left({L}\right), \precsim}\right)$ be an inclusion ordered set where :$\mathit{Ids}\left({L}\right)$ denotes the set of all ideals in $L$, :$\mathord\precsim = \mathord\subseteq \cap \left({\mathit{Ids}\left({L}\right) \times \mathit{Ids}\left({L}\right)}\right)$ Let $P = \struct {\mathcal P\left({S}\right), \precsim'}$ be an inclusion ordered set where :$\mathcal P\left({S}\right)$ denotes the power set of $S$, :$\mathord\precsim' = \mathord\subseteq \cap \left({\mathcal{P}\left({S}\right) \times \mathcal{P}\left({S}\right)}\right)$ Then $I$ is continuous lattice subframe of $P$.


Smallest Perfect Square Dissection

The smallest perfect square dissection is of an integer square of side $112$ into $21$ parts.


Smallest Number Expressible as Sum of at most Three Triangular Numbers in 4 ways

$21$ is the smallest number which can be expressed as the sum of at most $3$ triangular numbers in $4$ ways.


Necessary Condition for Twice Differentiable Functional to have Minimum

Let $J \sqbrk y$ be a twice differentiable functional. Let $\delta J \sqbrk {\hat y; h} = 0$. Suppose, for $y = \hat y$ and all admissible $h$: :$\delta^2 J \sqbrk {y; h} \ge 0$  Then $J$ has a minimum for $y=\hat y$ if  {{explain|if what?}


Projective Plane of Order 4 is Unique

The finite projective plane of order $4$ is unique.


Numbers Equal to Number of Digits in Factorial

For $n \in \set {1, 22, 23, 24}$ the number of digits in the decimal representation of $n!$ is equal to $n$.


Square of Small-Digit Palindromic Number is Palindromic

Let $n$ be an integer such that the sum of the squares of the digits of $n$ in decimal representation is less than $10$. Let $n$ be palindromic. Then $n^2$ is also palindromic. The sequence of such numbers begins: :$0, 1, 2, 3, 11, 22, 101, 111, 121, 202, 212, 1001, 1111, \dots$ {{OEIS|A057135}}


Numbers whose Sigma is Square

The sequence of positive integers whose $\sigma$ value is square starts as follows: {{begin-eqn}} {{eqn | l = \map \sigma 3       | r = 4       | c =  }} {{eqn | l = \map \sigma {22}       | r = 36       | c =  }} {{eqn | l = \map \sigma {66}       | r = 144       | c =  }} {{eqn | l = \map \sigma {70}       | r = 144       | c =  }} {{eqn | l = \map \sigma {81}       | r = 121       | c =  }} {{eqn | l = \map \sigma {94}       | r = 144       | c =  }} {{eqn | l = \map \sigma {115}       | r = 144       | c =  }} {{eqn | l = \map \sigma {119}       | r = 144       | c =  }} {{eqn | l = \map \sigma {170}       | r = 324       | c =  }} {{end-eqn}} {{OEIS|A006532}}


Three times Number whose Sigma is Square

Let $n \in \Z_{>0}$ be a positive integer. Let the $\sigma$ value of $n$ be square. Let $3$ not be a divisor of $n$. Then the $\sigma$ value of $3 n$ is square.


Product of Coprime Numbers whose Sigma is Square has Square Sigma

Let $m, n \in \Z_{>0}$ be a positive integer. Let the $\sigma$ value of $m$ and $n$ both be square. Let $m$ and $n$ be coprime. Then the $\sigma$ value of $m n$ is square.


Sequence of Differences on Generalized Pentagonal Numbers

Recall the generalised pentagonal numbers $GP_n$ for $n = 0, 1, 2, \ldots$ Consider the sequence defined as $\Delta_n = GP_{n + 1} - GP_n$: :$1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, \ldots$ {{OEIS|A026741}} Then: :The values of $\Delta_n$ for odd $n$ consist of the odd numbers :The values of $\Delta_n$ for even $n$ consist of the natural numbers.


Euler's Pentagonal Numbers Theorem

Consider the infinite product: :$\displaystyle P = \prod_{n \mathop \in \Z_{>0} } \paren {1 - x^n}$ Then $P$ can be expressed as: :$\displaystyle P = \sum_{n \mathop \in \Z_{>0} } \paren {-1}^{\ceiling {n / 2} } x^{GP_n}$ where: :$\ceiling {n / 2}$ denotes the ceiling of $n / 2$ :$GP_n$ denotes the $n$th generalized pentagonal number. That is: :$P = 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + \cdots$


Intersection of Semilattice Ideals is Ideal/Set of Sets

Let $\struct {S, \preceq}$ be a bounded below join semilattice. Let $\II$ be a set of ideals in $\struct {S, \preceq}$. Then $\bigcap \II$ is an ideal in $\struct {S, \preceq}$.


Compact Element iff Existence of Finite Subset that Element equals Intersection and Includes Subset

Let $X, E$ be sets. Let $P = \left({\mathcal P\left({X}\right), \precsim}\right)$ be an inclusion ordered set where :$\mathcal P\left({X}\right)$ denotes the power set of $X$, :$\mathord\precsim = \mathord\subseteq \cap \left({\mathcal{P}\left({X}\right) \times \mathcal{P}\left({X}\right)}\right)$ Let $L = \left({S, \preceq}\right)$ be a continuous lattice subframe of $P$. Then $E$ is compact element in $L$ {{iff}} :$\exists F \in \mathit{Fin}\left({X}\right): E = \bigcap \left\{ {I \in S: F \subseteq I}\right\} \land F \subseteq E$ where $\mathit{Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$.


Legendre's Condition

Let $y =\map y x$ be a real function, such that: :$\map y a = A,\quad \map y b = B$ Let $J \sqbrk y$ be a functional, such that: :$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$ where :$F \in C^2 \closedint a b$  {{WRT}} all its variables, and $C$ stands for differentiability class. Then a necessary condition for $J \sqbrk y$ to have a minimum at $y = \hat y$ is: :$\bigintlimits {F_{y'y'} } {y \mathop = \hat y} {} \ge 0 \quad \forall x \in \closedint a b$


Euler's Pentagonal Numbers Theorem/Corollary 1

Let $n \in \Z_{>0}$ be a strictly positive integer. Let $\map \sigma n$ denote the $\sigma$ function on $n$. Then: :$\map \sigma n = \displaystyle \sum_{1 \mathop \le n - GP_k \mathop < n} -\paren {-1}^{\ceiling {k / 2} } \map \sigma {n - GP_k} + n \sqbrk {\exists k \in \Z: GP_k = n}$


Euler's Pentagonal Numbers Theorem/Corollary 2

Let $n \in \Z_{>0}$ be a strictly positive integer. Let $\map p n$ denote the number of partitions on $n$. Then: :$\map p n = \displaystyle \sum_{1 \mathop \le n - GP_k \mathop < n} -\paren {-1}^{\ceiling {k / 2} } \map p {n - GP_k} + \sqbrk {\exists k \in \Z: GP_k = n}$


Integer as Sum of 4 Cubes

Let $n \in \Z$ be an integer. Let $n \not \equiv 4 \pmod 9$ and $n \not \equiv 5 \pmod 9$. Then it is possible to express $n$ as the sum of no more than $4$ cubes which may be either positive or negative.


Period of Reciprocal of 23 is of Maximal Length

The decimal expansion of the reciprocal of $19$ has the maximum period, that is: $22$: :$\dfrac 1 {23} = 0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$ {{OEIS|A021027}}


Smallest Integer not Sum of Two Ulam Numbers

The smallest integer greater than $1$ which is not the sum of two Ulam numbers is $23$.


Numbers with Square-Free Binomial Coefficients

For every $n$ greater than $23$, there exists a binomial coefficient $\dbinom n k$ that is not square-free. More specifically, the list of numbers $n$ such that $\dbinom n k$ are squarefree for all $k = 0, \dots, n$ is given by: :$1, 2, 3, 5, 7, 11, 23$ {{OEIS|A048278}}


Infima Inheriting Ordered Subset of Complete Lattice is Complete Lattice

Let $L = \struct {X, \preceq}$ be a complete lattice. Let $S = \struct {T, \precsim}$ be an infima inheriting ordered subset of $L$. Then $S$ is a complete lattice.


Gelfond's Constant minus Pi

Gelfond's constant minus $\pi$ is very close to $20$: :$e^\pi - \pi \approx 20$


Sums of Consecutive Sequences of Squares that equal Squares

The $24$th square pyramidal number is the only one which is square: :$1^2 + 2^2 + 3^2 + \cdots + 24^2 = 70^2$ while there are several Sum of Sequence of Squares which are square, for example: :$18^2 + 19^2 + \cdots + 28^2 = 77^2$ and: :$25^2 + 26^2 + \cdots + 624^2 = 9010^2$


Smallest Scalene Obtuse Triangle with Integer Sides and Area

The smallest scalene obtuse triangle with integer sides and area has sides of length $4, 13, 15$.


Image of Compact Subset under Directed Suprema Preserving Closure Operator

Let $L = \left({S, \preceq}\right)$ be a bounded below algebric lattice. Let $c: S \to S$ be a closure operator that preserves directed suprema. Then: : $c \left[{K\left({L}\right)}\right] = K \left({\left({c\left[{S}\right], \precsim}\right)}\right)$ where :$K \left({L}\right)$ denotes the compact subset of $L$, :$c \left[{S}\right]$ denotes the image of $S$ under $c$, :$\mathord \precsim = \mathord \preceq \cap \left({c \left[{S}\right] \times c \left[{S}\right]}\right)$


Smallest Positive Integer with 5 Fibonacci Partitions

The smallest positive integer which can be partitioned into distinct Fibonacci numbers in $5$ different ways is $24$.


Sigma Function of Non-Square Semiprime

Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$. Then: :$\map \sigma n = \paren {p + 1} \paren {q + 1}$ where $\map \sigma n$ denotes the sigma function.


Sigma Function of Square-Free Integer

Let $n$ be an integer such that $n \ge 2$. Let $n$ be square-free. Let the prime decomposition of $n$ be: :$\displaystyle n = \prod_{1 \mathop \le i \mathop \le r} p_i =  p_1 p_2 \cdots p_r$ Let $\sigma \left({n}\right)$ be the sigma function of $n$. That is, let $\sigma \left({n}\right)$ be the sum of all positive divisors of $n$. Then: :$\displaystyle \sigma \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} p_i + 1$


Spheres in 24 Dimensions in Leech Lattice

Let a set of identical spheres in a $24$-dimensional space be arranged in a Leech lattice. Then each sphere will touch $196 \, 560$ other spheres. This is believed to be the densest possible sphere packing in $24$ dimensions.


Cross-Sections of Leech Lattice

Except for dimensions $10$, $11$ and $13$, the densest possible sphere packing in all dimensions lower than $24$ can be obtained by cross-sections of the $24$-dimensional space Leech lattice arrangement.


Difference Triangle for Sequence of Fifth Powers

The difference triangle for the sequence of fifth powers ends on the fifth line with instances of $5!$, where $!$ denotes factorial.


Ideals form Arithmetic Lattice

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below lattice. Let $I = \left({\mathit{Ids}\left({L}\right), \precsim}\right)$ be an inclusion ordered set where :$\mathit{Ids}\left({L}\right)$ denotes the set of all ideals in $L$, :$\mathord\precsim = \mathord\subseteq \cap \left({\mathit{Ids}\left({L}\right) \times \mathit{Ids}\left({L}\right)}\right)$ Then $I$ is an arithmetic lattice.


Power of n equalling (n - 1)! + 1

There is exactly one solution to the equation in the integers: :$\paren {n - 1}! + 1 = n^k$ for $k > 1$, and that is: :$n = 5$ :$k = 2$


Square which is 2 Less than Cube

$25$ is the only square number which is $2$ less than a cube: :$5^2 + 2 = 3^3$


Ideals form Algebraic Lattice

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $I = \left({\mathit{Ids}\left({L}\right), \precsim}\right)$ be an inclusion ordered set where :$\mathit{Ids}\left({L}\right)$ denotes the set of all ideals in $L$, :$\mathord\precsim = \mathord\subseteq \cap \left({\mathit{Ids}\left({L}\right) \times \mathit{Ids}\left({L}\right)}\right)$ Then $I$ is algebraic lattice.


Definition:Powerful Number

A '''powerful number''' is a positive integer such that each of its prime factors appears with multiplicity at least $2$. That is, each of its prime factors occurs at least squared.


Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite

Let   :$\map P x : \closedint a b \to \R$  :$\map h x : \closedint a b \to \R$. Let $\map h x$ be continuously differentiable $\forall x \in \closedint a b$. Suppose: :$\forall x \in \closedint a b: \map P x > 0$ Then: :$\ds \forall \map h x : \map h a = \map h b = 0 : \int_a^b \paren {P h'^2 + Q h^2} \rd x > 0$ {{iff}} the interval $\closedint a b$ contains no points conjugate to $a$.


Consecutive Odd Powerful Numbers

$25$ and $27$ are the only known pair of consecutive odd powerful numbers. It is not known whether there are any more.


Square Cullen Numbers

The numbers: :$1, 9, 25$ are Cullen numbers which are also square.


Numbers Partitioned into Six Hexagonal Numbers

The integers $11$ and $26$ cannot be represented by the sum of less than $6$ hexagonal numbers.


Continuous Lattice Subframe of Algebraic Lattice is Algebraic Lattice

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below algebraic lattice. Let $P = \struct {T, \precsim}$ be a continuous lattice subframe of $L$. Then $P$ is algebraic lattice.


Period of Reciprocal of 27 is Smallest with Length 3

$27$ is the smallest positive integer the decimal expansion of whose reciprocal has a period of $3$: :$\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$ {{OEIS|A021027}}


Period of Reciprocal of 7 is of Maximal Length

$7$ is the smallest integer $n$ the decimal expansion of whose reciprocal has the maximum period $n - 1$, that is: $6$: :$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$ {{OEIS|A020806}} :300px


Sequence of Smallest Numbers whose Reciprocal has Period n

Let $\sequence {s_n}$ be the sequence defined as: :$s_n$ is the smallest positive integer the decimal expansion of whose reciprocal has a period of $n$ for $n = 0, 1, 2, \ldots$ Then $\sequence {s_n}$ begins: :$1, 3, 11, 27, 101, 41, 7, 239, 73, 81, 451, \ldots$ {{OEIS|A003060}}


Sequence of Successive Longest Collatz Sequence Generators

The sequence of integers which generate a Collatz process which is longer than that of any smaller integers begins: {{begin-eqn}} {{eqn | l = 1       | o = :       | c = $0$ steps }} {{eqn | l = 2       | o = :       | c = $1$ step }} {{eqn | l = 3       | o = :       | c = $7$ steps }} {{eqn | l = 6       | o = :       | c = $8$ steps }} {{eqn | l = 7       | o = :       | c = $16$ steps }} {{eqn | l = 9       | o = :       | c = $19$ steps }} {{eqn | l = 18       | o = :       | c = $20$ steps }} {{eqn | l = 25       | o = :       | c = $23$ steps }} {{eqn | l = 27       | o = :       | c = $111$ steps }} {{end-eqn}} {{OEIS|A006877}}


Dissection of Nonagon into Triangles with Chords

There are $27$ different ways to dissect a convex nonagon into triangles using $6$ chords, not counting reflections and rotations as different.


Image of Directed Suprema Preserving Closure Operator is Algebraic Lattice

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below algebraic lattice. Let $c:S \to S$ be a closure operator that preserves directed suprema. Let $C = \left({c\left[{S}\right], \precsim}\right)$ be an ordered subset of $L$. Then $C$ is algebraic lattice.


Dissection of Polygon into Triangles with Chords

The number of different ways $k$ a convex $n$-sided polygon can be divided into triangles using chords, not counting reflections and rotations as different, is given for the first few $n$ as follows: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="right" style = "padding: 2px 10px" | $k$ |- | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $5$ | align="right" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="right" style = "padding: 2px 10px" | $3$ |- | align="right" style = "padding: 2px 10px" | $7$ | align="right" style = "padding: 2px 10px" | $4$ |- | align="right" style = "padding: 2px 10px" | $8$ | align="right" style = "padding: 2px 10px" | $12$ |- | align="right" style = "padding: 2px 10px" | $9$ | align="right" style = "padding: 2px 10px" | $27$ |- | align="right" style = "padding: 2px 10px" | $10$ | align="right" style = "padding: 2px 10px" | $82$ |- | align="right" style = "padding: 2px 10px" | $11$ | align="right" style = "padding: 2px 10px" | $228$ |- | align="right" style = "padding: 2px 10px" | $12$ | align="right" style = "padding: 2px 10px" | $733$ |- | align="right" style = "padding: 2px 10px" | $13$ | align="right" style = "padding: 2px 10px" | $2282$ |} {{OEIS|A000207}}


Dissection of Polygon into Triangles with Chords counting Isometries

The number of different ways $k$ a convex $n$-sided polygon can be divided into triangles using chords, counting reflections and rotations as different, is given for the first few $n$ as follows: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="right" style = "padding: 2px 10px" | $k$ |- | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $2$ |- | align="right" style = "padding: 2px 10px" | $5$ | align="right" style = "padding: 2px 10px" | $5$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="right" style = "padding: 2px 10px" | $14$ |- | align="right" style = "padding: 2px 10px" | $7$ | align="right" style = "padding: 2px 10px" | $42$ |- | align="right" style = "padding: 2px 10px" | $8$ | align="right" style = "padding: 2px 10px" | $132$ |- | align="right" style = "padding: 2px 10px" | $9$ | align="right" style = "padding: 2px 10px" | $429$ |- | align="right" style = "padding: 2px 10px" | $10$ | align="right" style = "padding: 2px 10px" | $1430$ |- | align="right" style = "padding: 2px 10px" | $11$ | align="right" style = "padding: 2px 10px" | $4862$ |} {{OEIS|A000108}} These are the Catalan numbers.


Boundary of Compact Closed Set is Compact

Let $X$ be a topological space. Let $K\subset X$ be a compact subspace of $X$. Let $K$ be closed in $X$. Then its boundary $\partial K$ is compact.


Discrete Subgroup of Hausdorff Group is Closed

Let $G$ be a Hausdorff topological group. Let $H$ be a discrete subgroup of $G$. Then $H$ is closed in $G$.


Group is Hausdorff iff Discrete Subgroups are Closed

A topological group is Hausdorff {{iff}} its discrete subgroups are closed.


Closure of Subgroup is Group

Let $G$ be a topological group. Let $H\leq G$ be a subgroup. Let $\overline H$ denote its closure. Then $\overline H$ is a subgroup of $G$.


Image of Group Homomorphism is Hausdorff Implies Kernel is Closed

Let $G$ and $H$ be topological groups. Let $f: G \to H$ be a morphism. Let its image $\Img f$ be Hausdorff. Then its kernel $\map \ker f$ is closed in $G$.


Group is Connected iff Subgroup and Quotient are Connected

Let $G$ be a topological group. Let $H \le G$ be a subgroup. {{TFAE}} :$(1):\quad$ $G$ is connected :$(2):\quad$ $H$ is connected and the left quotient space $G / H$ is connected :$(3):\quad$ $H$ is connected and the right quotient space $G / H$ is connected.


Group Acts by Homeomorphisms Implies Projection on Quotient Space is Open

Let $G$ be a group acting by homeomorphisms on a topological space $X$. Then the projection map $\pi: X \to X / G$ is open.


Neighborhood iff Contains Neighborhood

Let $X$ be a topological space. Let $x\in X$. Let $V\subset X$ be a subset. Then the following are equivalent: :$V$ is a neighborhood of $x$ in $X$ :$V$ contains a neighborhood of $x$ in $X$


Perfect Number ends in 6 or 28 preceded by Odd Digit

Let $n$ be an even perfect number. Then $n$ ends either in $6$ or $28$ preceded by an odd digit.


Open Subset of Locally Connected Space is Locally Connected

Let $T - \left({S, \tau}\right)$ be a locally connected topological space. Let $U \subset S$ be open in $T$. Then $U$ is locally connected.


Component of Locally Connected Space is Open

Let $T = \left({S, \tau}\right)$ be a locally connected topological space. Let $G$ be a component of $T$. Then $G$ is open.


Path Component of Locally Path-Connected Space is Open

Let $T = \left({S, \tau}\right)$ be a locally path-connected topological space. Let $G$ be a path component of $T$. Then $G$ is open in $T$.


Connected and Locally Path-Connected Implies Path Connected

Let $T$ be a connected and locally path-connected topological space. Then $T$ is path-connected.


Components are Open iff Union of Open Connected Sets

Let $T = \struct {S, \tau}$ be a topological space. {{TFAE}} :$(1): \quad$ The components of $T$ are open. :$(2): \quad S$ is a union of open connected sets of $T$.


Path Components are Open iff Union of Open Path-Connected Sets

Let $T = \left({S, \tau}\right)$ be a topological space. {{TFAE}}: :(1) $\quad$ The path components of $T$ are open. :(2) $\quad S$ is a union of open path-connected sets of $T$.


Open Subset of Locally Path-Connected Space is Locally Path-Connected

Let $T = \struct {S, \tau}$ be a locally path-connected topological space. Let $U \subset S$ be open in $T$. Then $U$ is locally path-connected in $T$.


Even Perfect Number is Hexagonal

All perfect numbers which are even are hexagonal.


Hexagonal Number is Triangular Number

Let $H_n$ be the $n$th hexagonal number. Then $H_n$ is the $2 n - 1$th triangular number.


Perfect Number which is Sum of Equal Powers of Two Numbers

$28$ is the only perfect number which is the sum of equal powers of exactly $2$ positive integers: :$28 = 1^3 + 3^3$


Congruence Modulo 3 of Power of 2

Let $n \in \Z_{\ge 0}$ be a positive integer. Then: :$2^n \equiv \paren {-1}^n \pmod 3$ That is: :$\exists q \in \Z: 2^n = 3 q + \paren {-1}^n$


Odd Power of 2 is Congruent to 2 Modulo 3

Let $n \in \Z_{\ge 0}$ be an odd positive integer. Then: :$2^n \equiv 2 \pmod 3$


Open Set in Open Subspace

Let $X$ be a topological space. Let $U\subset X$ be an open subset. Let $V\subset U$ be a subset. Then $V$ is open in $U$ {{iff}} $V$ is open in $X$.


Equivalence of Definitions of Locally Compact Hausdorff Space

{{TFAE|def = Locally Compact Hausdorff Space}} Let $T = \struct{S, \tau}$ be a Hausdorff topological space.


Neighborhood in Compact Hausdorff Space Contains Compact Neighborhood

Let $X$ be a compact Hausdorff topological space. Let $x\in X$. Let $U$ be a neighborhood of $x$. Then $U$ contains a compact neighborhood of $x$.


Are All Perfect Numbers Even?/Progress/Form

An odd perfect number $n$ is of the form: :$n = p^a q^b r^c \cdots$ where: :$p, q, r, \ldots$ are prime numbers of the form $4 k + 1$ for some $k \in \Z_{>0}$ :$a$ is also of the form $4 k + 1$ for some $k \in \Z_{>0}$ :$b, c, \ldots$ are all even.


Image of Idempotent and Directed Suprema Preserving Mapping is Complete Lattice

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Let $f:S \to S$ be a mapping that is idempotent and preserves directed suprema. Let $F = \struct {f \sqbrk S, \precsim}$ be an ordered subset of $L$. Then $F$ inherits directed suprema and is complete lattice.


Are All Perfect Numbers Even?/Progress/Minimum Size

It had been established by $1986$ that an odd perfect number, if one were to exist, would have over $200$ digits. By $1997$ that lower bound had been raised to $300$ digits. By $2012$ that lower bound had been raised again to $1500$ digits.


Are All Perfect Numbers Even?/Progress/Prime Factors

An odd perfect number has: :at least $8$ distinct prime factors :at least $11$ distinct prime factors if $3$ is not one of them :at least $101$ prime factors (not necessarily distinct) :its greatest prime factor is greater than $1 \, 000 \, 000$ :its second largest prime factor is greater than $1000$ :at least one of the prime powers factoring it is greater than $10^{62}$ :if less than $10^{9118}$ then it is divisible by the $6$th power of some prime.


Prime-Generating Quadratics of form 2 a squared plus p/3

The quadratic form: :$2 a^2 + 3$ yields prime numbers for $a = 0, 1, 2$.


Prime-Generating Quadratics of form 2 a squared plus p/5

The quadratic form: :$2 a^2 + 5$ yields prime numbers for $a = 0, 1, \ldots, 4$.


Prime-Generating Quadratics of form 2 a squared plus p/11

The quadratic form: :$2 a^2 + 11$ yields prime numbers for $a = 0, 1, \ldots, 10$.


Prime-Generating Quadratics of form 2 a squared plus p/29

The quadratic form: :$2 a^2 + 29$ yields prime numbers for $a = 0, 1, \ldots, 28$.


Directed Suprema Preserving Mapping is Increasing

Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice. Let $f:S \to S$ be a mapping that preserves directed suprema. Then $f$ is an increasing mapping.


Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma 1

Let the function $\map h x$ satisfy the equation: :$-\map {\dfrac \d {\d x} } {P h'} + Q h = 0$ Let $\map h x$ have the boundary conditions: :$\map h a = \map h b = 0$ Then: :$\displaystyle \int_a^b \paren {P h'^2 + Q h^2} \rd x = 0$


Sequence of 4 Consecutive Square-Free Triplets

The following sets of $4$ consecutive triplets of integers, with one integer between each triplet, are square-free: :$29, 30, 31; 33, 34, 35; 37, 38, 39; 41, 42, 43$ :$101, 102, 103; 105, 106, 107; 109, 110, 111; 113, 114, 115$


Sequence of Prime Primorial minus 1

For prime $p$, let $p \#$ denote the $p$th primorial, defined in the sense that $p \#$ is the product of all primes less than or equal to $p$. The sequence $\left\langle{p}\right\rangle$ such that $p \# - 1$ is prime begins: :$3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, \ldots$ {{OEIS|A006794}}


Image of Pair under Mapping

Let $S, T$ be sets. Let $f: S \to T$ be a mapping. Then: :$\forall x, y \in S: f \sqbrk {\set {x, y} } = \set {\map f x, \map f y}$


Image under Increasing Mapping equal to Special Set is Complete Lattice

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Let $f: S \to S$ be an increasing mapping. Let $P = \struct {M, \precsim}$ be an ordered subset of $L$ such that :$M = \set {x \in S: x = \map f x}$ Then $P$ is complete lattice.


Hensel's Lemma/First Form

Let $p$ be a prime number. Let $k > 0$ be a positive integer. Let $f \left({X}\right) \in \Z \left[{X}\right]$ be a polynomial. Let $x_k \in \Z$ such that: :$f \left({x_k}\right) \equiv 0 \pmod {p^k}$ :$f' \left({x_k}\right) \not \equiv 0 \pmod p$ Then for every integer $l \ge 0$, there exists an integer $x_{k + l}$ such that: :$f \left({x_{k + l} }\right) \equiv 0 \pmod {p^{k + l} }$ :$x_{k + l}\equiv x_k \pmod {p^k}$ and any two integers satisfying these congruences are congruent modulo $p^{k + l}$. Moreover, for all $l\geq0$ and any solutions $x_{k + l}$ and $x_{k + l + 1}$: :$x_{k + l + 1} \equiv x_{k + l} - \dfrac {f \left({x_{k + l} }\right)} {f' \left({x_{k + l} }\right)} \pmod {p^{k + l + 1} }$ :$x_{k + l + 1} \equiv x_{k + l} \pmod {p^{k + l} }$


Sigma Function of Power of 2

Let $n \in \Z_{>0}$ be a power of $2$. Then: :$\map \sigma n = 2 n - 1$


Equivalence of Definitions of Saturation Under Equivalence Relation

Let $\sim$ be an equivalence relation on a set $S$. Let $T \subset S$ be a subset. {{TFAE|def = Saturation (Equivalence Relation)|view = saturation}}


Equivalence of Definitions of Saturated Set Under Equivalence Relation

Let $\sim$ be an equivalence relation on a set $S$. Let $T\subset S$ be a subset. {{TFAE|def = Saturated Set (Equivalence Relation)|view = saturated set|context = Equivalence Relation}}


Compact Subset is Bounded Below Join Semilattice

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $P = \left({K\left({L}\right), \precsim}\right)$ be an ordered subset of $L$, where $K\left({L}\right)$ denotes the compact subset of $L$. Then $P$ is a bounded below join semilattice.


Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma 2

Let $\map h x : \closedint a b \to \R$ be continuously differentiable $\forall x \in \closedint a b$. Suppose the function $\map h x$ satisfies the equation: :$-\map {\dfrac \d {\d x} } {\paren {t P + \paren {1 - t} } h'} + t Q h = 0$ subject to the boundary conditions: :$\map h {a, t} = \map h {b, t} = 0$ Then: :$\displaystyle \int_a^b \paren {\paren {P h'^2 + Q h^2} t + \paren {1 - t} h'^2} \rd x = 0$


Jacobi's Necessary Condition

Let $J$ be a functional, such that: :$J \sqbrk y = \ds \int_a^b \map F {x, y, y'} \rd x$ Let $\map y x$ correspond to the minimum of $J$. Let: :$F_{y'y'}>0$ along $\map y x$. Then the open interval $\openint a b$ contains no points conjugate to $a$.


Stolz-Cesàro Theorem

Let $\sequence {a_n}$ be a sequence. {{explain|Domain of $\sequence {a_n}$ -- $\R$ presumably but could it be $\C$?}} Let $\sequence {b_n}$ be a sequence of (strictly) positive real numbers such that: :$\displaystyle \sum_{i \mathop = 0}^\infty b_n = \infty$ If: :$\displaystyle \lim_{n \mathop \to \infty} \dfrac {a_n} {b_n} = L \in \R$ then also: :$\displaystyle \lim_{n \mathop \to \infty} \dfrac {a_1 + a_2 + \cdots + a_n} {b_1 + b_2 + \cdots + b_n} = L$


Bottom is Compact

Let $L$ be a bounded below ordered set. Then $\bot$ is a compact element where $\bot$ is the smallest element in $L$.


Saturation Under Equivalence Relation in Terms of Graph

Let $\RR \subset S \times S$ be an equivalence relation on a set $S$. Let $\pr_1, \pr_2 : S \times S \to S$ denote the projections. Let $T\subset S$ be a subset. Let $\overline T$ denote its saturation. Then the following hold: :$\overline T = \map {\pr_1} {\RR \cap \map {\pr_2^{-1} } T}$ :$\overline T = \map {\pr_2} {\RR \cap \map {\pr_1^{-1} } T}$


Nonnegative Quadratic Functional implies no Interior Conjugate Points

If the quadratic functional  :$\ds \int_a^b \paren {P h'^2 + Q h^2} \rd x$ where: :$\forall x \in \closedint a b: \map P x > 0$ is nonnegative for all $\map h x$: :$\map h a = \map h b = 0$ then the closed interval $\closedint a b$ contains no inside points conjugate to $a$. In other words, the open interval $\openint a b$ contains no points conjugate to $a$. {{explain|Rewrite the above so it makes better sense. For example, should the "nonnegative" comment be above the condition on $\map P x$?}}


Projection of Subset is Open iff Saturation is Open

Let $\sim$ be an equivalence relation on a topological space $X$. Let $X / \sim$ be the quotient space. Let $p$ denote the quotient mapping. Let $U \subset X$. Then the following are equivalent: :$\map p U$ is open in $X / \sim$ :The saturation of $U$ is open in $X$


Multiple of 6 is Semiperfect

Let $n \in \Z_{>0}$ be a multiple of $6$. Then $n$ is semiperfect.


Open Projection and Closed Graph Implies Quotient is Hausdorff

Let $\mathcal R\subset X\times X$ be an equivalence relation on a topological space $X$. Let $X/\mathcal R$ be the quotient space. Let $p$ denote the quotient mapping. Let: :$\mathcal R$ be closed in $X\times X$ :$p$ be open Then $X/\mathcal R$ is Hausdorff.


Subgroup is Closed iff Quotient is Hausdorff

Let $G$ be a topological group. Let $H \le G$ be a subgroup. Let $G / H$ be their quotient. Then the following are equivalent: :$H$ is closed in $G$ :$G / H$ is Hausdorff


Higher Homotopy Groups are Abelian

Let $T = \left({S, \tau}\right)$ be a topological space. Let $x_0 \in S$. Let $n \ge 2$ be a integer. Let $\pi_n \left({T, x_0}\right)$ be the $n$th homotopy group with base point $x_0$. Then $\pi_n \left({T, x_0}\right)$ is abelian.


Multiple of Semiperfect Number is Semiperfect

Let $n \in \Z_{>0}$ be a semiperfect number. Let $k \in \Z_{>0}$ be a (strictly) positive integer. Then $k n$ is also a semiperfect number.


Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset

Let $L = \left({S, \preceq}\right)$ be an algebric lattice. Let $c:S \to S$ be a closure operator that preserves directed suprema. Then $c\left[{K\left({L}\right)}\right] \subseteq K\left({\left({c\left[{S}\right], \precsim}\right)}\right)$ where :$K\left({L}\right)$ denotes the compact subset of $L$, :$c\left[{S}\right]$ denotes the image of $S$ under $c$, :$\mathord\precsim = \mathord\preceq \cap \left({c\left[{S}\right] \times c\left[{S}\right]}\right)$


Integers such that all Coprime and Less are Prime

The following positive integers have the property that all positive integers less than and coprime to it, excluding $1$, are prime: :$1, 2, 3, 4, 6, 8, 12, 18, 24, 30$ {{OEIS|A048597}} There are no other positive integers with this property.


Separable Extension is Contained in Galois Extension

Let $E/F$ be a separable finite field extension. Then there exists a finite field extension $L/E$ such that $L/F$ is Galois.


Galois Extension is Galois over Intermediate Field

Let $L / F$ be a Galois Extension. Let $E$ be an intermediate field. Then $L / E$ is Galois.


Frobenius Endomorphism on Field is Injective

Let $p$ be a prime number. Let $F$ be a field of characteristic $p$. Then the Frobenius endomorphism $\Frob: F \to F$ is injective.


Galois Field is Perfect

Let $\GF$ be a Galois field. Then $\GF$ is perfect.


Algebraically Closed Field is Perfect

Let $F$ be an algebraically closed field. Then $F$ is perfect.


Area of Smallest Rectangle accommodating Re-Entrant Knight's Tour

The area of the smallest rectangular chessboard on which a re-entrant knight's tour is possible is $30$ squares. This can be configured either as a $5 \times 6$ chessboard or a $3 \times 10$ chessboard.


Area of Smallest Square accommodating Re-Entrant Knight's Tour

The area of the smallest square chessboard on which a re-entrant knight's tour is possible is $6 \times 6 = 36$ squares.


Subgroup of Index Least Prime Divisor is Normal

Let $G$ be a finite group of order $n>1$. Let $p$ be the least prime divisor of $n$. Let $H$ be a subgroup of index $p$. Then $H$ is normal.


Alternating Group is Simple

Let $n \ge 5$ be an integer. Then the $n$th alternating group $A_n$ is simple.


Universal Property of Quotient Ring

Let $R,S$ be commutative rings. Let $I\trianglelefteq R$ be an ideal. Let $\pi : R\to R/I$ be the projection. Let $f:R\to S$ be a ring homomorphism with $f(I)=0$. Then there exists a unique ring homomorphism $\overline f:R/I \to S$ such that $f = \overline f \circ \pi$. :$\xymatrix{ R \ar[d]_\pi \ar[r]^{\forall f} & S\\ R/I \ar[ru]_{\exists ! \bar f} }$


Universal Property of Quotient Space

Let $X$ and $Y$ be topological spaces. Let $\sim$ be an equivalence relation on $X$. Let $\pi : X \to X/\sim$ be the quotient mapping. Let $f : X \to Y$ be continuous and $\sim$-invariant. Then there exists a unique continuous map $\overline f : X/\!\sim \to Y$ such that $f = \overline f \circ \pi$.


Universal Property of Quotient Group

Let $G,H$ be a groups. Let $N\trianglelefteq G$ be an normal subgroup. Let $\pi : G\to G/N$ be the projection. Let $f:G\to H$ be a group homomorphism with $N\subset\ker f$. Then there exists a unique group homomorphism $\overline f:G/N \to H$ such that $f = \overline f \circ \pi$. $\xymatrix{ G \ar[d]^\pi \ar[r]^{\forall f} & H\\ G/N \ar[ru]_{\exists ! \bar f} }$


Sum of Successive Powers in 2 ways

$31$ and $8191$ can be expressed as the sum of successive powers starting from $1$ in in $2$ different ways.


Period of Reciprocal of 31 is of Odd Length

:$\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$ {{OEIS|A021035}}


Properties of Periodic Part of Reciprocal of 31

We have that the decimal expansion of the reciprocal of $31$ is: {{:Period of Reciprocal of 31 is of Odd Length}} Then: {{begin-eqn}} {{eqn | l = 032258 \times 2       | r = 64 \, 516 }} {{eqn | l = 032258 \times 4       | r = 129 \, 032 }} {{eqn | l = 032258 \times 5       | r = 161 \, 290 }} {{eqn | l = 032258 \times 7       | r = 225 \, 806 }} {{eqn | l = 032258 \times 8       | r = 258 \, 064 }} {{eqn | l = 032258 \times 9       | r = 290 \, 322 }} {{eqn | l = 032258 \times 14       | r = 451 \, 612 }} {{eqn | l = 032258 \times 16       | r = 516 \, 128 }} {{eqn | l = 032258 \times 18       | r = 580 \, 644 }} {{eqn | l = 032258 \times 19       | r = 612 \, 902 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 03225 + 80645 + 16129       | r = 99 \, 999 }} {{eqn | l = 032 + 258 + 065 + 416 + 129       | r = 900 }} {{end-eqn}}


Sequence of Prime Primorial plus 1

For prime $p$, let $p \#$ denote the $p$th primorial, defined in the sense that $p \#$ is the product of all primes less than or equal to $p$. The sequence $\left\langle{p}\right\rangle$ such that $p \# + 1$ is prime begins: :$2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113, \ldots$ {{OEIS|A005234}}


Smallest Adjacent Happy Numbers

The smallest adjacent happy numbers are $31$ and $32$.


Increasing Mapping Preserves Lower Bounds

Let $L = \left({S, \preceq}\right)$, $L' = \left({S', \preceq'}\right)$ be ordered sets. Let $f:S \to S'$ be an increasing mapping. Let $x \in S$, $X \subseteq S$ such that :$x$ is lower bound for $X$. Then $f \left({x}\right)$ is lower bound for $f \left[{X}\right]$.


Order Isomorphism Preserves Lower Bounds

Let $L = \left({S, \preceq}\right)$, $L' = \left({S', \preceq'}\right)$ be ordered sets. Let $f: S \to S'$ be an order isomorphism between $L$ and $L'$. Let $x \in S$, $X \subseteq S$. Then $x$ is lower bound for $X$ {{iff}} $f \left({x}\right)$ is lower bound for $f \left[{X}\right]$.


Order Embedding is Increasing Mapping

Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be ordered sets. Let $f:S_1 \to S_2$ be an order embedding. Then $f$ is increasing mapping.


Order Isomorphism Preserves Upper Bounds

Let $L = \struct {S, \preceq}$, $L' = \struct {S', \preceq'}$ be ordered sets. Let $f: S \to S'$ be an order isomorphism between $L$ and $L'$. Let $x \in S$, $X \subseteq S$. Then $x$ is an upper bound for $X$ {{iff}} $\map f x$ is an upper bound for $f \sqbrk X$.


Increasing Mapping Preserves Upper Bounds

Let $L = \left({S, \preceq}\right)$, $L' = \left({S', \preceq'}\right)$ be ordered sets. Let $f:S \to S'$ be an increasing mapping. Let $x \in S$, $X \subseteq S$ such that :$x$ is upper bound for $X$. Then $f\left({x}\right)$ is upper bound for $f\left[{X}\right]$.


Order Isomorphism Preserves Infima and Suprema

Let $L = \left({S, \preceq}\right)$, $L' = \left({S', \preceq'}\right)$ be ordered sets. Let $f:S \to S'$ be an order isomorphism between $L$ and $L'$. Then $f$ preserves infima and suprema.


Number of Regions by dividing Circle by Chords

Let $n$ points be marked on the circumference of a circle $C$. Let chords be drawn between each pair of these points. For each $n$, the maximum number $C \left({n}\right)$ of regions into which $C$ can be divided is as follows: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="right" style = "padding: 2px 10px" | $C \left({n}\right)$  |- | align="right" style = "padding: 2px 10px" | $1$ | align="right" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $2$ | align="right" style = "padding: 2px 10px" | $2$ |- | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $4$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $8$ |- | align="right" style = "padding: 2px 10px" | $5$ | align="right" style = "padding: 2px 10px" | $16$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="right" style = "padding: 2px 10px" | $31$ |- | align="right" style = "padding: 2px 10px" | $7$ | align="right" style = "padding: 2px 10px" | $57$ |- | align="right" style = "padding: 2px 10px" | $8$ | align="right" style = "padding: 2px 10px" | $99$ |- | align="right" style = "padding: 2px 10px" | $9$ | align="right" style = "padding: 2px 10px" | $163$ |- | align="right" style = "padding: 2px 10px" | $10$ | align="right" style = "padding: 2px 10px" | $256$ |} {{OEIS|A000127}} {{ProofWanted|There is an equation which defines this -- I think it can be found in Graham, Knuth & Patashnik but that's just too far for me to reach at the moment without me getting out of this chair.}}


Prime Factors of 2^64 - 1

The prime decomposition of $2^{64} - 1$ is given by: :$2^{64} - 1 = 3 \times 5 \times 17 \times 257 \times 641 \times 65 \, 537 \times 6 \, 700 \, 417$


Prime Decomposition of 5th Fermat Number

The prime decomposition of the $5$th Fermat number is given by: {{begin-eqn}} {{eqn | l = 2^{\paren {2^5} } + 1       | r = 4 \, 294 \, 967 \, 297       | c = Sequence of Fermat Numbers }} {{eqn | r = 641 \times 6 \, 700 \, 417       | c =  }} {{eqn | r = \paren {5 \times 2^7 + 1} \times \paren {3 \times 17449 \times 2^7 + 1}       | c =  }} {{end-eqn}}


Mapping Assigning to Element Its Compact Closure Preserves Infima and Directed Suprema

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below algebraic lattice. Let $C = \left({K\left({L}\right), \preceq'}\right)$ be an ordered subset of $L$ where $K\left({L}\right)$ denotes the compact subset of $L$. Let $P = \left({\mathcal P\left({K\left({L}\right)}\right), \precsim}\right)$ be an inclusion ordered set of power set of $K\left({L}\right)$. Then there exists $f:S \to \mathcal P\left({K\left({L}\right)}\right)$ such that $f$ preserves infima and directed suprema and is an injection and $\forall x \in S: f\left({x}\right) = x^{\mathrm{compact} }$ where $x^{\mathrm{compact} }$ denotes the compact closure of $x$.


Power of 2 is Difference between Two Powers

Let $n \in \Z_{>0}$ be a power of $2$. Then $n$ is the difference between powers of two positive integers greater than or equal to $2$. {{questionable|This is so trivial I wonder whether something got lost in translation.}}


Smallest Sequence of Three Consecutive Semiprimes

The smallest triple of consecutive semiprimes is: :$33, 34, 35$


Integers not Sum of Distinct Triangular Numbers

The sequence of integers which cannot be expressed as the sum of distinct triangular numbers is: :$2, 5, 8, 12, 23, 33$ {{OEIS|A053614}}


Powers of 2 and 5 without Zeroes

The following $n \in \Z$ are such that both $2^n$ and $5^n$ have no zeroes in their decimal representation: :$0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 33$ {{OEIS|A007496}}


Vector Cross Product Distributes over Addition

The vector cross product is distributive over addition. That is, in general: :$\mathbf a \times \paren {\mathbf b + \mathbf c} = \paren {\mathbf a \times \mathbf b} + \paren {\mathbf a \times \mathbf c}$ for $\mathbf a, \mathbf b, \mathbf c \in \R^3$.


Jacobi's Equation is Variational Equation of Euler's Equation

The Variational equation of Euler's equation is Jacobi's equation.


Palindromes in Base 10 and Base 2

The following $n \in \Z$ are palindromic in both decimal and binary: :$0, 1, 3, 5, 7, 9, 33, 99, 313, 585, 717, 7447, 9009, 15 \, 351, 32 \, 223, 39 \, 993, \ldots$ {{OEIS|A007632}}


Bottom in Compact Closure

Let $L = \left({S, \preceq}\right)$ be a bounded below ordered set. Let $x \in S$. Then $\bot \in x^{\mathrm{compact} }$ where $\bot$ denotes the smallest element in $L$, :$ x^{\mathrm{compact} }$ denotes the compact closure of $x$.


Product of Two Distinct Primes has 4 Positive Divisors

Let $n \in \Z_{>0}$ be a positive integer which is the product of $2$ distinct primes. Then $n$ has exactly $4$ positive divisors.


Product of Two Distinct Primes is Multiplicatively Perfect

Let $n \in \Z_{>0}$ be a positive integer which is the product of $2$ distinct primes. Then $n$ is multiplicatively perfect.


Cube of Prime has 4 Positive Divisors

Let $n \in \Z_{>0}$ be a positive integer which is the cube of a prime number. Then $n$ has exactly $4$ positive divisors.


Cube of Prime is Multiplicatively Perfect

Let $n \in \Z_{>0}$ be a positive integer which is the cube of a prime number. Then $n$ is multiplicatively perfect.


Integers Differing by 2 with Same Sigma

Let $\sigma: \Z_{>0} \to \Z_{>0}$ be the $\sigma$ function, defined on the strictly positive integers. The equation: :$\sigma \left({n}\right) = \sigma \left({n + 2}\right)$ is satisfied by integers in the sequence: :$33, 54, 284, 366, 834, 848, 918, 1240, 1504, 2910, 2913, 3304, \ldots$ {{OEIS|A007373}}


Compact Closure is Directed

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $x \in S$. Then $x^{\mathrm{compact} }$ is directed where $x^{\mathrm{compact} }$ denotes the compact closure of $x$.


Sum of 2 Lucky Numbers in 4 Ways

The number $34$ is the smallest positive integer to be the sum of $2$ lucky numbers in $4$ different ways.


35 Hexominoes

There exist $35$ distinct free hexominoes: :600px


Fixed Point of Permutation is Fixed Point of Power

Let $S_n$ denote the symmetric group on $n$ letters. Let $\sigma \in S_n$. Let $i \in \Fix \sigma$, where $\Fix \sigma$ denotes the set of fixed elements of $\sigma$. Then for all $m \in \Z$: :$i \in \Fix {\sigma^m}$


Hexominoes cannot form Rectangle

While there are a total of $210$ squares in a complete set of hexominoes, it is impossible to build them into a rectangle of side lengths $a$ and $b$ where $a \times b = 210$.


Number of Heptominoes

There exist $108$ distinct free heptominoes, one of which has a hole: :600px


369 Octominoes

There exist $369$ distinct free octominoes, $6$ of which have a hole: :600px


1285 9-ominoes

There exist $1285$ distinct free $9$-ominoes, $37$ of which have a hole: :600px


Mapping Assigning to Element Its Compact Closure is Order Isomorphism

Let $L = \struct {S, \vee, \preceq}$ be a bounded below algebraic join semilattice. Let $C = \struct {\map K L, \preceq'}$ be an ordered subset of $L$ where $\map K L$ denotes the compact subset of $L$. Let $I = \struct {\map {\mathit {Ids} } C, \precsim}$ be an inclusion ordered set where $\map {\mathit {Ids} } C$ denotes the set of all ideals in $C$. Let $f: S \to \map {\mathit {Ids} } C$ be a mapping such that :$\forall x \in S: \map f x = x^{\mathrm {compact} }$ where $x^{\mathrm {compact} }$ denotes the compact closure of $x$. Thus $f$ is order isomorphism between $L$ and $I$.


Maximum Length of Non-Crossing Knight's Move

The maximum length of a non-crossing knight's tour on a standard chessboard is $35$ moves.


Prime Factors of 35, 36, 4734 and 4735

The integers: :$35, 4374$ have the same prime factors between them as the integers: :$36, 4375$


Equivalence Class of Fixed Element

Let $S_n$ denote the symmetric group on $n$ letters. Let $\sigma \in S_n$. Let $\RR_\sigma$ denote the equivalence defined in Permutation Induces Equivalence Relation. Let $i \in \N^*_{\le n}$. Then: :$i \in \Fix \sigma$ {{iff}} $\eqclass i {\RR_\sigma} = \set i$ where: :$\eqclass i {\RR_\sigma}$ denotes the equivalence class of $i$ under $\RR_\sigma$ :$\Fix \sigma$ denotes the set of fixed elements of $\sigma$.


Compact Closure is Increasing

Let $\left({S, \preceq}\right)$ be an ordered set. Let $x, y \in S$ such that :$x \preceq y$ Then $x^{\mathrm{compact} } \subseteq y^{\mathrm{compact} }$ where $x^{\mathrm{compact} }$ denotes the compact closure of $x$.


Sum of Entries in Row of Pascal's Triangle

The sum of all the entries in the $n$th row of Pascal's triangle is equal to $2^n$.


Columns of Pascal's Triangle contain Simplicial Polytopic Numbers

The columns of Pascal's triangle contain the simplicial polytopic numbers: : Column $0$: repeated instances of number $1$ : Column $1$: the (strictly) positive integers : Column $2$: the triangular numbers : Column $3$: the tetrahedral numbers : Column $4$: the pentatope numbers and so on.


Sum of Entries in Lesser Diagonal of Pascal's Triangle equal Fibonacci Number

The sum of the entries in the $n$th lesser diagonal of Pascal's triangle equals the $n + 1$th Fibonacci number.


Extension of Infima Preserving Mapping to Complete Lattice Preserves Infima

Let $L_1 = \struct {S_1, \preceq_1}$, $L_2 = \struct {S_2, \preceq_2}$ be ordered sets. Let $L_3 = \struct {S_3, \preceq_3}$ be a complete lattice. Suppose that. :$L_2$ is an infima inheriting ordered subset of $L_3$. Let $f: S_1 \to S_2$ be a mapping such that: :$f$ preserves infima. Then $f: S_1 \to S_3$ preserves infima.


Extension of Directed Suprema Preserving Mapping to Complete Lattice Preserves Directed Suprema

Let $L_1 = \left({S_1, \preceq_1}\right)$, $L_2 = \left({S_2, \preceq_2}\right)$ be ordered sets. Let $L_3 = \left({S_3, \preceq_3}\right)$ be a complete lattice. Suppose that :$L_2$ is directed suprema inheriting ordered subset of $L_3$. Let $f:S_1 \to S_2$ be a mapping such that :$f$ preserves directed suprema. Then $f:S_1 \to S_3$ preserves directed suprema.


Reciprocal as Summation of Binomial Coefficients of Reciprocals

:$\forall n \in \Z_{>0}: \dfrac 1 n = \displaystyle \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \dbinom {n - 1} k \dfrac 1 {k + 1}$ where $\dbinom {n - 1} k$ denotes a binomial coefficient. That is, for example: {{begin-eqn}} {{eqn | l = \dfrac 1 1       | r = 1 }} {{eqn | l = \dfrac 1 2       | r = 1 - \dfrac 1 2 }} {{eqn | l = \dfrac 1 3       | r = 1 - 2 \times \dfrac 1 2 + \dfrac 1 3 }} {{eqn | l = \dfrac 1 4       | r = 1 - 3 \times \dfrac 1 2 + 3 \times \dfrac 1 3 - \dfrac 1 4 }} {{eqn | l = \dfrac 1 5       | r = 1 - 4 \times \dfrac 1 2 + 6 \times \dfrac 1 3 - 4 \times \dfrac 1 4 + \dfrac 1 5 }} {{end-eqn}}


Complement of Irreducible Topological Subset is Prime Element

Let $T = \left({S, \tau}\right)$ be a topological space. Let $X$ be an irreducible subset of $S$ such that :$\complement_S\left({X}\right) \in \tau$ Let $L = \left({\tau, \preceq}\right)$ be an inclusion ordered set of topology $\tau$. Then $\complement_S\left({X}\right)$ is prime element in $L$.


Element of Leibniz Harmonic Triangle as Sum of Elements on Diagonal from Below

Consider the Leibniz harmonic triangle: {{:Definition:Leibniz Harmonic Triangle}} Let $\tuple {n, m}$ be the element in the $n$th row and $m$th column. Then: :$\tuple {n, m} = \displaystyle \sum_{k \mathop \ge 0} \tuple {n + 1 + k, m + k}$


Element of Pascal's Triangle is Sum of Diagonal or Column starting above it going Upwards

Consider Pascal's triangle: :{{Definition:Pascal's Triangle}} Let $\tuple {n, m}$ be the element in the $n$th row and $m$th column. Then: :$\tuple {n, m} = \displaystyle \sum_{k \mathop \ge 0} \tuple {n - k - 1, m - 1}$ and: :$\tuple {n, m} = \displaystyle \sum_{k \mathop \ge 0} \tuple {n - k, m - k - 1}$


Power of Moved Element is Moved

Let $S_n$ denote the symmetric group on $n$ letters. Let $\sigma \in S_n$. Then for all $m \in \Z$: :$i \notin \Fix \sigma \implies \sigma^m \paren i \notin \Fix \sigma$ where $\Fix \sigma$ denotes the set of fixed elements of $\sigma$.


Integer both Square and Triangular

Consider a pair of (strictly) positive integers $a$ and $b$ such that $a < b$. Then: :$\dfrac a b$ is a best rational approximation to the square root of $2$ {{iff}}: :$\paren {a b}^2$ is both square and triangular.


Relative Complement is Decreasing

Let $X, Y, S$ be set such that $X \subseteq Y \subseteq S$ Then $\complement_S\left({X}\right) \supseteq \complement_S\left({Y}\right)$


Sum of Sequence of Cubes/Sequence

The sequence of integers which are the sum of the sequence of the first $n$ cubes begins: :$0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, \ldots$


Square Numbers which are Sigma values

The sequence of square numbers which are the $\sigma$ value of a (strictly) positive integer begins: :$1, 4, 36, 121, 144, 256, 324, 400, 576, 784, 900, 961, \ldots$ {{OEIS|A038688}}


Join and Meet in Inclusion Ordered Set of Topology

Let $T = \left({S, \tau}\right)$ be a topological space. Let $L = \left({\tau, \preceq}\right)$ be an inclusion ordered set of $\tau$. Let $X, Y \in \tau$. Then $X \vee Y = X \cup Y$ and $X \wedge Y = X \cap Y$


Join in Inclusion Ordered Set

Let $P = \left({X, \subseteq}\right)$ be an inclusion ordered set. Let $A, B \in X$ such that :$A \cup B \in X$ Then $A \vee B = A \cup B$


2-Digit Numbers divisible by both Product and Sum of Digits

The $2$-digit positive integers which are divisible by both the sum and product of their digits are: :$12, 24, 36$


Sequence of 7 Consecutive Integers including Multiple of Prime greater than 41

Every sequence of $7$ consecutive integers greater than $36$ includes a multiple of a prime number greater than $41$.


Cyclic Permutation of 3-Digit Multiple of 37

Let $n$ be a $3$-digit multiple of $37$. Let $m$ be an integer formed by cyclically permuting the digits of $n$. Then $m$ is also a multiple of $37$.


Definition:Juggler Sequence

Let $m \in \Z_{\ge 0}$ be a positive integer. The '''juggler sequence on $m$''' is defined recursively as: :$J_m \left({n}\right) = \begin{cases} m & : n = 0 \\ \left\lfloor{\sqrt {J_m \left({n - 1}\right)} }\right\rfloor & : n \text{ even} \\ \left\lfloor{\sqrt {\left({J_m \left({n - 1}\right)}\right)^3} }\right\rfloor & : n \text{ odd} \end{cases}$ where: :$\left\lfloor{x}\right\rfloor$ denotes the floor of $x$ :$\sqrt x$ denotes the positive square root of $x$.


Period of Reciprocal of 37 has Length 3

$37$ is the $2$nd positive integer the decimal expansion of whose reciprocal has a period of $3$: :$\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$ {{OEIS|A021041}}


Centered Hexagonal Number as Sum of Triangular Numbers

Let $C_n$ be the $n$th centered hexagonal number. Then: :$C_n = 6 T_{n - 1} + 1$ where $T_{n - 1}$ denotes the $n - 1$th triangular number.


Closed Form for Centered Hexagonal Numbers

Let $C_n$ be the $n$th centered hexagonal number. Then: :$C_n = 3 n \paren {n - 1} + 1$


Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers

The positive even integers which cannot be expressed as the sum of $2$ composite odd numbers are: :$2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 26, 28, 32, 38$ {{OEIS|A118081}}


Upper Closure is Compact in Topological Lattice

Let $L = \left({S, \preceq, \tau}\right)$ be a topological lattice. Suppose that :for every subset $X$ of $S$ if $X$ is open, then $X$ is upper. Let $x \in S$. Then $x^\succeq$ is compact where $x^\succeq$ denotes the upper closure of $x$.


Positive Real Numbers Closed under Multiplication

The set $\R_{>0}$ of strictly positive real numbers is closed under multiplication: :$\forall a, b \in \R_{> 0}: a \times b \in \R_{> 0}$


Positive Power Function on Non-negative Reals is Strictly Increasing

Let $a \in \Q_{> 0}$ be a strictly positive rational number. Let $f_a: \R_{\ge 0} \to \R$ be the real function defined as: :$\map {f_a} x = x^a$ Then $f_a$ is strictly increasing.


Smallest Consecutive Even Numbers such that Added to Divisor Count are Equal

$30$ is the smallest positive even integer $n$ with the property: {{begin-eqn}} {{eqn | l = n + \map \tau n       | r = m       | c =  }} {{eqn | l = \paren {n + 2} + \map \tau {n + 2}       | r = m       | c =  }} {{eqn | l = \paren {n + 4} + \map \tau {n + 4}       | r = m       | c =  }} {{end-eqn}} where: :$m \in \Z_{>0}$ is some positive integer :$\map \tau n$ is the $\tau$ function: the number of divisors of $n$. In this case, where $n = 30$, we have that $m = 38$.


2-Digit Positive Integer equals Product plus Sum of Digits iff ends in 9

Let $n$ be a $2$-digit positive integer. Then: :$n$ equals the sum added to the product of its digits {{iff}}: :the last digit of $n$ is $9$.


Number of Convex Polygons from Complete Set of Hexiamonds

The number of convex polygons that can be assembled from the complete set of $12$ hexiamonds is $39$.


Infimum of Open Set is Way Below Element in Complete Scott Topological Lattice

Let $T = \left({S, \preceq, \tau}\right)$ be a complete topological lattice with Scott topology. Let $X$ be an open subset of $S$, Let $x \in X$. Then $\inf X \ll x$ where $\ll$ denotes the way below relation.


Letters of Names of Numbers in Alphabetical Order/English

When written in English, the only integer the letters of whose name appear in alphabetical order is: :$40$: '''forty'''


Letters of Names of Numbers in Alphabetical Order/French

When written in French, the only integers the letters of whose name appear in alphabetical order are: {{begin-eqn}} {{eqn | o =        | l = 2:       | c = '''deux''' }} {{eqn | o =        | l = 5:       | c = '''cinq''' }} {{eqn | o =        | l = 10:       | c = '''dix''' }} {{eqn | o =        | l = 100:       | c = '''cent''' }} {{end-eqn}}


Cyclic Permutations of 5-Digit Multiples of 41

Let $n$ be a multiple of $41$ with $5$ digits. Let $m$ be an integer formed by cyclically permuting the digits of $n$. Then $m$ is a multiple of $41$.


Prime-Generating Quadratic of form 2 x squared minus 1000 x minus 2609

The quadratic function: :$2 x^2 - 1000 x - 2609$ has $602$ prime values among its first $1000$ values. Some of those prime numbers are negative.


Scott Topology equals to Scott Sigma

Let $\left({T, \preceq, \tau}\right)$ be a up-complete topological lattice with Scott topology. Then $\tau = \sigma\left({\left({T, \preceq}\right)}\right)$ where $\sigma\left({L}\right)$ denotes the Scott sigma of $L$.


Complement of Lower Closure is Prime Element in Inclusion Ordered Set of Scott Sigma

Let $L = \struct {S, \preceq, \tau}$ be a complete Scott topological lattice. Let $D = \struct {\map \sigma L, \precsim}$ be an inclusion ordered set of the Scott sigma of $L$. Let $x \in S$. Then: :$\relcomp S {x^\preceq}$ is a prime element in $D$ and: :$\relcomp S {x^\preceq} \ne S$


Topological Closure of Singleton is Irreducible

Let $T = \left({S, \tau}\right)$ be a topological space. Let $x$ be a point of $T$. Then: : $\left\{ {x}\right\}^-$ is irreducible where $\left\{ {x}\right\}^-$ denotes the topological closure of $\left\{ {x}\right\}$.


Element equals to Supremum of Infima of Open Sets that Element Belongs implies Topological Lattice is Continuous

Let $L = \left({S, \preceq, \tau}\right)$ be a complete Scott topological lattice. Let :$\forall x \in S: x = \sup \left\{ {\inf X: x \in X \in \sigma\left({L}\right)}\right\}$ Then $L$ is continuous.


Proper Class is not Element of Class

Let $\mathrm P$ be a proper class. Then $\mathrm P$ is not an element of any class, that is: :$\neg \exists A : \mathrm P \in A$


Prime-Generating Quadratic of form x squared - 79 x + 1601

The quadratic function: :$x^2 - 79 x + 1601$ gives prime values for integer $x$ such that $0 \le x \le 79$. The primes generated are repeated once each.


Continuous Group Action is by Homeomorphisms

Let $G$ be a topological group acting continuously on a topological space $X$. Then $G$ acts by homeomorphisms.


Discrete Group Acts Continuously iff Acts by Homeomorphisms

Let $G$ be a discrete group acting on a topological space $X$. Then the following are equivalent: :$G$ acts continuously :$G$ acts by homeomorphisms


Euler Phi Function of 3

:$\map \phi 3 = 2$


Number of Distinct Parenthesizations on Word

Let $w_n$ denote an arbitrary word of $n$ elements. The number of distinct parenthesizations of $w_n$ is the Catalan number $C_{n - 1}$: :$C_{n - 1} = \dfrac 1 n \dbinom {2 \paren {n - 1} } {n - 1}$


Number of Paths on Graph along X-axis using Diagonal Steps

The number of different paths that can be taken on a Cartesian plane from the origin to $\tuple {2 n + 2, 0}$, using only diagonal steps, and never touching the $x$-axis except at the beginning and the end of the path, is the Catalan number $C_n$: :400px


Non-Integral Value of Göbel's Sequence

Consider Göbel's sequence defined recursively as: :$x_n = \begin{cases} 1 & : n = 0 \\ \ds \paren {1 + \sum_{k \mathop = 0}^{n - 1} {x_k}^2} / n & : n > 0 \end{cases}$ The smallest $n$ such that $x_n$ is not an integer is $43$.


Non-Integral Value of 3-Göbel Sequence

Consider the $3$-Göbel sequence defined recursively as: :$x_n = \begin {cases} 1 & : n = 0 \\ \displaystyle \paren {1 + \sum_{k \mathop = 0}^{n - 1} {x_k}^3} / n & : n > 0 \end {cases}$ The smallest $n$ such that $x_n$ is not an integer is $88$.


Best Approximation from Below to 1 as Sum of Minimal Number of Unit Fractions

:$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 7 + \dfrac 1 {43} = 1 -  \dfrac 1 {1806}$ This is the best approximation from below to $1$ as the minimal sum of unit fractions.


Cuboid with Integer Edges and Face Diagonals

The smallest cuboid whose edges and the diagonals of whose faces are all integers has edge lengths $44$, $117$ and $240$. Its space diagonal, however, is not an integer. {{expand|Add the definition of Definition:Euler Brick, and rewrite and rename as appropriate.}}


Largest n such that 1 to n can be Partitioned for no Element to be Sum of 2 Elements in Same Set

$44$ is the largest integer $n$ such that the set of integers from $1$ to $n$ can be partitioned into $4$ subsets such that no integer in any of these subsets is the sum of $2$ other integers in the same subset: :$\set {1, 3, 5, 15, 17, 19, 26, 28, 40, 42, 44}$ :$\set {2, 7, 8, 18, 21, 24, 27, 33, 37, 38, 43}$ :$\set {4, 6, 13, 20, 22, 23, 25, 30, 32, 39, 41}$ :$\set {9, 10, 11, 12, 14, 16, 29, 31, 34, 35, 36}$


Conjugate Transpose is Involution

Let $\mathbf A$ be a complex-valued matrix. Let $\mathbf A^*$ denote the Hermitian conjugate of $\mathbf A$. Then the operation of Hermitian conjugate is an involution: :$\paren {\mathbf A^*}^* = \mathbf A$


Number as Sum of Distinct Primes greater than 11

Every number greater than $45$ can be expressed as the sum of distinct primes greater than $11$.


Subclass of Set is Set

Let $A$ be a set. Let $\map \phi x$ be a condition in which $x$ is taken to be a set. Then there exists a set that consists of all of the elements of $A$ that satisfies this condition. In ZF, this result is known as the Axiom of Specification.


Hexagonal Number as 4 times Triangular Number plus n

Let $H_n$ be the $n$th hexagonal number. Then: :$H_n = 4 T_{n - 1} + n$ where $T_{n - 1}$ is the $n - 1$th triangular number.


Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Dependent on N Functions

Let $K$ be a (real) functional, such that: :$\ds K \sqbrk {\mathbf h} = \int_a^b \paren {\mathbf h' \mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$ where: :$\mathbf h$ is an $N$-dimensional vector :$\mathbf Q$ is a $N \times N$ matrix :$\mathbf P$ is a $N\times N$ symmetric positive definite matrix. Let $\closedint a b$ be such that it does not contain a point conjugate to $a$. Then: :$\forall \mathbf h: \map {\mathbf h} a = \map {\mathbf h} b = 0: K \sqbrk {\mathbf h} > 0$ {{iff}} the above holds. {{explain|"the above" -- reference the specific statement, there is plenty of "above" and it may not be immediately clear exactly what}}


Class is Proper iff Bijection from Class to Proper Class

Let $A$ be a class. Let $\mathrm P$ be a proper class. Then $A$ is proper if and only if there exists a bijection from $A$ to $\mathrm P$.


Infinite Product of Analytic Functions

Let $D \subset \C$ be an open connected set. Let $\left\langle{f_n}\right\rangle$ be a sequence of analytic functions $f_n: D \to \C$ that are not identically zero. Let $\displaystyle \sum_{n \mathop = 1}^\infty \left({f_n - 1}\right)$ converge locally uniformly absolutely on $D$. Then: :$(1): \quad f = \displaystyle \prod_{n \mathop = 1}^\infty f_n$ converges locally uniformly absolutely on $D$ :$(2): \quad f$ is analytic :$(3): \quad$ For each $z \in D$, $f_n \left({z}\right) = 0$ for finitely many $n \in \N$ :$(4): \quad$ For each $z \in D$, $\operatorname {mult}_z \left({f}\right) = \displaystyle \sum_{n \mathop = 1}^\infty \operatorname{mult}_z \left({f_n}\right)$ where $\operatorname{mult}$ denotes multiplicity.


Uniformly Continuous Function Preserves Uniform Convergence

Let $X$ be a set. Let $M$ and $N$ be metric spaces. Let $(g_n)$ be a sequence of mappings $g_n:X\to M$. Let $g_n$ converge uniformly to $g:X\to M$. Let $f:M\to N$ be uniformly continuous. Then $(f\circ g_n)$ converges uniformly to $f\circ g$.


Complex Exponential is Uniformly Continuous on Half-Planes

Let $a\in\R$. Then $\exp$ is uniformly continuous on the half-plane $\set {z \in \C : \map \Re z \le a}$.


Local Uniform Convergence Implies Compact Convergence

Let $X$ be a topological space. Let $M$ be a metric space. Let $(f_n)$ be a sequence of mappings $f_n : X\to M$. Let $f_n$ converge locally uniformly to $f:X\to M$. Then $f_n$ converges compactly to $f$.


Bounds for Complex Logarithm

Let $\ln$ denote the complex logarithm. Let $z \in \C$ with $\cmod z \le \dfrac 1 2$. Then: :$\dfrac 1 2 \cmod z \le \cmod {\map \ln {1 + z} } \le \dfrac 3 2 \cmod z$


Reciprocal of Absolutely Convergent Product is Absolutely Convergent

Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\sequence {1 + a_n}$ be a sequence of nonzero elements of $\mathbb K$. Let the infinite product $\ds \prod_{n \mathop = 1}^\infty \left({1 + a_n}\right)$ converge absolutely to $a \in \mathbb K \setminus \set 0$. Then $\ds \prod_{n \mathop = 1}^\infty \frac 1 {1 + a_n}$  converges absolutely to $1 / a$.


Equivalence of Definitions of Absolute Convergence of Product

Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\sequence{a_n}$ be a sequence in $\mathbb K$. {{TFAE|def = Absolute Convergence of Product}} === Definition 1 === {{:Definition:Absolute Convergence of Product/General Definition/Definition 1}} === Definition 2 === {{:Definition:Absolute Convergence of Product/General Definition/Definition 2}}


Factors in Convergent Product Converge to One

Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ be convergent. Then $a_n \to 1$.


Logarithm of Convergent Product of Real Numbers

The following are equivalent: :The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converges to $a \in \R_{\ne 0}$. :The series $\displaystyle \sum_{n \mathop = 1}^\infty \ln a_n$ converges to $\ln a$.


Factors in Absolutely Convergent Product Converge to One

Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be absolutely convergent. Then: :$a_n \to 0$


Logarithm of Divergent Product of Real Numbers

Let $\left\langle{a_n}\right\rangle$ be a sequence of strictly positive real numbers.


Logarithm of Infinite Product of Real Numbers

Let $(a_n)$ be a sequence of strictly positive real numbers.


Absolutely Convergent Product Does not Diverge to Zero

Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be absolutely convergent. Then it is not divergent to $0$.


Absolute Value of Absolutely Convergent Product is Absolutely Convergent

Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converge absolutely to $a\in\mathbb K$. Then $\displaystyle \prod_{n \mathop = 1}^\infty \norm{a_n}$ converges absolutely to $\norm{a}$.


Logarithm of Divergent Product of Real Numbers/Infinity

{{TFAE}}: :$(1): \quad$ The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $+\infty$. :$(2): \quad$ The series $\displaystyle \sum_{n \mathop = 1}^\infty \log a_n$ diverges to $+\infty$.


Product of Convergent Products is Convergent

Let $\struct {\mathbb K, \norm{\,\cdot\,} }$ be a valued field. Let $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converge to $a$. Let $\displaystyle \prod_{n \mathop = 1}^\infty b_n$ converge to $b$. Then $\displaystyle \prod_{n \mathop = 1}^\infty a_nb_n$ converges to $ab$.


Product of Convergent and Divergent Product is Divergent

Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field. Let $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ be convergent. Let $\displaystyle \prod_{n \mathop = 1}^\infty b_n$ be divergent. Then $\displaystyle \prod_{n \mathop = 1}^\infty a_n b_n$ is divergent.


Trimorphic Number is not necessarily Automorphic

Let $n \in \Z_{>0}$ be a trimorphic number. Then it is not necessarily the case that $n$ is also an automorphic number.


Reciprocal of 49 shows Powers of 2 in Decimal Expansion

The decimal expansion of the reciprocal of $49$ contains the powers of $2$: :$\dfrac 1 {49} = 0 \cdotp \dot 02040 \, 81632 \, 65306 \, 12244 \, 89795 \, 91836 \, 73469 \, 38775 \, 5 \dot 1$ {{OEIS|A007450}}


Numbers the Multiple of whose Reciprocal are Cyclic Permutations

Let $m \in \Z_{>0}$. Consider the reciprocal of $m$. Let $n \in \Z$ such that $1 \le n < m$. Then: :The digits in the decimal expansion of the rational number $\dfrac n m$ form a cyclic permutation of the digits in the decimal expansion of $\dfrac 1 m$ {{iff}}: :$(1): \quad m$ is the integer power of a prime number $p$ :$(2): \quad$ The period of recurrence of the decimal expansion of $1 / p$ is $p - 1$ :$(3): \quad p$ is not a divisor of $n$.


Uniform Product of Continuous Functions is Continuous

Let $X$ be a metric space. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\left\langle{f_n}\right\rangle$ be a sequence of bounded continuous mappings $f_n: X \to \mathbb K$. Let the product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge uniformly to $f$. Then $f$ is continuous.


Logarithm of Infinite Product of Complex Functions

Let $X$ be a weakly locally compact topological space. Let $\sequence {f_n}$ be a sequence of everywhere nonzero continuous mappings $f_n: X \to \C$. Then the following are equivalent: :$(1): \quad$ The product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converges locally uniformly to $f$. :$(2): \quad$ The series $\displaystyle \sum_{n \mathop = 1}^\infty \ln f_n$ converges locally uniformly to $\ln f + 2k\pi i$ for some mapping $k:K\to\Z$.


Logarithmic Derivative of Infinite Product of Analytic Functions

Let $D\subset\C$ be open. Let $(f_n)$ be a sequence of analytic functions $f_n:D\to\C$. Let none of the $f_n$ be identically zero on any open subset of $D$. Let the product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$. Then $\displaystyle \frac{f'}f = \sum_{n \mathop = 1}^\infty \frac{f_n'}{f_n}$ and the series converges locally uniformly in $D\setminus\{z\in D : f(z) = 0\}$.


Galois Connection implies Upper Adjoint is Surjection iff Lower Adjoint is Injection

Let $L = \struct {S, \preceq}$ and $R = \struct {T, \precsim}$ be ordered sets. Let $g: S \to T$ and $d: T \to S$ be mappings such that $\struct {g, d}$ is a Galois connection. Then $g$ is a surjection {{iff}} $d$ is an injection.


Derivative of Uniform Limit of Analytic Functions

Let $U$ be an open subset of $\C$. Let $\left\langle{f_n}\right\rangle_{n \mathop \in \N}$ be a sequence of analytic functions $f_n : U \to \C$. Let $\left\langle{f_n}\right\rangle$ converge locally uniformly to $f$ on $U$. Then the sequence $\left\langle{f_n'}\right\rangle_{n \mathop \in \N}$ converges locally uniformly to $f'$.


Derivative of Infinite Product of Analytic Functions

Let $D \subset \C$ be open. Let $\left\langle{f_n}\right\rangle$ be a sequence of analytic functions $f_n: D \to \C$. Let the product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$. Then: :$\displaystyle f' = \sum_{n \mathop = 1}^\infty f_n'\cdot \prod_{\substack{k \mathop = 1 \\ k\mathop \ne n} }^\infty f_k$ and the series converges locally uniformly in $D$.


Uniformly Convergent Sequence Multiplied with Function

Let $X$ be a set. Let $V$ be a normed vector space over $\mathbb K$. Let $\sequence {f_n}$ be a sequence of mappings $f_n: X \to V$. Let $\sequence {f_n}$ be uniformly convergent. Let $g: X \to \mathbb K$ be bounded. Then $\sequence {f_n g}$ is uniformly convergent.


Logarithmic Derivative of Product of Analytic Functions

Let $D \subset \C$ be open. Let $f, g: D \to \C$ be analytic. Let $z \in D$ with $f \left({z}\right) \ne 0 \ne g \left({z}\right)$. Then: :$\dfrac{\left({f g}\right)' \left({z}\right)} {\left({f g}\right) \left({z}\right)} = \dfrac{f' \left({z}\right)} {f \left({z}\right)} + \dfrac {g' \left({z}\right)} {g \left({z}\right)}$ {{explain|Link to a clarifying definition of what $\left({f g}\right)$ is -- presumably it's the pointwise product, but in the context there's nothing to say it can't be $\left({f \circ g}\right)$, that is, $f$ of $g$ of $z$.}}


Uniformly Convergent Sequence on Dense Subset

Let $X$ be a metric space. Let $Y \subset X$ be dense. Let $V$ be a Banach space. Let $\sequence {f_n}$ be a sequence of continuous mappings $f_n : X\to V$. Let $\sequence {f_n}$ be uniformly convergent on $Y$. Then $\sequence {f_n}$ is uniformly convergent on $X$.


Tail of Uniformly Convergent Product Converges Uniformly to One

Let $X$ be a compact topological space. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\sequence {f_n}$ be a sequence of continuous mappings $f_n: X \to \mathbb K$. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge uniformly on $X$. Then for all $N \in \N$, $\displaystyle \prod_{n \mathop = N}^\infty f_n$ converges uniformly and the sequence $\displaystyle \prod_{n \mathop = N}^\infty f_n$ converges uniformly to $1$.


Factors in Uniformly Convergent Product Converge Uniformly to One

Let $X$ be a set. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\left\langle{f_n}\right\rangle$ be a sequence of bounded mappings $f_n: X \to \mathbb K$. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge uniformly on $X$. Then $f_n$ converges uniformly to $1$.


Quintuplets of Consecutive Integers which are not Sigma Values

The elements of the following $5$-tuples of consecutive integers have the property that they are not values of the $\sigma$ function $\map \sigma n$ for any $n$: :$\tuple {49, 50, 51, 52, 53}$ :$\tuple {115, 116, 117, 118, 119}$ :$\tuple {145, 146, 147, 148, 149}$


Positive Integers which are not Sigma Values

The following positive integers are not the values of the $\sigma$ function $\sigma \left({n}\right)$ for any $n$: :$2, 5, 9, 10, 11, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 37, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 58, 59, 61, 64, 65, \ldots$ {{OEIS|A007369}}


Uniformly Absolutely Convergent Product is Uniformly Convergent

Let $X$ be a set. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\mathbb K$ be complete. Let $\left\langle{f_n}\right\rangle$ be a sequence of bounded mappings $f_n: X \to \mathbb K$. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge uniformly absolutely on $X$. Then it converges uniformly.


Uniformly Convergent Product Satisfies Uniform Cauchy Criterion

Let $X$ be a compact topological space. Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field. Let $\sequence {f_n}$ be a sequence of continuous mappings $f_n: X \to \mathbb K$. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge uniformly on $X$. Then $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ satisfies the uniform Cauchy condition for products.


Absolute Value of Uniformly Convergent Product

Let $X$ be a compact topological space. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\left\langle{f_n}\right\rangle$ be a sequence of continuous mappings $f_n: X \to \mathbb K$. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge uniformly to $f$. Then $\displaystyle \prod_{n \mathop = 1}^\infty \norm{f_n}$ converges uniformly to $ \norm{f}$.


Squares whose Digits can be Separated into 2 other Squares

The decimal representation of the following square numbers can be split into two parts which are each themselves square: {{begin-eqn}} {{eqn | l = 7^2       | r = 49       | c = $4 = 2^2$,       | cc= $9 = 3^2$ }} {{eqn | l = 13^2       | r = 169       | c = $16 = 4^2$,       | cc= $9 = 3^2$ }} {{eqn | l = 19^2       | r = 361       | c = $36 = 6^2$,       | cc= $1 = 1^2$ }} {{eqn | l = 35^2       | r = 1225       | c = $1 = 1^2$,       | cc= $225 = 15^2$ }} {{eqn | l = 38^2       | r = 1444       | c = $144 = 12^2$,       | cc= $4 = 2^2$ }} {{eqn | l = 41^2       | r = 1681       | c = $16 = 4^2$,       | cc= $81 = 9^2$ }} {{eqn | l = 57^2       | r = 3249       | c = $324 = 18^2$,       | cc= $9 = 3^2$ }} {{eqn | l = 65^2       | r = 4225       | c = $4 = 2^2$,       | cc= $225 = 15^2$ }} {{eqn | l = 70^2       | r = 4900       | c = $4 = 2^2$,       | cc= $900 = 30^2$ }} {{eqn | l = 125^2       | r = 15 \, 625       | c = $1 = 1^2$,       | cc= $5625 = 75^2$ }} {{eqn | l = 130^2       | r = 16 \, 900       | c = $16 = 4^2$,       | cc= $900 = 30^2$ }} {{eqn | l = 190^2       | r = 36 \, 100       | c = $36 = 6^2$,       | cc= $100 = 10^2$ }} {{eqn | l = 205^2       | r = 42 \, 025       | c = $4 = 2^2$,       | cc= $2025 = 45^2$ }} {{eqn | l = 223^2       | r = 49 \, 729       | c = $49 = 7^2$,       | cc= $729 = 27^2$ }} {{end-eqn}} {{OEIS|A048375}}


Zeroes of Infinite Product of Analytic Functions

Let $D \subset \C$ be an open connected set. Let $\left\langle{f_n}\right\rangle$ be a sequence of analytic functions $f_n: D \to \C$. Let $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$. Let $z_0\in D$. Then: :$(1): \quad$ $f$ is identically zero {{Iff}} some $f_n$ is identically zero  :$(2): \quad$ $f_n \left({z_0}\right) = 0$ for finitely many $n \in \N$ :$(3): \quad$ If $f$ is not identically zero, $\operatorname {mult}_{z_0} \left({f}\right) = \displaystyle \sum_{n \mathop = 1}^\infty \operatorname{mult}_{z_0} \left({f_n}\right)$ where $\operatorname{mult}$ denotes multiplicity.


Infinite Product of Analytic Functions is Analytic

Let $D \subset \C$ be an open set. Let $\left\langle{f_n}\right\rangle$ be a sequence of analytic functions $f_n: D \to \C$. Let $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$. Then $f$ is analytic.


Sum of 2 Squares in 2 Distinct Ways

Let $m, n \in \Z_{>0}$ be distinct positive integers that can be expressed as the sum of two distinct square numbers. Then $m n$ can be expressed as the sum of two square numbers in at least two distinct ways.


Bounds for Finite Product of Real Numbers

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Then: :$\displaystyle \sum_{k \mathop = 1}^n a_k \le \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \map \exp {\sum_{k \mathop = 1}^n a_k}$


Bounds for Complex Exponential

Let $\exp$ denote the complex exponential. Let $z \in \C$ with $\cmod z \le \dfrac 1 2$. Then :$\dfrac 1 2 \cmod z \le \cmod {\exp z - 1} \le \dfrac 3 2 \cmod z$


Sum of 2 Squares in 2 Distinct Ways/Sequence

The sequence of positive integers which can be expressed as the sum of two square numbers in two or more different ways begins: {{begin-eqn}} {{eqn | l = 50       | r = 7^2 + 1^2       | rr= = 5^2 + 5^2 }} {{eqn | l = 65       | r = 8^2 + 1^2       | rr= = 7^2 + 4^2 }} {{eqn | l = 85       | r = 9^2 + 2^2       | rr= = 7^2 + 6^2 }} {{eqn | l = 125       | r = 11^2 + 2^2       | rr= = 10^2 + 5^2 }} {{eqn | l = 130       | r = 11^2 + 3^2       | rr= = 9^2 + 7^2 }} {{eqn | l = 145       | r = 12^2 + 1^2       | rr= = 9^2 + 8^2 }} {{eqn | l = 170       | r = 13^2 + 1^2       | rr= = 11^2 + 7^2 }} {{end-eqn}}


Sum of 2 Squares in 2 Distinct Ways/Examples/50

$50$ is the smallest positive integer which can be expressed as the sum of two square numbers in two distinct ways: {{begin-eqn}} {{eqn | l = 50       | r = 5^2 + 5^2 }} {{eqn | r = 7^2 + 1^2 }} {{end-eqn}}


Lower Adjoint at Element is Minimum of Preimage of Singleton of Element implies Composition is Identity

Let $L = \struct {S, \preceq}, R = \struct {T, \precsim}$ be ordered sets. Let $g: S \to T, d: T \to S$ be mappings such that :$\forall t \in T: \map d t = \min \set {g^{-1} \sqbrk {\set t} }$ Then $g \circ d = I_T$ where $I_T$ denotes the identity mapping of $T$.


Bounds for Weierstrass Elementary Factors

Let $E_p: \C \to \C$ denote the $p$th Weierstrass elementary factor: :$\map {E_p} z = \begin{cases} 1 - z & : p = 0 \\ \paren {1 - z} \map \exp {z + \dfrac {z^2} 2 + \cdots + \dfrac {z^p} p} & : \text{otherwise}\end{cases}$ Let $z \in \C$.


Order of Product of Entire Functions

Let $f, g: \C \to \C$ be entire functions of order $\alpha$ and $\beta$. Then $f g$ has order at most $\map \max {\alpha, \beta}$.


Numbers Partitioned into up to 4 Squares in 5 Ways

The following positive integers can be expressed as the sum of no more than $4$ squares in $5$ distinct ways: :$50, 52, 54, 58, \ldots$


Order of Product of Entire Function with Polynomial

Let $f:\C\to\C$ be an entire function of order $\omega$. Let $P:\C\to\C$ be a nonzero polynomial. Then $f\cdot P$ has order $\omega$.


Order of Shifted Entire Function

Let $f: \C \to \C$ be an entire function of order $\alpha$. Let $a \in \C$. Then $f \left({z + a}\right)$ has order $\alpha$.


Zerofree Entire Function of Finite Order is Exponential of Polynomial

Let $f:\C\to\C$ be an entire function of finite order. Let $f$ have no zeroes. Then $f=\exp P$ for some polynomial $P$.


Borel-Carathéodory Lemma

Let $D \subset \C$ be an open set with $0 \in D$. Let $R > 0$ be such that $\map B {0, R} \subset D$. {{explain|Domain of $R$, definition of $B$}} Let $f: D \to \C$ be analytic with $\map f 0 = 0$. Let $\map \Re {\map f z} \le M$ for $\cmod z \le R$. Let $0 < r < R$. Then for $\cmod z \le r$: :$(1): \quad \cmod {\map f z} \le \dfrac {2 M r} {R - r}$ :$(2): \quad \cmod {\map {f^{\paren k} } z} \le \dfrac {2 M R k!} {\paren {R - r}^{k + 1} }$ for all $k \ge 1$


Order of Sum of Entire Functions

Let $f, g: \C \to \C$ be entire functions of order $\alpha$ and $\beta$. Then $f + g$ has order at most $\map \max {\alpha, \beta}$, with equality if $\alpha \ne \beta$.


Exponent of Convergence is Less Than Order

Let $f: \C \to \C$ be an entire function. Let $\omega$ be its order. Let $\tau$ be its exponent of convergence. Then $\tau \le \omega$.


Class is Proper iff Bijection from Class to Proper Class/Corollary

$A$ is proper if and only if there exists a bijection from $\mathrm P$ to $A$.


Injection from Proper Class to Class

Let $A$ be a class. Let $\mathrm P$ be a proper class. Let $f: \mathrm P \to A$ be an injection. Then $A$ is proper.


Polynomial has Order Zero

Let $P:\C\to\C$ be a polynomial function. Then $P$ has order $0$.


Order is Maximum of Exponent of Convergence and Degree

Let $f: \C \to \C$ be an entire function. Let $\omega$ be its order. Let $\tau$ be its exponent of convergence. Let $h$ be the degree of the polynomial in its canonical factorization. Then: :$\omega = \map \max {\tau, h}$


Recurring Parts of Multiples of One Fifty-Third

The multiples of $\dfrac 1 {53}$ from $\dfrac 1 {53}$ to $\dfrac {52} {53}$ can be divided into $4$ sets of equal size: :one where the digits of the recurring part consists of a cyclic permutation of $01886 \, 79245 \, 283$ :one where the digits of the recurring part consists of a cyclic permutation of $03773 \, 58490 \, 566$ :one where the digits of the recurring part consists of a cyclic permutation of $07547 \, 16981 \, 132$ :one where the digits of the recurring part consists of a cyclic permutation of $09433 \, 96226 \, 415$.


Order of Reciprocal of Entire Function

Let $f: \C \to \C$ be an entire function of order $\rho$. Let $f$ have no zeroes. Then $1/f$ has order $\rho$.


Jacobi's Necessary Condition/Dependent on N Functions

Let $J$ be a functional, such that: :$J \sqbrk {\mathbf y} = \displaystyle \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ where $\mathbf y = \paren {\sequence {y_i}_{1 \le i \le N} }$ is an N-dimensional real vector. Let $\map {\mathbf y} x$ correspond to the minimum of $J$. Let the $N\times N$ matrix $\mathbf P = F_{y_i' y_j'}$ be positive definite along $\map {\mathbf y} x$. Then the open interval $\openint a b$ contains no points conjugate to $a$.


Probability of no 2 People out of 53 Sharing the Same Birthday

Let there be $53$ people in a room. The probability that no $2$ of them have the same birthday is approximately $\dfrac 1 {53}$.


Primes not Sum of or Difference between Powers of 2 and 3

The sequence of prime numbers which cannot be expressed as either sum of or the difference between a power of $2$ and a power of $3$ begins: :$53, 71, 103, 107, 109, 149, 151, \ldots$ {{OEIS|A007644}}


Triangular Fibonacci Numbers

The only Fibonacci numbers which are also triangular are: :$0, 1, 3, 21, 55$ {{OEIS|A039595}}


Repdigit Triangular Numbers

The only repdigit numbers which are also triangular are: :$55, 66, 666$ {{OEIS|A045914}}


Increasing and Ordering on Mappings implies Mapping is Composition

Let $L = \left({S, \preceq}\right), R = \left({T, \precsim}\right)$ be ordered sets. Ley $g:S \to T, d:T \to S$ be mappings such that :$g$ and $d$ are increasing mappings and :$d \circ g \preceq I_S$ and $I_T \precsim g \circ d$ where $\preceq, \precsim$ denotes the orderings on mappings. Then $d \circ \left({g \circ d}\right)$ and $g = \left({g \circ d}\right) \circ g$


Closed Form for Square Pyramidal Numbers

The closed-form expression for the $n$th square pyramidal number is: :$S_n = \dfrac {n \paren {n + 1} \paren {2 n + 1} } 6$


Closed Form for Pentagonal Pyramidal Numbers

The closed-form expression for the $n$th pentagonal pyramidal number is: :$Q_n = \dfrac {n^2 \paren {n + 1} } 2$


Surjection from Class to Proper Class

Let $A$ be a class. Let $\mathrm P$ be a proper class. Let $f: A \to \mathrm P$ be a surjection. Then $A$ is proper.


Image of Set under Mapping is Set

Let $A$ be a class. Let $\mathrm U$ denote the universal class. Let $f: A \to \mathrm U$ be a class mapping. Let $S$ be a subset of $A$. {{explain|Note that further work is needed on the Subset page to clarify that a "subset" of a "proper class" is indeed a "set". Some words are already on that page, but this needs to be formalised and separated out into its own transcluded "class theoretic" page.}} Then the image $f \sqbrk S$ is also a set. If $A$ is a set, then this result is known as the Axiom of Replacement in Zermelo-Fraenkel set theory.