hcju/proofwikips huggingface dataset:
Conditions for Functional to be Extremum of Two-variable Functional over Canonical Variable p
Let $y = \map y x$ and $\map F {x, y, y'}$ be real functions. Let $\dfrac {\partial^2 F} {\partial {y'}^2} \ne 0$. Let $\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$ Let $\ds J \sqbrk {y, p} = \int_a^b \paren {-\map H {x, y, p} + p y'} \rd x$, where $H$ is the Hamiltonian of $J \sqbrk y$. Then $\ds J \sqbrk y = \bigvalueat {J \sqbrk {y, p} } {\frac {\delta J \sqbrk{y, p} } {\delta p} \mathop = 0}$
Image of Mapping of Infima is Generator Set of Filter
Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $f, g:\N \to S$ be mappings such that :$\forall n \in \N: g\left({n}\right) = \inf \left\{ {f\left({m}\right): m \in \N \land m \le n}\right\}$ Let $F$ be a filter such that :$f\left[{\N}\right]$ is generator set of $F$, where $f\left[{\N}\right]$ denotes the image of $f$. Then $g\left[{\N}\right]$ is generator set of $F$.
Subset of Set is Coarser than Set
Let $\left({S, \preceq}\right)$ be a preordered set. Let $A, B$ be subset of $S$ such that :$A \subseteq B$ Then $A$ is coarser than $B$.
Definition:Ramanujan-Nagell Number
A '''Ramanujan-Nagell number''' is a positive integer of the form $2^m - 1$ which is also triangular.
Existence of Product of Three Distinct Primes between n and 2n
Let $n \in \Z$ be an integer such that $n > 15$. Then between $n$ and $2 n$ there exists at least one integer which is the product of $3$ distinct prime numbers.
Second Column and Diagonal of Pascal's Triangle consist of Triangular Numbers
The $2$nd column and $2$nd diagonal of Pascal's triangle consists of the set of triangular numbers.
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 $I$  :$\exists x \in S: I = x^\preceq$ 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  $x$ 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 $19$  $a + 4 b$ 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 $L$  :$\exists F \in \mathit{Fin}\left({X}\right): E = \bigcap \left\{ {I \in S: F \subseteq I}\right\} \land F \subseteq E$ 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 :$F \in C^2 \closedint a b$   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 $24$.