hcju/proofwikips huggingface dataset:
123456789 x 9 + 10 = 1111111111
{{begin-eqn}} {{eqn | l = 1 \times 9 + 2       | r = 11 }} {{eqn | l = 12 \times 9 + 3       | r = 111 }} {{eqn | l = 123 \times 9 + 4       | r = 1111 }} {{eqn | l = 1234 \times 9 + 5       | r = 11111 }} {{eqn | l = 12345 \times 9 + 6       | r = 111111 }} {{eqn | l = 123456 \times 9 + 7       | r = 1111111 }} {{eqn | l = 1234567 \times 9 + 8       | r = 11111111 }} {{eqn | l = 12345678 \times 9 + 9       | r = 111111111 }} {{eqn | l = 123456789 \times 9 + 10       | r = 1111111111 }} {{end-eqn}} The above pattern is an instance of the identity: :$\displaystyle \left({b - 1}\right) \sum_{j \mathop = 1}^n j b^{n - j} + n + 1 = \sum_{j \mathop = 0}^n b^j$ where $b = 10$ and $n$ goes from $1$ to $9$.
Algebraic Numbers form Field
The set of real numbers $\R$ forms a field under addition and multiplication: $\left({\R, +, \times}\right)$. {{wtd|correct the statement}}
Linearly Independent over the Rational Numbers iff Linearly Independent over the Integers
Let $z_1, z_2, \ldots, z_n$ be complex numbers. Then: :$z_1, z_2, \ldots, z_n$ are linearly independent over the rational numbers $\Q$ : :$z_1, z_2, \ldots, z_n$ are linearly independent over the integers $\Z$.
Sums of Squares of Diagonals of Order 3 Magic Square
Consider the order 3 magic square: {{:Magic Square/Examples/Order 3}} The sums of the squares of the diagonals, when expressed as $3$-digit decimal numbers, are equal to the sums of the squares of those same diagonals of that same order 3 magic square when reversed. {{improve|Find a way to describe the "diagonals" accurately, as what is being demonstrated here does not match the description.}}
Lambert W of Non-Zero Algebraic Number is Transcendental
Let $W$ denote the (general) Lambert W function. Let $a$ be a non-zero algebraic number. Then, $W \left({a}\right)$ is transcendental.
Way Below in Lattice of Power Set
Let $X$ be a set. Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \preceq}\right)$ be a lattice of power set of $X$ where $\mathord\preceq = \mathord\subseteq \cap \left({\mathcal P\left({X}\right) \times \mathcal P\left({X}\right)}\right)$ Let $x, y \in \mathcal P\left({X}\right)$. Then $x \ll y$  :for every a set $Y$ of subsets of $X$ such that $y \subseteq \bigcup Y$ ::then there exists a finite subset $Z$ of $Y$: $x \subseteq \bigcup Z$ where $\ll$ denotes the way below relation.
Divisibility by 10
An integer $N$ expressed in decimal notation is divisible by $10$  the least significant digit of $N$ is $0$. That is: :$N = \sqbrk {a_n \ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $10$  : :$a_0 = 0$
Difference of Two Squares cannot equal 2 modulo 4
Let $n \in \Z_{>0}$ be of the form $4 k + 2$ for some $k \in \Z$. Then $n$ cannot be expressed in the form: :$n = a^2 - b^2$ for $a, b \in \Z$.
Compact Closure is Set of Finite Subsets in Lattice of Power Set
Let $X$ be a set. Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \preceq}\right)$ be the lattice of power set of $X$ where $\mathord\preceq = \mathord\subseteq \cap \mathcal P\left({X}\right) \times \mathcal P\left({X}\right)$ Let $x \in \mathcal P\left({X}\right)$. Then $x^{\mathrm{compact} } = \mathit{Fin}\left({x}\right)$ where $\mathit{Fin}\left({x}\right)$ denotes the set of all finite subsets of $x$.
10 is Only Triangular Number that is Sum of Consecutive Odd Squares
$10$ is the only triangular number which is the sum of two consecutive odd squares: :$10 = 1^2 + 3^2$
10 Consecutive Integers contain Coprime Integer
Let $n \in \Z$ be an integer. Let $S := \set {n, n + 1, n + 2, \ldots, n + 9}$ be the set of $10$ consecutive integers starting from $n$. Then at least one element of $S$ is coprime to every other element of $S$.
Two Fifths as Pandigital Fraction
There are $3$ ways $\dfrac 2 5$ can be expressed as a pandigital fraction: :$\dfrac 2 5 = \dfrac {6894} {17235}$ :$\dfrac 2 5 = \dfrac {8694} {21735}$ :$\dfrac 2 5 = \dfrac {9486} {23715}$
Two Sevenths as Pandigital Fraction
There are $6$ ways $\dfrac 2 7$ can be expressed as a pandigital fraction: :$\dfrac 2 7 = \dfrac {3654} {12789}$ :$\dfrac 2 7 = \dfrac {3674} {12859}$ :$\dfrac 2 7 = \dfrac {5342} {18697}$ :$\dfrac 2 7 = \dfrac {7418} {25963}$ :$\dfrac 2 7 = \dfrac {9786} {34251}$ :$\dfrac 2 7 = \dfrac {9862} {34517}$
Two Ninths as Pandigital Fraction
There are $2$ ways $\dfrac 2 9$ can be expressed as a pandigital fraction: :$\dfrac 2 9 = \dfrac {3924} {17658}$ :$\dfrac 2 9 = \dfrac {7596} {34182}$
Closed Form for Tetrahedral Numbers
The closed-form expression for the $n$th tetrahedral number is: :$H_n = \dfrac {n \paren {n + 1} \paren {n + 2} } 6$
Factorial as Product of Two Factorials
Apart from the general pattern, following directly from the definition of the factorial: :$\paren {n!}! = n! \paren {n! - 1}!$ the only known factorial which is the product of two factorials is: :$10! = 6! \, 7!$
Two Thirds as Pandigital Fraction
$\dfrac 2 3$ cannot be expressed as a pandigital fraction.
Three Quarters as Pandigital Fraction
$\dfrac 3 4$ cannot be expressed as a pandigital fraction.
Three Fifths as Pandigital Fraction
$\dfrac 3 5$ cannot be expressed as a pandigital fraction.
Three Sevenths as Pandigital Fraction
$\dfrac 3 7$ cannot be expressed as a pandigital fraction.
Three Eighths as Pandigital Fraction
$\dfrac 3 8$ cannot be expressed as a pandigital fraction.
Four Sevenths as Pandigital Fraction
$\dfrac 4 7$ cannot be expressed as a pandigital fraction.
Five Sevenths as Pandigital Fraction
$\dfrac 5 7$ cannot be expressed as a pandigital fraction.
Six Sevenths as Pandigital Fraction
$\dfrac 6 7$ cannot be expressed as a pandigital fraction.
Five Sixths as Pandigital Fraction
$\dfrac 5 6$ cannot be expressed as a pandigital fraction.
Five Eighths as Pandigital Fraction
$\dfrac 5 8$ cannot be expressed as a pandigital fraction.
Seven Eighths as Pandigital Fraction
$\dfrac 7 8$ cannot be expressed as a pandigital fraction.
Four Ninths as Pandigital Fraction
$\dfrac 4 9$ cannot be expressed as a pandigital fraction.
Five Ninths as Pandigital Fraction
$\dfrac 5 9$ cannot be expressed as a pandigital fraction.
Seven Ninths as Pandigital Fraction
$\dfrac 7 9$ cannot be expressed as a pandigital fraction.
Eight Ninths as Pandigital Fraction
$\dfrac 8 9$ cannot be expressed as a pandigital fraction.
Generating Function for Triangular Numbers
Let $T_n$ denote the $n$th triangular number. Then the generating function for $\sequence {T_n}$ is given as: :$\displaystyle \map G z = \frac z {\paren {1 - z}^3}$
Lattice of Power Set is Algebraic
Let $X$ be a set. Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \preceq}\right)$ be the lattice of power set of $X$ where $\mathord\preceq = \mathord\subseteq \cap \left({\mathcal P\left({X}\right) \times \mathcal P\left({X}\right)}\right)$ Then $L$ is algebraic.
Difference of Squares of Sum and Difference
:$\forall a, b \in \R: \left({a + b}\right)^2 - \left({a - b}\right)^2 = 4 a b$
Is there a Limit to the Multiplicative Persistence of a Number?
It has been conjectured that there may be an upper limit to the multiplicative persistence of a natural number.
Non-Empty Compact Closure is Directed
Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice. Let $x \in S$ such that :$x^{\mathrm{compact} }$ is a non-empty set, where $x^{\mathrm{compact} }$ denotes the compact closure of $x$. Then :$x^{\mathrm{compact} }$ is directed.
Set of Cardinality not Greater than Cardinality of Finite Set is Finite
Let $X, Y$ be sets such that :$\left\vert X \right\vert \le \left\vert Y \right\vert$ and :$Y$ is finite, where $\left\vert X \right\vert$ denotes the cardinality of $X$. Then $X$ is finite.
Finite iff Cardinality Less than Aleph Zero
Let $X$ be a set. Then $X$ is finite  $\card X < \aleph_0$ where: :$\card X$ denotes the cardinality of $X$ :$\aleph_0 = \card \N$ by Aleph Zero equals Cardinality of Naturals.
P-adic Valuation of Difference of Powers with Coprime Exponent
Let $x, y \in \Z$ be distinct integers. Let $n \ge 1$ be a natural number. Let $p$ be a prime number. Let: :$p \divides x - y$ and: :$p \nmid x y n$. Then  :$\map {\nu_p} {x^n - y^n} = \map {\nu_p} {x - y}$
One of 4 Consecutive Numbers Greater than 11 is Divisible by Prime Greater than 11
Let $n \in \Z$ such that $n > 11$. Then at least one of the set: :$\set {n, n + 1, n + 2, n + 3}$ is divisible by a prime number greater than $11$.
Ratio of Consecutive Lucas Numbers
For $n \in \N$, let $L_n$ be the $n$th Lucas number. Then: :$\displaystyle \lim_{n \mathop \to \infty} \frac {L_{n + 1} } {L_n} = \phi$ where $\phi = \dfrac {1 + \sqrt 5} 2$ is the golden mean.
Ordered Set of All Mappings is Ordered Set
Let $L = \struct {S, \preceq}$ be an ordered set. Let $X$ be a set. Then $L^X$ is also an ordered set.
Closed Form for Lucas Numbers
The Lucas numbers have a closed-form solution: :$L_n = \phi^n + \paren {1 - \phi}^n = \paren {\dfrac {1 + \sqrt 5} 2}^n + \paren {\dfrac {1 - \sqrt 5} 2}^n$ where $\phi$ is the golden mean. Putting $\hat \phi = 1 - \phi = -\dfrac 1 \phi$ this can be written: :$L_n = \phi^n + \hat \phi^n$
Fibonacci Number 2n equals Fibonacci Number n by Lucas Number n
Let $F_n$ denote the $n$th Fibonacci number. Let $L_n$ denote the $n$th Lucas number. Then: :$F_{2 n} = F_n L_n$
Fibonacci Number 3n in terms of Fibonacci Number n and Lucas Number 2n
Let $F_n$ denote the $n$th Fibonacci number. Let $L_n$ denote the $n$th Lucas number. Then: :$F_{3 n} = F_n \paren {L_{2 n} + \paren {-1}^n}$
Relation between Square of Fibonacci Number and Square of Lucas Number
Let $F_n$ denote the $n$th Fibonacci number. Let $L_n$ denote the $n$th Lucas number. Then: :$5 {F_n}^2 + 4 \paren {-1}^n = {L_n}^2$
12 Pentominoes
There exist $12$ distinct free pentominoes: :600px
18 Fixed Pentominoes
There exist $18$ distinct fixed pentominoes: :600px
Differential of Differentiable Functional is Unique
The differential of a differentiable functional is unique.
Ordered Subset of Ordered Set is Ordered Set
Let $L = \struct {S, \preceq}$ be an ordered set. Let $\struct {S', \preceq'}$ be an ordered subset of $L$. Then $\struct {S', \preceq'}$ is an ordered set.
Finite Subsets form Ideal
Let $X$ be a set. Let $\mathit{Fin}\left({X}\right)$ be the set of all finite subsets of $X$. Then $\mathit{Fin}\left({X}\right)$ is ideal in $\left({\mathcal P\left({X}\right), \subseteq}\right)$ where $\mathcal P\left({X}\right)$ denotes the power set of $X$.
Product of Proper Divisors
Let $n$ be an integer such that $n \ge 1$. Let $\map P n$ denote the product of the proper divisors of $n$. Then: :$\map P n = n^{\map \tau n / 2 - 1}$ where $\map \tau n$ denotes the $\tau$ function of $n$.
Product of Divisors
Let $n$ be an integer such that $n \ge 1$. Let $\map D n$ denote the product of the divisors of $n$. Then: :$\map D n = n^{\map \tau n / 2}$ where $\map \tau n$ denotes the divisor counting function of $n$.
Square of Reversal of Small-Digit Number
Let $n$ be an integer whose decimal representation consists of sufficiently small digits. Then the reversal of the square of $n$ is the square of the reversal of $n$.
12 times Sigma of 12 equals 14 times Sigma of 14
$x = 12$ and $y = 14$ are solutions to the indeterminate equation: :$x \ \sigma \left({x}\right) = y \ \sigma \left({y}\right)$ where $\sigma$ denotes the $\sigma$ function.
12 Knights to Attack or Occupy All Squares on Chessboard
On a standard chessboard, it is a maximum of $12$ knights are needed to ensure all squares are either occipied or under attack.
12 Identical Spheres can touch One Other Sphere
A total of $12$ identical spheres can touch one other such sphere. Each of the outer spheres touch the sphere in the center and $3$ other spheres.
Number of Multidimensional Spheres that can touch One Other Sphere
Let $S_n$ be a spheres of $n$ dimensions with a given radius $r$. Let $\map T n$ denote the number of instances of $S_n$ that can touch one other such instance of $S_n$. The sequence of $\map T n$ begins as follows: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="right" style = "padding: 2px 10px" | $\map T n$  |- | align="right" style = "padding: 2px 10px" | $0$ | align="right" style = "padding: 2px 10px" | $0$ |- | 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" | $6$ |- | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $12$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $24$ |- | align="right" style = "padding: 2px 10px" | $5$ | align="right" style = "padding: 2px 10px" | $40$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="right" style = "padding: 2px 10px" | $72$ |- | align="right" style = "padding: 2px 10px" | $7$ | align="right" style = "padding: 2px 10px" | $126$ |- | align="right" style = "padding: 2px 10px" | $8$ | align="right" style = "padding: 2px 10px" | $240$ |- | align="right" style = "padding: 2px 10px" | $9$ | align="right" style = "padding: 2px 10px" | $272$ |} {{OEIS|A001116}}
Ordering on Closure Operators iff Images are Including
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Let $f, g: S \to S$ be closure operators on $L$. Then: :$f \preceq g$  $g \sqbrk S \subseteq f \sqbrk S$ where :$\preceq$ denotes the ordering on mappings :$f \sqbrk S$ denotes the image of $f$.
Multiple of Perfect Number is Abundant
Let $n$ be a perfect number. Let $m$ be a positive integer such that $m > 1$. Then $m n$ is abundant.
Multiple of Abundant Number is Abundant
Let $n$ be an abundant number. Let $m$ be a positive integer such that $m > 1$. Then $m n$ is abundant.
Divisor of Perfect Number is Deficient
Let $n$ be a perfect number. Let $n = r s$ where $r$ and $s$ are positive integers such that $r > 1$ and $s > 1$. Then $r$ and $s$ are both deficient.
Divisor of Deficient Number is Deficient
Let $n$ be a perfect number. Let $n = k d$ where $r$ is a positive integer. Then $k$ is deficient.
Abundancy Index of Product is greater than Abundancy Index of Proper Factors
Let $n \in \Z_{>0}$ be a composite number such that $n = r s$, where $r, s \in \Z_{>1}$. Then: :$\dfrac {\map \sigma n} n > \dfrac {\map \sigma r} r$ and consequently also: :$\dfrac {\map \sigma n} n > \dfrac {\map \sigma s} s$ where $\map \sigma n$ denotes the $\sigma$ function of $n$. That is, the abundancy index of a composite number is strictly greater than the abundancy index of its divisors.
Condition for Differentiable Functional to have Extremum
Let $S$ be a set of mappings. Let $y, h \in S: \R \to \R$ be real functions.  Let $J \sqbrk y: S \to \R$ be a differentiable functional. Then a necessary condition for the differentiable functional $J \sqbrk {y; h}$ to have an extremum for $y = \hat y$ is: :$\bigvalueat {\delta J \sqbrk {y; h} } {y \mathop = \hat y} = 0$
Ordered Set of Closure Systems is Ordered Set
Let $L = \left({S, \preceq}\right)$ be an ordered set. Then $\operatorname{ClSystems}\left({L}\right)$ is an ordered set, where $\operatorname{ClSystems}\left({L}\right)$ denotes the ordered set of closure systems.
Necessary Condition for Integral Functional to have Extremum for given function
Let $S$ be a set of real mappings such that: :$S = \set {\map y x: \paren {y: S_1 \subseteq \R \to S_2 \subseteq \R}, \paren {\map y x \in C^1 \closedint a b}, \paren {\map y a = A, \map y b = B} }$ Let $J \sqbrk y: S \to S_3 \subseteq \R$ be a functional of the form: :$\displaystyle \int_a^b \map F {x, y, y'} \rd x$  Then a necessary condition for $J \sqbrk y$ to have an extremum (strong or weak) for a given function $\map y x$ is that $\map y x$ satisfy Euler's equation: :$F_y - \dfrac \d {\d x} F_{y'} = 0$
Image of Closure Operator Inherits Infima
Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $f$ be a closure operator on $L$. Then $R = \left({f\left[{S}\right], \precsim}\right)$ inherits infima, where :$\mathord\precsim = \mathord\preceq \cap \left({f\left[{S}\right] \times f\left[{S}\right]}\right)$ :$f\left[{S}\right]$ denotes the image of $f$.
Necessary Condition for Integral Functional to have Extremum for given function/Lemma
Let $\map \alpha x, \map \beta x$ be real functions. Let $\map \alpha x, \map \beta x$ be continuous on $\closedint a b$. Let: :$\forall \map h x \in C^1: \displaystyle \int_a^b \paren {\map \alpha x \map h x + \map \beta x \map {h'} x} \d x = 0$ subject to the boundary conditions: :$\map h a = \map h b = 0$ Then $\map \beta x$ is differentiable. Furthermore: :$\forall x \in \closedint a b: \map {\beta'} x = \map \alpha x$
If Definite Integral of a(x)h(x) vanishes for any C^0 h(x) then C^0 a(x) vanishes
Let $\map \alpha x$ be a continuous real function on the closed real interval $\closedint a b$. Let $\displaystyle \int_a^b \map \alpha x \map h x \rd x = 0$ for every real function $\map h x \in C^0 \closedint a b$ such that $\map h a = 0$ and $\map h b = 0$. {{explain|the notation $C^0 \closedint a b$}} Then $\map \alpha x = 0$ for all $x \in \closedint a b$.
Closure Operator does not Change Infimum of Subset of Image
Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $c: S \to S$ be a closure operator on $L$. Let $X$ be a subset of $c\left[{S}\right]$ such that :$X$ admits an infimum, where $c\left[{S}\right]$ denotes the image of $c$. Then $\inf X = c\left({\inf X}\right)$
Infimum in Ordered Subset
Let $L = \struct {S, \preceq}$ be an ordered set. Let $R = \struct {T, \preceq'}$ be an ordered subset of $L$. Let $X \subseteq T$ such that :$X$ admits an infimum in $L$. Then $\inf_L X \in T$  :$X$ admits an infimum in $R$ and $\inf_R X = \inf_L X$
Operator Generated by Closure System is Closure Operator
Let $L = \left({X, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $S = \left({T, \precsim}\right)$ be a closure system of $L$. Then $\operatorname{operator}\left({S}\right)$ is closure operator, where $\operatorname{operator}\left({S}\right)$ denotes the operator generated by $S$.
Regular Dodecahedron is Dual of Regular Icosahedron
The regular dodecahedron is the dual of the regular icosahedron.
Image of Operator Generated by Closure System is Set of Closure System
Let $L = \left({X, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $S = \left({T, \precsim}\right)$ be a closure system of $L$. Then $\operatorname{operator}\left({S}\right)\left[{X}\right] = T$ where $\operatorname{operator}\left({S}\right)$ denotes the operator generated by $S$.
Regular Icosahedron as Pentagonal Antiprism with Pyramidal Endcaps
The regular icosahedron can be considered as a regular pentagonal antiprism with two regular pentagonal pyramid as end caps. {{stub|Need to construct an icosahedron.}}
Construction of Rhombic Dodecahedron
The rhombic dodecahedron can be constructed as follows: Take a cube $K$ embedded in $3$-dimensional space. Place $6$ more cubes, each congruent with $K$, so that one face of each coincides with a different face of $K$. Join the vertices of $K$ to the centers of the adjacent cubes to describe square pyramids whose apices are the centers of the adjacent cubes and whose bases are the faces of $K$. The polyhedron formed by the $6$ square pyramids so formed, together with $K$, is a rhombic dodecahedron.
Recurring Parts of Multiples of One Thirteenth
The multiples of $\dfrac 1 {13}$ from $\dfrac 1 {13}$ to $\dfrac {12} {13}$ can be divided into two sets of equal size: :one where the digits of the recurring part consists of a cyclic permutation of $076923$ :one where the digits of the recurring part consists of a cyclic permutation of $153846$. :300px
Twelve Factorial plus One is divisible by 13 Squared
:$12! + 1$ is divisible by $13^2$.
Torus can be cut into 13 Pieces with 3 Plane Cuts
A torus can be cut into as many as $13$ separate pieces by $3$ plane cuts.
Operator Generated by Image of Closure Operator is Closure Operator
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Let $c: S \to S$ be a closure operator on $L$. Then $\map {\operatorname {operator} } {\struct {c \sqbrk S, \precsim} } = c$ where :$\mathord \precsim = \mathord \preceq \cap \paren {c \sqbrk S \times c \sqbrk S}$ :$\map {\operatorname {operator} } {\struct {c \sqbrk S, \precsim} }$ denotes the operator generated by $\struct {c \sqbrk S, \precsim}$
Pell's Equation/Examples/13
:$x^2 - 13 y^2 = 1$ has the smallest positive integral solution: :$x = 649$ :$y = 180$
Pell's Equation/Examples/29
:$x^2 - 29 y^2 = 1$ has the smallest positive integral solution: :$x = 9801$ :$y = 1820$
Vanishing First Variational Derivative implies Euler's Equation for Vanishing Variation
Let $\map y x$ be a real function such that $\map y a = A$ and $\map y b = B$. Let $J \sqbrk y$ be a functional of the form: :$\displaystyle J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$ Then: :$\dfrac {\delta J} {\delta y} = 0 \implies F_y - \dfrac \d {\d x} F_{y'} = 0$
Supremum in Ordered Subset
Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $R = \left({T, \preceq'}\right)$ be an ordered subset of $L$. Let $X \subseteq T$ such that :$X$ admits an supremum in $L$. Then $\sup_L X \in T$  :$X$ admits an supremum in $R$ and $\sup_R X = \sup_L X$
Partial Quotients of Continued Fraction Expansion of Irrational Square Root
Let $n \in \Z$ such that $n$ is not a square. Let the continued fraction expansion of $\sqrt n$ be expressed as: :$\left[{a_0, a_1, a_2, \ldots}\right]$ Then the partial quotients of this continued fraction expansion can be calculated as: :$a_r = \left\lfloor{\dfrac{\left\lfloor{\sqrt n}\right\rfloor + P_r} {Q_r} }\right\rfloor$ where: :$P_r = \begin{cases} 0 & : r = 0 \\ a_{r - 1} Q_{r - 1} - P_{r - 1} & : r > 0 \\ \end{cases}$ :$Q_r = \begin{cases} 1 & : r = 0 \\ \dfrac {n - {P_r}^2} {Q_{r - 1} } & : r > 0 \\ \end{cases}$
Operator Generated by Closure System Preserves Directed Suprema iff Closure System Inherits Directed Suprema
Let $L = \left({X, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $S = \left({Y, \precsim}\right)$ be a closure system on $L$. Then $\operatorname{operator}\left({S}\right)$ preserves directed suprema  $S$ inherits directed suprema. where $\operatorname{operator}\left({S}\right)$ denotes the operator generated by $S$.
Closure Operator Preserves Directed Suprema iff Image of Closure Operator Inherits Directed Suprema
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $h:S \to S$ be a closure operator on $L$. Then $h$ preserves directed suprema  $\left({h \left[{S}\right], \precsim}\right)$ inherits directed suprema. where :$h \left[{S}\right]$ denotes the image of $h$, :$\mathord\precsim = \mathord\preceq \cap \left({h \left[{S}\right] \times h \left[{S}\right]}\right)$
Thirteen Archimedean Polyhedra
There exist exactly $13$ distinct Archimedean polyhedra: :Truncated tetrahedron :Cuboctahedron :Truncated Cube :Truncated octahedron :Rhombicuboctahedron :Truncated cuboctahedron :Snub cube :Icosidodecahedron :Truncated dodecahedron :Truncated icosahedron :Rhombicosidodecahedron :Truncated icosidodecahedron :Snub dodecahedron
Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions
Let $\mathbf y$ be an $n$-dimensional real vector. Let $J \sqbrk {\mathbf y}$ be a functional of the form: $\displaystyle J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ Let: :$\mathbf y \in C^1 \closedint a b$ where $C^1 \closedint a b$ denotes that $\mathbf y$ is continuously differentiable in $\closedint a b$ Let $\mathbf y$ satisfy boundary conditions: :$\map {\mathbf y} a = \mathbf A$  :$\map {\mathbf y} b = \mathbf B$ where $\mathbf A$, $\mathbf B$ are real vectors. Then a necessary condition for $J \sqbrk {\mathbf y}$ to have an extremum (strong or weak) for a given $\mathbf y$ is that they satisfy Euler's equations: $F_{\mathbf y} - \dfrac \d {\d x} F_{\mathbf y'} = 0$
Regular Prism is Semiregular Polyhedron
A regular prism is a semiregular polyhedron.
Regular Antiprism is Semiregular Polyhedron
A regular antiprism is a semiregular polyhedron.
Finer Supremum Precedes Supremum
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $X, Y$ be subsets of $S$ such that :$X$ is finer than $Y$. Then $\sup X \preceq \sup Y$ where $\sup X$ denotes the supremum of $X$.