hcju/proofwikips huggingface dataset:
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 $X$  $f \left({x}\right)$ 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 $X$  $\map f x$ 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: :$i \in \Fix \sigma$  $\eqclass i {\RR_\sigma} = \set i$ 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 $2$ : :$\paren {a b}^2$ 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 : :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.