hcju/proofwikips huggingface dataset:
Set of Integers Bounded Above by Real Number has Greatest Element
Let $\Z$ be the set of integers. Let $\le$ be the usual ordering on the real numbers $\R$. Let $\O \subset S \subseteq \Z$ such that $S$ is bounded above in $\struct {\R, \le}$. Then $S$ has a greatest element.
Set of Integers Bounded Below by Real Number has Smallest Element
Let $\Z$ be the set of integers. Let $\le$ be the usual ordering on the real numbers $\R$. Let $\O \subset S \subseteq \Z$ such that $S$ is bounded below in $\struct {\R, \le}$. Then $S$ has a smallest element.
Real Number lies between Unique Pair of Consecutive Integers
Let $x$ be a real number.
Supremum of Set of Integers is Integer
Let $S \subset \Z$ be a non-empty subset of the set of integers. Let $S$ be bounded above in the set of real numbers. Then its supremum $\sup S$ is an integer.
Definition:Constructed Semantics/Instance 1/Rule of Idempotence
The Rule of Idempotence: :$(p \lor p) \implies p$ is a tautology in Instance 1 of constructed semantics.
Definition:Constructed Semantics/Instance 1/Rule of Addition
The Rule of Addition: :$q \implies (q \lor p)$ is a tautology in Instance 1 of constructed semantics.
Definition:Constructed Semantics/Instance 1/Rule of Commutation
The Rule of Commutation: :$\left({p \lor q}\right) \implies \left({q \lor p}\right)$ is a tautology in Instance 1 of constructed semantics.
Supremum of Set of Integers equals Greatest Element
Let $S \subset \Z$ be a non-empty subset of the set of integers. Let $S$ be bounded above in the set of real numbers $\R$. Then $S$ has a greatest element, and it is equal to the supremum $\sup S$.
Definition:Constructed Semantics/Instance 1/Factor Principle
The Factor Principle: :$\left({p \implies q}\right) \implies \left({\left({r \lor p}\right) \implies \left ({r \lor q}\right)}\right)$ is a tautology in Instance 1 of constructed semantics.
Weak Inequality of Integers iff Strict Inequality with Integer plus One
Let $a, b \in \Z$ be integers.  :$(1): \quad a \le b$ :$(2): \quad a < b + 1$ where: :$\le$ is the ordering on the integers :$<$ is the strict ordering on the integers.
Set of Orbits forms Partition
Let $G$ be a group. Let $X$ be a set. Let $G$ act on $X$. Then the set of orbits of the group action forms a partition of $X$.
Interior of Closure is Regular Open
Let $T = \left({S, \tau}\right)$ be a topological space. Let $H \subseteq S$. Then $H^{- \circ}$ is regular open.
Jordan's Lemma
Consider a complex-valued, continuous function $f$ defined on the contour: :$C_r = \left\{r e^{i \theta}: 0 \le \theta \le \pi \right\}, \ r > 0$ If the function $f$ is of the form: :$f \left({z}\right) = e^{i a z} g \left({z}\right), \ a > 0, \ z \in C_r$ Then: :$\displaystyle \left\vert{\int_{C_r} f \left({z}\right) \rd z}\right\vert \le \frac \pi a \max_{0 \le \theta \le \pi} \left\vert{g \left(re^{i \theta}\right)}\right\vert$
Lobachevsky Integral Formula
Let $f$ be a continuous function, periodic in $\pi$. Then: :$\displaystyle \int_0^\infty \frac {\sin x} x f \left({x}\right) \rd x = \int_0^{\frac \pi 2} f \left({x}\right) \rd x$
Dirichlet Series of Convolution of Arithmetic Functions
Let $f, g: \N \to \C$ be arithmetic functions. Let $h = f * g$ be their Dirichlet convolution. Let $F, G, H$ be their Dirichlet series. Let $s$ be a complex number such that $\map F s$ and $\map G s$ converge absolutely. Then $\map H s$ converges absolutely to $\map F s \times \map G s$.
Riemann Zeta Function in terms of Dirichlet Eta Function
Let $\zeta$ be the Riemann zeta function. Let $\eta$ be the Dirichlet eta function. Let $s \in \C$ be a complex number with real part $\sigma > 1$. Then $\zeta(s) = \dfrac 1 {1 - 2^{1-s}} \eta \left({s}\right)$.
Functional Equation for Riemann Zeta Function
Let $\zeta$ be the Riemann zeta function. Let $\map \zeta s$ have an analytic continuation for $\map \Re s > 0$. Then: :$\displaystyle \map \Gamma {\frac s 2} \pi^{-s/2} \map \zeta s = \map \Gamma {\frac {1 - s} 2} \pi^{\frac {s - 1} 2} \map \zeta {1 - s}$ where $\Gamma$ is the gamma function
Integral Representation of Riemann Zeta Function in terms of Fractional Part
Let $\zeta$ be the Riemann zeta function. Let $s\in\C$ be a complex number with real part $\sigma>1$. Then  :$\displaystyle \zeta \left({s}\right) = \frac s {s - 1} - s \int_1^\infty \left\{ {x}\right\} x^{-s - 1} \rd x$ where $\left\{{x}\right\}$ denotes the fractional part of $x$.
Binomial Form of Relation between Riemann Zeta Function and Dirichlet Eta Function
Let $\zeta$ be the Riemann zeta function. Let $s\in\C$ be a complex number with real part $\Re(s)>1$. Then: $\displaystyle \zeta \left({s}\right) = \frac 1 {1 - 2^{1-s} } \sum_{n \mathop = 0}^\infty \left({\frac 1 {2^{n+1} } \sum_{k \mathop = 0}^n \left({-1}\right)^k \binom n k \left({k + 1}\right)^{-s} }\right)$
Analytic Continuations of Riemann Zeta Function to Complex Plane
The Riemann zeta function $\zeta$ has a unique analytic continuation to $\C\setminus\{1\}$.
Analytic Continuations of Riemann Zeta Function to Right Half-Plane
The Riemann zeta function has a unique analytic continuation to $\{s \in \C : \Re(s) > 0\}\setminus\{1\}$, the half-plane $\Re(s)>0$ minus the point $s=1$.
Dirichlet Series of Inverse of Arithmetic Function
Let $f : \N \to\C$ be an arithmetic function. Let $g : \N \to \C$ be an Dirichlet inverse of $f$. Let $F, G$ be their Dirichlet series. Let $s \in \C$ such that both $F(s)$ and $G(s)$ converge absolutely. Then $F(s) \cdot G(s) = 1$.
Invertibility of Arithmetic Functions
Let $f: \N \to \C$ be an arithmetic function. Then $f$ has a Dirichlet inverse : :$\map f 1 \ne 0$
Equivalence of Definitions of Consistent Set of Formulas
{{TFAE|def = Consistent (Logic)/Proof System/Propositional Logic|view = Consistent Proof System for Propositional Logic}} Let $\LL_0$ be the language of propositional logic. Let $\mathscr P$ be a proof system for $\LL_0$. Let $\FF$ be a collection of logical formulas.
Analytic Continuation of Riemann Zeta Function using Mellin Transform of Fractional Part
Let $\zeta$ denote the Riemann zeta function. The analytic continuation of $\zeta$ to the half-plane $\map \Re s > 0$ is given by: :$\displaystyle \frac s {s - 1} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x$ where $x^{-s - 1}$ takes the principle value $e^{-\map \ln x \paren {s + 1} }$
Analytic Continuation of Riemann Zeta Function using Dirichlet Eta Function
Let $\zeta$ be the Riemann zeta function. Let $\eta$ be the Dirichlet Eta Function. Then: :$\dfrac 1 {1 - 2^{1 - s} } \map \eta s$ defines an analytic continuation of $\zeta$ to the half-plane $\map \Re s > 0$ minus $s = 1$.
Analytic Continuation of Riemann Zeta Function using Jacobi Theta Function
Let $\zeta$ be the Riemann zeta function. Then :$\displaystyle\frac{\pi^{s/2}}{\Gamma \left({\frac s 2}\right)} \cdot\left( - \frac 1 {s \left({1 - s}\right)} + \int_1^\infty \left({x^{s / 2 - 1} + x^{- \left({s + 1}\right) / 2} }\right) \omega \left({x}\right) \ \mathrm d x \right)$ defines an analytic continuation of $\zeta$ to the half-plane $\Re(s)>0$ minus $s=1$.
Complex Modulus of Real Number equals Absolute Value
Let $x \in \R$ be a real number. Then the complex modulus of $x$ equals the absolute value of $x$.
Equivalence of Definitions of Ceiling Function
Let $x$ be a real number. {{TFAE|def = Ceiling Function}}
Characterizing Property of Infimum of Subset of Real Numbers
Let $S \subset \R$ be a non-empty subset of the real numbers. Let $S$ be bounded below. Let $\alpha \in \R$.  :$(1): \quad \alpha$ is the infimum of $S$ :$(2): \quad \alpha$ is a lower bound for $S$ ::and: :::$\forall \epsilon \in \R_{> 0}$ there exists $x \in S$ with $x < \alpha + \epsilon$
Characterizing Property of Supremum of Subset of Real Numbers
Let $S \subset \R$ be a non-empty subset of the real numbers. Let $S$ be bounded above. Let $\omega \in \R$.  :$(1): \quad \omega$ is the supremum of $S$ :$(2): \quad \omega$ is an upper bound for $S$ ::and: :::$\forall \epsilon \in \R_{> 0}$ there exists $x \in S$ with $x > \omega - \epsilon$
Limit of Positive Real Sequence is Positive
Let $\sequence {x_n}$ be a sequence of positive real numbers. Let $x_n$ converge to $L$. Then $L \ge 0$.
Real Sequence with Nonzero Limit is Eventually Nonzero
Let $\sequence {x_n}$ be a real sequence. Let $\sequence {x_n}$ converge to $a \ne 0$. Then: :$\exists N \in \N: \forall n \ge N: x_n \ne 0$ That is, eventually every term of $\sequence {x_n}$ becomes non-zero.
Mittag-Leffler Expansion for Cotangent Function
:$\displaystyle \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$ where: :$z \in \C$ is not an integer :$\cot$ is the cotangent function.
Laurent Series Expansion for Cotangent Function
:$\displaystyle \pi \cot \pi z = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n - 1}$ where: :$z \in \C$ such that $\cmod z < 1$ :$\zeta$ is the Riemann Zeta function.
Equivalence of Definitions of Removable Discontinuity of Real Function
Let $A \subseteq \R$ be a subset of the real numbers. Let $f : A \to \R$ be a real function. Let $f$ be discontinuous at $a\in A$. {{TFAE|def = Removable Discontinuity of Real Function|view = removable discontinuity}}
Equivalence of Definitions of Polynomial Function on Subset of Ring
Let $R$ be a commutative ring with unity. Let $S \subset R$ be a subset. {{TFAE|def = Polynomial Function/Ring |view = polynomial function}}
Equivalence of Definitions of Polynomial Ring in Multiple Variables
Let $R$ be a commutative ring with unity. The following definitions of polynomial ring are equivalent in the following sense: : For every two constructions, there exists an $R$-isomorphism which sends indeterminates to indeterminates. {{explain|this statement has to be made more precise}} === Definition 1: As the monoid ring on a free monoid on a set === {{:Definition:Polynomial Ring/Monoid Ring on Free Monoid on Set}}
Monomials form Basis of Polynomial Ring/One Variable
Let $R$ be a commutative ring with unity. Let $R \sqbrk X$ be a polynomial ring over $R$ in the variable $X$. Then the monomials of $R \sqbrk X$ are a basis of $R \sqbrk X$ as a module over $R$.
Equivalence of Definitions of Polynomial in Ring Element
Let $R$ be a commutative ring. Let $S$ be a subring with unity of $R$. Let $x\in R$. {{TFAE|def = Polynomial in Ring Element}}
Equality of Monomials of Polynomial Ring in One Variable
Let $R$ be a commutative ring with unity. Let $R \sqbrk X$ be a polynomial ring in one variable $X$ over $R$. Let $k, l \in \N$ be distinct natural numbers. Then the mononomials $X^k$ and $X^l$ are distinct, where $X^k$ denotes the $k$th power of $X$.
Ring Isomorphic to Polynomial Ring is Polynomial Ring/One Variable
Let $R$ be a commutative ring with unity. Let $R \sqbrk X$ be a polynomial ring in one variable $X$ over $R$. Let $\iota : R \to R \sqbrk X$ denote the canonical embedding. Let $S$ be a commutative ring with unity and $f: R \sqbrk X \to S$ be a ring isomorphism. Then $\struct {S, f \circ \iota, \map f X}$ is a polynomial ring in one variable $\map f X$ over $R$.
Equality of Monomials of Polynomial Ring in Multiple Variables
Let $R$ be a commutative ring with unity. Let $I$ be a set. Let $R \sqbrk {\sequence {x_i}_{i \mathop \in I} }$ be a polynomial ring in $I$ variables $\sequence {x_i}_{i \mathop \in I}$ over $R$. Let $a, b : I \to \N$ be distinct mappings with finite support. Then the monomials $\ds \prod_{i \mathop \in I} X_i^{a_i}$ and $\ds \prod_{i \mathop \in I} X_i^{b_i}$ are distinct, where: :$X_i^k$ denotes the $k$th power of $X_i$ :$\prod$ denotes product with finite support
Mittag-Leffler Expansion for Cosecant Function
:$\displaystyle \pi \cosec \pi z = \frac 1 z + 2\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac z {z^2 - n^2}$
Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions
Let $\closedint a b$ be a closed real interval. Let $f$ be a bounded real function defined on $\closedint a b$. Let $P$ and $Q$ be finite subdivisions of $\closedint a b$. Let $\map L P$ be the lower sum of $f$ on $\closedint a b$ with respect to $P$. Let $\map U Q$ be the upper sum of $f$ on $\closedint a b$ with respect to $Q$. Then $\map L P \le \map U Q$.
Set Finite iff Surjection from Initial Segment of Natural Numbers
Let $S$ be a set. Then $S$ is finite  for some $n \in \N$ there exists a surjection $f: \N_{< n} \to S$. Here, $\N_{< n}$ denotes an initial segment of $\N$.
Set Finite iff Injection to Initial Segment of Natural Numbers
Let $S$ be a set. Then $S$ is finite  for some $n \in \N$ there exists an injection $f: S \to \N_{< n}$. Here, $\N_{< n}$ denotes an initial segment of $\N$.
Finite Product of Finite Sets is Finite
Let $\sequence {S_n}$ be a sequence of finite sets. Let $\displaystyle \prod_{k \mathop = 1}^n S_k$ be their Cartesian product. Then $\displaystyle \prod_{k \mathop = 1}^n S_k$ is also a finite set.
Existence of Integers with Multiplicative Persistence Greater than 11
It is not known whether there exists a number $n$ such that: :$P \left({n}\right) = 12$ where $P \left({n}\right)$ denotes the multiplicative persistence of $n$.
Smallest Strong Fibonacci Pseudoprime of Type I
The smallest strong Fibonacci pseudoprime of type I is $443 \, 372 \, 888 \, 629 \, 441$.
Infimum of Upper Sums Never Smaller than Lower Sum
:$\inf_P \map U P \ge \map L S$
Supremum of Lower Sums Never Greater than Upper Sum
:$\sup_P \map L P \le \map U S$
Properties of 5,559,060,566,555,523
$3^{33} = 5 \, 559 \, 060 \, 566 \, 555 \, 523$ has the following properties: : It has a remarkably large number of $5$s (half of its digits). : Multiply it by $2$, $4$ or $6$, and the result has $10$ of a particular digit. : Multiply it by $8$, and $9$ of the digits of the result are $4$.
First Occurrence of Prime Gap of 864
The first occurrence of a prime gap of $864$ is between $6 \, 505 \, 941 \, 701 \, 960 \, 039$ and $6 \, 505 \, 941 \, 701 \, 960 \, 903$.
Hardy-Ramanujan Number/Examples/48,988,659,276,962,496
The $5$th Hardy-Ramanujan number $\map {\operatorname {Ta} } 5$ is $48 \, 988 \, 659 \, 276 \, 962 \, 496$: {{begin-eqn}} {{eqn | l = 48 \, 988 \, 659 \, 276 \, 962 \, 496       | r = 38 \, 787^3 + 365 \, 757^3       | c =  }} {{eqn | r = 107 \, 839^3 + 362 \, 753^3       | c =  }} {{eqn | r = 205 \, 292^3 + 342 \, 952^3       | c =  }} {{eqn | r = 221 \, 424^3 + 336 \, 588^3       | c =  }} {{eqn | r = 231 \, 518^3 + 331 \, 954^3       | c =  }} {{end-eqn}}
Reciprocal of 19 from Sum of Powers of 2 Backwards
The decimal expansion of the reciprocal of $19$ can be constructed by summing the powers of $2$, offset progressively backwards by $1$ digit: <pre>                          1                         2                        4                       8                     16                    32                   64                 128                256               512             1024            2048           4096          8192        16384       32768      65536    131072 + 262144  ...... -------------------------- .......1052631578947368421 </pre> while: :$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$
Multiples of Reciprocals with Maximum Period form Magic Square
Let $p$ be a prime number whose reciprocal has a decimal expansion which has the maximum period, that is, $p - 1$. Let the first $p - 1$ digits of $\dfrac 1 p$ be expressed as an integer $n$ (with one or more leading zeroes as it is presented). Then the first $p - 1$ multiples of $n$, when listed in order of size, arrange themselves as a magic square such that: :the sum of the elements of each row :the sum of the elements in each column are the same.
General Fibonacci Sequence whose Terms are all Composite
The general Fibonacci sequence $\left\langle{a_n}\right\rangle$ defined as: :$a_n = \begin{cases} r & : n = 0 \\ s & : n = 1 \\ a_{n - 2} + a_{n - 1} & : n > 1 \end{cases}$ where: :$r = 62 \, 638 \, 280 \, 004 \, 239 \, 857$ :$s = 49 \, 463 \, 435 \, 743 \, 205 \, 655$ is such that: : $r$ and $s$ are coprime : all its terms are composite.
Superset of Infinite Set is Infinite
Let $S$ be an infinite set. Let $T \supseteq S$ be a superset of $S$. Then $T$ is also infinite.
Prime Decomposition of 2^58+1
The number $2^{58} + 1$ has the prime decomposition: :$2^{58} + 1 = 5 \times 107 \, 367 \, 629 \times 536 \, 903 \, 681$
Sum of Squares as Product of Factors with Square Roots
:$x^2 + y^2 = \left({x + \sqrt {2 x y} + y}\right) \left({x - \sqrt {2 x y} + y}\right)$
Prime Decomposition of Repunits
The prime decomposition of the repunits is as follows: {{begin-eqn}} {{eqn | l = R_2       | o =        | r = 11       | c =  }} {{eqn | l = R_3       | r = 3 \times 37       | c =  }} {{eqn | l = R_4       | r = 11 \times 101       | c =  }} {{eqn | l = R_5       | r = 41 \times 271       | c =  }} {{eqn | l = R_6       | r = 11 \times 111 \times 91       | c =  }} {{eqn | r = 3 \times 7 \times 11 \times 13 \times 37       | c =  }} {{eqn | l = R_7       | r = 239 \times 4649       | c =  }} {{eqn | l = R_8       | r = 1111 \times 10 \, 001       | c =  }} {{eqn | r = 11 \times 73 \times 101 \times 137       | c =  }} {{eqn | l = R_9       | r = 111 \times 1 \, 001 \, 001       | c =  }} {{eqn | r = 3^2 \times 37 \times 333 \, 667       | c =  }} {{eqn | l = R_{10}       | r = 11 \times 11 \, 111 \times 9091       | c =  }} {{eqn | r = 11 \times 41 \times 271 \times 9091       | c =  }} {{eqn | l = R_{11}       | r = 21 \, 649 \times 513 \, 239       | c =  }} {{eqn | l = R_{12}       | r = 111 \times 1111 \times 900 \, 991       | c =  }} {{eqn | r = 3 \times 7 \times 11 \times 13 \times 37 \times 101 \times 9901       | c =  }} {{eqn | l = R_{13}       | r = 53 \times 79 \times 265 \, 371 \, 653       | c =  }} {{eqn | l = R_{14}       | r = 11 \times 1 \, 111 \, 111 \times 909 \, 091       | c =  }} {{eqn | r = 11 \times 239 \times 4649 \times 909 \, 091       | c =  }} {{eqn | l = R_{15}       | r = 111 \times 11 \, 111 \times 90 \, 090 \, 991       | c =  }} {{eqn | r = 3 \times 31 \times 37 \times 41 \times 271 \times 2 \, 906 \, 161       | c =  }} {{eqn | l = R_{16}       | r = 11 \, 111 \, 111 \times 100 \, 000 \, 001       | c =  }} {{eqn | r = 11 \times 17 \times 73 \times 101 \times 137 \times 5 \, 882 \, 353       | c =  }} {{eqn | l = R_{17}       | r = 2 \, 071 \, 723 \times 5 \, 363 \, 222 \, 357       | c =  }} {{eqn | l = R_{18}       | r = 111 \, 111 \, 111 \times 1 \, 000 \, 000 \, 001       | c =  }} {{eqn | r = 3^2 \times 7 \times 11 \times 13 \times 19 \times 37 \times 52 \, 579 \times 333 \, 667       | c =  }} {{eqn | l = R_{19}       | r = 1 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111       | c =  }} {{eqn | l = R_{20}       | r = 1111 \times 11111 \times 900 \, 099 \, 009 \, 991       | c =  }} {{eqn | r = 11 \times 41 \times 101 \times 271 \times 3541 \times 9091 \times 27 \, 961       | c =  }} {{end-eqn}} {{expand|Go as far as we want}}
Repunit cannot be Square
A repunit (apart from the trivial $1$) cannot be a square.
Second Principle of Recursive Definition
Let $\N$ be the natural numbers. Let $T$ be a set. Let $a \in T$. For each $n \in \N_{>0}$, let $G_n: T^n \to T$ be a mapping. Then there exists exactly one mapping $f: \N \to T$ such that: :$\forall x \in \N: f(x) = \begin{cases} a & : x = 0\\ G_n \left({f(0), \ldots, f(n)}\right) & : x = n + 1 \end{cases}$
Repunit cannot be Cube
A repunit (apart from the trivial $1$) cannot be a cube.
Condition for Repunits to be Coprime
Let $R_p$ and $R_q$ be repunit numbers with $p$ and $q$ digits respectively. Then $R_p$ and $R_q$ are coprime  $p$ and $q$ are coprime.
Divisors of Repunit with Composite Index
Let $R_n$ be a repunit number with $n$ digits. Let $n$ be composite such that $n = r s$ where $1 < r < n$ and $1 < s < n$. Then $R_r$ and $R_s$ are both divisors of $R_n$.
Square of Small Repunit is Palindromic
The squares of repunits with up to $9$ digits are palindromic.
Square Number ending in 9 Digits in Reverse Order
The next integer after $1 \, 111 \, 111 \, 111$ whose square ends in $987 \, 654 \, 321$ is: :$2 \, 380 \, 642 \, 361^2 = 5 \, 667 \, 458 \, 050 \, 987 \, 654 \, 321$
5 Numbers such that Sum of any 3 is Square
This set of $5$ integers has the property that the sum of any $3$ of them is square: {{begin-eqn}} {{eqn | l = 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931       | o =  }} {{eqn | l = 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947       | o =  }} {{eqn | l = 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222       | o =  }} {{eqn | l = 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747       | o =  }} {{eqn | l = 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347       | o =  }} {{end-eqn}}
Total Number of Reachable Positions on Rubik's Cube
The total number $N$ of reachable positions on Rubik's cube is: {{begin-eqn}} {{eqn | l = N       | r = 43 \, 252 \, 003 \, 274 \, 489 \, 856 \, 000       | c =  }} {{eqn | r = \dfrac {8! \times 12! \times 3^8 \times 2^{12} } {2 \times 3 \times 3}       | c =  }} {{end-eqn}}
Smallest Number which is Multiplied by 99 by Appending 1 to Each End
The smallest positive integer which is multiplied by $99$ when $1$ is appended to each end is: :$112 \, 359 \, 550 \, 561 \, 797  \, 732 \, 809$
Smallest Multiply Perfect Number of Order 6
The number $154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600$ is multiply perfect of order $6$: {{begin-eqn}} {{eqn | l = \sigma \left({154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600}\right)       | r = 926 \, 073 \, 336 \, 514 \, 623 \, 897 \, 600       | c =  }} {{eqn | r = 6 \times 154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600       | c =  }} {{end-eqn}} It is the smallest positive integer to be so.
Sequence of 9 Consecutive Integers each with 48 Divisors
The $9$ integers beginning $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044$ each has $48$ divisors.
Polydivisible Number/Examples/3,608,528,850,368,400,786,036,725
The largest polydivisible number has $25$ digits: :$3 \, 608 \, 528 \, 850 \, 368 \, 400 \, 786 \, 036 \, 725$
No Polydivisible Number with 26 Digits Exists
There exists no polydivisible number with $26$ digits or more.
Probability of All Players receiving Complete Suit at Bridge
The probability of all $4$ players in a game of Bridge being dealt a complete suit is $1$ in $2 \, 235 \, 197 \, 406 \, 895 \, 366 \, 368 \, 301 \, 560 \, 000$.
Powers of 10 Expressible as Product of 2 Zero-Free Factors
The powers of $10$ which can be expressed as the product of $2$ factors neither of which has a zero in its decimal representation are: {{begin-eqn}} {{eqn | l = 10^1       | r = 2 \times 5       | c =  }} {{eqn | l = 10^2       | r = 4 \times 25 }} {{eqn | l = 10^3       | r = 8 \times 125 }} {{eqn | l = 10^4       | r = 16 \times 625 }} {{eqn | l = 10^5       | r = 32 \times 3125 }} {{eqn | l = 10^6       | r = 64 \times 15 \, 625 }} {{eqn | l = 10^7       | r = 128 \times 78 \, 125 }} {{eqn | l = 10^9       | r = 512 \times 1 \, 953 \, 125 }} {{eqn | l = 10^{18}       | r = 262 \, 144 \times 3 \, 814 \, 697 \, 265 \, 625 }} {{eqn | l = 10^{33}       | r = 8 \, 589 \, 934 \, 592 \times 116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125 }} {{end-eqn}}
Sequence of Palindromic Sophie Germain Primes
The number $N = 191 \, 918 \, 080 \, 818 \, 091 \, 909 \, 090 \, 909 \, 190 \, 818 \, 080 \, 819 \, 191$ has the property that: :$N$ is a palindromic Sophie Germain prime :$2 N + 1$ is also a palindromic Sophie Germain prime :$2 \left({2 N + 1}\right) + 1$ is also a palindromic prime, but not a Sophie Germain prime.
Power of 2 containing no Digit 2
$2^{168}$ contains no $2$ anywhere in its decimal representation.
Order of Fischer-Griess Monster
The order of the Fischer-Griess Monster $\mathrm M$ is given as: {{begin-eqn}} {{eqn | l = \order {\mathrm M}       | r = 2^{46} \times 3^{20} \times 5^9 \times 7^6 \times 11^2 \times 13^3 \times 17 \times 19 \times 23 \times 29 \times 31 \times 41 \times 47 \times 59 \times 71       | c =  }} {{eqn | r = 808 \, 017 \, 424 \, 794 \, 512 \, 875 \, 886 \, 459 \, 904 \, 961 \, 710 \, 757 \, 005 \, 754 \, 368 \, 000 \, 000 \, 000       | c =  }} {{end-eqn}}
Smallest Cube whose Sum of Divisors is Cube
The smallest cube $N$ such that $\map \sigma N$ is also a cube is: :$27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209$ where $\map \sigma N$ denotes the $\sigma$ function of $N$: the sum of the divisors of $N$