hcju/proofwikips huggingface dataset:
Summation over Interval equals Indexed Summation
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $a, b \in \Z$ be integers. Let $\left[{a \,.\,.\, b}\right]$ be the integer interval between $a$ and $b$. Let $f: \left[{a \,.\,.\, b}\right] \to \mathbb A$ be a mapping. Then the summation over the finite set $\left[{a \,.\,.\, b}\right]$ equals the indexed summation from $a$ to $b$: :$\displaystyle\sum_{k \mathop \in \left[{a \,.\,.\, b}\right]} f \left({k}\right) = \sum_{k \mathop = a}^b f \left({k}\right)$
Hardy-Ramanujan Number/Examples/1729
The $2$nd Hardy-Ramanujan number $\map {\operatorname {Ta}} 2$ is $1729$: {{begin-eqn}} {{eqn | l = 1729       | r = 12^3 + 1^3       | c =  }} {{eqn | r = 10^3 + 9^3       | c =  }} {{end-eqn}}
Cardinality of Integer Interval
Let $a, b \in \Z$ be integers. Let $\left[{a \,.\,.\, b}\right]$ denote the integer interval between $a$ and $b$. Then $\left[{a \,.\,.\, b}\right]$ is finite and its cardinality equals: :$\begin{cases}  b - a + 1 & : b \ge a - 1 \\  0 & : b \le a - 1  \end{cases}$
Hardy-Ramanujan Number/Examples/6,963,472,309,248
The $4$th Hardy-Ramanujan number $\operatorname {Ta} \left({4}\right)$ is $6 \, 963 \, 472 \, 309 \, 248$: {{begin-eqn}} {{eqn | l = 6 \, 963 \, 472 \, 309 \, 248       | r = 2421^3 + 19 \, 083^3       | c =  }} {{eqn | r = 5436^3 + 18 \, 948^3       | c =  }} {{eqn | r = 10 \, 200^3 + 18 \, 072^3       | c =  }} {{eqn | r = 13 \, 322^3 + 16 \, 630^3       | c =  }} {{end-eqn}}
Indexed Summation of Sum of Mappings
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $a, b$ be integers. Let $\closedint a b$ denote the integer interval between $a$ and $b$. Let $f, g: \closedint a b \to \mathbb A$ be mappings. Let $h = f + g$ be their pointwise sum. Then we have the equality of indexed summations: :$\displaystyle \sum_{i \mathop = a}^b \map h i = \sum_{i \mathop = a}^b \map f i + \sum_{i \mathop = a}^b \map g i$
Summation of Sum of Mappings on Finite Set
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S$ be a finite set. Let $f, g: S \to \mathbb A$ be mappings. Let $h = f + g$ be their sum. Then we have the equality of summations on finite sets: :$\displaystyle \sum_{s \mathop \in S} h \left({s}\right) = \sum_{s \mathop \in S} f \left({s}\right) + \sum_{s \mathop \in S} g \left({s}\right)$
Indexed Summation of Multiple of Mapping
Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$. Let $a, b$ be integers. Let $\left[{a \,.\,.\, b}\right]$ denote the integer interval between $a$ and $b$. Let $f: \left[{a \,.\,.\, b}\right] \to \mathbb A$ be a mapping. Let $\lambda \in \mathbb A$. Let $g = \lambda \cdot f$ be the product of $f$ with $\lambda$. Then we have the equality of indexed summations: :$\displaystyle \sum_{i \mathop = a}^b g \left({i}\right) = \lambda \cdot \sum_{i \mathop = a}^b f \left({i}\right)$
Summation of Multiple of Mapping on Finite Set
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S$ be a finite set. Let $f: S \to \mathbb A$ be a mapping. Let $\lambda \in \mathbb A$. Let $g = \lambda \cdot f$ be the product of $f$ with $\lambda$. Then we have the equality of summations on finite sets: :$\displaystyle \sum_{s \mathop \in S} g \left({s}\right) = \lambda \cdot \sum_{s \mathop \in S} f \left({s}\right)$
Linear Combination of Indexed Summations
Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$. Let $a,b$ be integers. Let $\left[{a \,.\,.\, b}\right]$ denote the integer interval between $a$ and $b$. Let $f, g : \left[{a \,.\,.\, b}\right] \to \mathbb A$ be mappings. Let $\lambda, \mu \in \mathbb A$. Let $\lambda \cdot f + \mu \cdot g$ be the sum of the product of $f$ with $\lambda$ and the product of $g$ with $\mu$. Then we have the equality of indexed summations: :$\displaystyle \sum_{i \mathop = a}^b \left( \lambda \cdot f(i) + \mu \cdot g(i) \right) = \lambda \cdot \sum_{i \mathop = a}^b f(i) + \mu \cdot \sum_{i \mathop = a}^b g(i)$
Triangle Inequality for Indexed Summations
Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$. Let $a,b$ be integers. Let $\left[{a \,.\,.\, b}\right]$ denote the integer interval between $a$ and $b$. Let $f : \left[{a \,.\,.\, b}\right] \to \mathbb A$ be a mapping. Let $|\cdot|$ denote the standard absolute value. Let $\vert f \vert$ be the absolute value of $f$. Then we have the inequality of indexed summations: :$\displaystyle \left\vert \sum_{i \mathop = a}^b f(i) \right\vert \leq \sum_{i \mathop = a}^b \vert f(i) \vert$
Triangle Inequality for Summation over Finite Set
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S$ be a finite set. Let $f : S \to \mathbb A$ be a mapping. Let $\left\vert{\, \cdot\,}\right\vert$ denote the standard absolute value. Let $\left\vert{f}\right\vert$ be the absoute value of $f$. Then we have the inequality of summations on finite sets: :$\displaystyle \left\vert \sum_{s \mathop \in S} f(s) \right\vert \leq \sum_{s \mathop \in S} \vert f(s) \vert$
Exchange of Order of Indexed Summations
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $a, b, c, d \in \Z$ be integers. Let $\closedint a b$ denote the integer interval between $a$ and $b$.
Exchange of Order of Summations over Finite Sets/Cartesian Product
Let $f: S \times T \to \mathbb A$ be a mapping. Then we have an equality of summations over finite sets: :$\displaystyle \sum_{s \mathop \in S} \sum_{t \mathop \in T} \map f {s, t} = \sum_{t \mathop \in T} \sum_{s \mathop \in S} \map f {s, t}$
Exchange of Order of Summations over Finite Sets
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S, T$ be finite sets. Let $S \times T$ be their cartesian product.
Sum over Complement of Finite Set
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S$ be a finite set. Let $f: S \to \mathbb A$ be a mapping. Let $T \subseteq S$ be a subset. Let $S \setminus T$ be its relative complement. Then we have the equality of summations over finite sets: :$\displaystyle \sum_{s \mathop \in S \setminus T} f \left({s}\right) = \sum_{s \mathop \in S} f \left({s}\right) - \sum_{t \mathop \in T} f \left({t}\right)$
Mapping Defines Additive Function of Subalgebra of Power Set
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S$ be a finite set. Let $f: S \to \mathbb A$ be a mapping. Let $B$ be an algebra of sets over $S$. Define $\Sigma: B \to \mathbb A$ using summation as: :$\Sigma \left({T}\right) = \displaystyle \sum_{t \mathop \in T} f \left({t}\right)$ for $T\subseteq S$. Then $\Sigma$ is an additive function on $B$.
Summation over Finite Set Equals Summation over Support
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S$ be a finite set. Let $f: S \to \mathbb A$ be a mapping. Let $\operatorname{Supp} \left({f}\right)$ be its support. Then we have an equality of summations over finite sets: :$\displaystyle \sum_{s \mathop \in S} f \left({s}\right) = \sum_{s \mathop \in \operatorname{Supp} \left({f}\right)} f \left({s}\right)$
Summation of Zero
Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$.
Summation of Zero/Indexed Summation
Let $a,b$ be integers. Let $\left[{a \,.\,.\, b}\right]$ denote the integer interval between $a$ and $b$. Let $0 : \left[{a \,.\,.\, b}\right] \to \mathbb A$ be the zero mapping. Then the indexed summation of $0$ from $a$ to $b$ equals zero: :$\displaystyle \sum_{i \mathop = a}^b 0(i) = 0$
Summation of Zero/Finite Set
Let $S$ be a finite set. Let $0 : S \to \mathbb A$ be the zero mapping. {{explain|Presumably the above is a constant mapping on $0$ -- needs to be made explicit.}} Then the summation of $0$ over $S$ equals zero: :$\displaystyle \sum_{s \mathop \in S} 0 \left({s}\right) = 0$
Universal Property of Polynomial Ring
Let $R$ be a commutative ring with unity. The different definitions of a polynomial ring $(R(x), \iota, x)$ on $R$ satisfy the universal property of a polynomial ring:
Pandigital Properties of 123,456,789
$123 \, 456 \, 789$ has the following properties: It is pandigital, and remains so when multiplied by $2$, $4$, $5$, $7$ and $8$: {{begin-eqn}} {{eqn | l = 123 \, 456 \, 789 \times 1       | r = 123 \, 456 \, 789       | c =  }} {{eqn | l = 123 \, 456 \, 789 \times 2       | r = 246 \, 913 \, 578       | c =  }} {{eqn | l = 123 \, 456 \, 789 \times 3       | r = 370 \, 370 \, 367       | c =  }} {{eqn | l = 123 \, 456 \, 789 \times 4       | r = 493 \, 827 \, 156       | c =  }} {{eqn | l = 123 \, 456 \, 789 \times 5       | r = 617 \, 283 \, 945       | c =  }} {{eqn | l = 123 \, 456 \, 789 \times 6       | r = 740 \, 740 \, 734       | c =  }} {{eqn | l = 123 \, 456 \, 789 \times 7       | r = 864 \, 197 \, 523       | c =  }} {{eqn | l = 123 \, 456 \, 789 \times 8       | r = 987 \, 654 \, 312       | c =  }} {{eqn | l = 123 \, 456 \, 789 \times 9       | r = 1 \, 111 \, 111 \, 101       | c =  }} {{end-eqn}} {{expand|Add some mathematical analysis explaining this phenomenon}}
Pandigital Integers remaining Pandigital on Multiplication
Certain pandigital integers remain pandigital when multiplying them by certain single-digit integers: {{begin-eqn}} {{eqn | l = 1 \, 098 \, 765 \, 432 \times 1       | r = 1 \, 098 \, 765 \, 432       | c = which is pandigital }} {{eqn | l = 1 \, 098 \, 765 \, 432 \times 2       | r = 2 \, 197 \, 530 \, 864       | c = which is pandigital }} {{eqn | l = 1 \, 098 \, 765 \, 432 \times 3       | r = 3 \, 296 \, 296 \, 296       | c =  }} {{eqn | l = 1 \, 098 \, 765 \, 432 \times 4       | r = 4 \, 395 \, 061 \, 728       | c = which is pandigital }} {{eqn | l = 1 \, 098 \, 765 \, 432 \times 5       | r = 5 \, 493 \, 827 \, 160       | c = which is pandigital }} {{eqn | l = 1 \, 098 \, 765 \, 432 \times 6       | r = 6 \, 592 \, 592 \, 592       | c =  }} {{eqn | l = 1 \, 098 \, 765 \, 432 \times 7       | r = 7 \, 691 \, 358 \, 024       | c = which is pandigital }} {{eqn | l = 1 \, 098 \, 765 \, 432 \times 8       | r = 8 \, 790 \, 123 \, 456       | c = which is pandigital }} {{eqn | l = 1 \, 098 \, 765 \, 432 \times 9       | r = 9 \, 888 \, 888 \, 888       | c =  }} {{end-eqn}} The sequence: :$1039675824, 1053826974, 1068253974, 1068379524, 1073968254, 1075396824, 1098765432, 1204756839, 1234567890, 1357802469$ contains all pandigital integers with at least $4$ nontrivial pandigital multiples, of which: :$1098765432, 1234567890$ has $5$. {{OEIS|A167476}} {{expand|Add some mathematical analysis explaining this phenomenon.}}
Algebra Defined by Ring Homomorphism is Algebra
Let $R$ be a commutative ring. Let $\struct {S, +, *}$ be a ring. Let $f : R \to S$ be a ring homomorphism. Let the image of $f$ be a subset of the center of $S$. Let $\struct {S_R, *}$ be the algebra defined by the ring homomorphism $f$. Then $\struct {S_R, *}$ is an algebra over $R$.
Algebra Defined by Ring Homomorphism is Associative
Let $R$ be a commutative ring. Let $\struct {S, +, *}$ be a ring with unity. Let $f: R \to S$ be a ring homomorphism. Let the image of $f$ be a subset of the center of $S$. Let $\struct {S_R, *}$ be the algebra defined by the ring homomorphism $f$. Then $\struct {S_R, *}$ is an associative algebra.
Algebra Defined by Ring Homomorphism on Commutative Ring is Commutative
Let $R$ be a commutative ring. Let $\left({S, +, *}\right)$ be a commutative ring. Let $f: R \to S$ be a ring homomorphism. Let $\left({S_R, *}\right)$ be the algebra defined by the ring homomorphism $f$. Then $\left({S_R, *}\right)$ is an commutative algebra.
Smallest Integer which is Sum of 3 Sixth Powers in 2 Ways
The smallest positive integer which can be expressed as the sum of $3$ sixth powers in $2$ different ways is: {{begin-eqn}} {{eqn | l = 160 \, 426 \, 514       | r = 3^6 + 19^6 + 22^6       | c =  }} {{eqn | r = 10^6 + 15^6 + 23^6       | c =  }} {{end-eqn}} Also note that: {{begin-eqn}} {{eqn | l = 854       | r = 3^2 + 19^2 + 22^2       | c =  }} {{eqn | r = 10^2 + 15^2 + 23^2       | c =  }} {{end-eqn}}
Infinite Number of Integers which are Sum of 3 Sixth Powers in 2 Ways
There exist an infinite number of positive integers which can be expressed as the sum of $3$ sixth powers in $2$ different ways.
Integer which is Sum of 3 Fourth Powers in 2 Ways and Products of Those Roots
The positive integer $256 \, 103 \, 393$ can be expressed as the sum of $3$ fourth powers in $2$ different ways: {{begin-eqn}} {{eqn | l = 256 \, 103 \, 393       | r = 22^4 + 93^4 + 116^4       | c =  }} {{eqn | r = 29^4 + 66^4 + 124^4       | c =  }} {{end-eqn}} Also note that: {{begin-eqn}} {{eqn | l = 237 \, 336       | r = 22 \times 93 \times 116       | c =  }} {{eqn | r = 29 \times 66 \times 124       | c =  }} {{end-eqn}}
Groups of Order 8
Let $G$ be a group of order $8$. Then $G$ is isomorphic to one of the following: :$\Z_8$ :$\Z_4 \oplus \Z_2$ :$\Z_2 \oplus \Z_2 \oplus \Z_2$ :$D_4$ :$\Dic 2$ where: :$\Z_n$ is the cyclic group of order $n$ :$D_4$ is the dihedral group of order $8$ :$\Dic 2$ is the dicyclic group of order $8$.
Triangular Numbers which are Product of 3 Consecutive Integers
The $6$ triangular numbers which can be expressed as the product of $3$ consecutive integers are: :$6, 120, 210, 990, 185 \, 836, 258 \, 474 \, 216$ {{OEIS|A001219}}
First Harmonic Number to exceed 20
The first harmonic number that is greater than $20$ is $H_{272 \, 400 \, 600}$. That is, the number of terms of the harmonic series required for its partial sum to exceed $20$ is $272 \, 400 \, 600$.
First Harmonic Number to exceed 10
The first harmonic number that is greater than $10$ is $H_{12 \, 367}$. That is, the number of terms of the harmonic series required for its partial sum to exceed $10$ is $12 \, 367$.
First Harmonic Number to exceed 100
The first harmonic number that is greater than $100$ is $H_n$ where $n \approx 1.5 \times 10^{43}$. That is, it takes approximately $1.5 \times 10^{43}$ terms of the harmonic series required for its partial sum to exceed $100$.
Number of Magic Squares of Order 5
Up to rotations and reflections, there are $275 \, 305 \, 224$ distinct magic squares of order $5$.
Universal Property of Field of Rational Fractions
Let $R$ be an integral domain. Let $(R(x), \iota, x)$ be the field of rational fractions over $R$. Let $(K, f, a)$ be an ordered triple, where: :$K$ is a field :$f : R \to K$ is a unital ring homomorphism :$a$ is an element of $K$. Then there exists a unique unital ring homomorphism $\bar f : R(x) \to K$ such that $\bar f\circ\iota = f$ and $\bar f(x) = a$. :$\xymatrix{ R \ar[d]^\iota \ar[r]^{\forall f} & K\\ R(x) \ar[ru]_{\exists ! \bar f} }$
Equivalence of Definitions of Unital Subalgebra
Let $R$ be a commutative ring. Let $\struct {A_R, *}$ be an unital algebra over $R$ whose unit is $1_A$. Let $\struct {B_R, *}$ be a subalgebra of $A_R$. {{TFAE|def = Unital Subalgebra}}
Polydivisible Number/Examples/381,654,729
The integer $381 \, 654 \, 729$ is the only polydivisible number which is pandigital in the sense of excluding zero.
Pandigital Product of Pandigital Pairs in 3 Ways
The pandigital integer $0 \, 429 \, 315 \, 678$ can be expressed as the product of a pandigital doubleton in $3$ different ways: {{begin-eqn}} {{eqn | l = 0 \, 429 \, 315 \, 678       | r = 04 \, 926 \times 87 \, 153 }} {{eqn | r = 07 \, 923 \times 54 \, 186}} {{eqn | r = 15 \, 846 \times 27 \, 093}} {{end-eqn}}
Smallest Integer which is Sum of 2 Fourth Powers in 2 Ways
The smallest positive integer which can be expressed as the sum of $2$ fourth powers in $2$ different ways is: {{begin-eqn}} {{eqn | l = 635 \, 318 \, 657       | r = 59^4 + 158^4       | c =  }} {{eqn | r = 133^4 + 134^4       | c =  }} {{end-eqn}}
Largest Pandigital Square less Zero
The largest pandigital square (in the sense where pandigital excludes the zero) is $923 \, 187 \, 456$: :$923 \, 187 \, 456 = 30 \, 384^2$
Pandigital Properties of 987,654,321
$987 \, 654 \, 321$ has the following properties: It is pandigital, and remains so when multiplied by $1$, $2$, $4$, $5$, $7$ and $8$: {{begin-eqn}} {{eqn | l = 987 \, 654 \, 321 \times 1       | r = 987 \, 654 \, 321       | c =  }} {{eqn | l = 987 \, 654 \, 321 \times 2       | r = 1 \, 975 \, 308 \, 642       | c =  }} {{eqn | l = 987 \, 654 \, 321 \times 3       | r = 2 \, 962 \, 962 \, 963       | c =  }} {{eqn | l = 987 \, 654 \, 321 \times 4       | r = 3 \, 950 \, 617 \, 284       | c =  }} {{eqn | l = 987 \, 654 \, 321 \times 5       | r = 4 \, 938 \, 271 \, 605       | c =  }} {{eqn | l = 987 \, 654 \, 321 \times 6       | r = 5 \, 925 \, 925 \, 925       | c =  }} {{eqn | l = 987 \, 654 \, 321 \times 7       | r = 6 \, 975 \, 308 \, 642       | c =  }} {{eqn | l = 987 \, 654 \, 321 \times 8       | r = 7 \, 901 \, 234 \, 568       | c =  }} {{eqn | l = 987 \, 654 \, 321 \times 9       | r = 8 \, 888 \, 888 \, 889       | c =  }} {{end-eqn}} {{expand|Add some mathematical analysis explaining this phenomenon}} Also: :$987 \, 654 \, 321 - 123 \, 456 \, 789 = 864 \, 197 \, 532$ which is also pandigital.
Largest 9-Digit Prime Number
The largest prime number with $9$ digits is $999 \, 999 \, 937$.
Smallest Pandigital Square with Zero
The smallest pandigital square (in the sense where pandigital includes the zero) is $1 \, 026 \, 753 \, 849$: :$1 \, 026 \, 753 \, 849 = 32 \, 043^2$
Sound Proof System is Consistent
Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathscr P$ be a proof system for $\mathcal L$. Suppose that $\mathscr P$ is sound for $\mathscr M$. Then $\mathscr P$ is consistent.
Rule of Explosion/Variant 3
:$p, \neg p \vdash q$
Smallest Integer which is Sum of 3 Fifth Powers in 2 Ways
The smallest positive integer which can be expressed as the sum of $3$ fifth powers in $2$ different ways: The positive integer $1 \, 375 \, 298 \, 099$ can be expressed as the sum of $3$ fifth powers in $2$ different ways: {{begin-eqn}} {{eqn | l = 1 \, 375 \, 298 \, 099       | r = 24^5 + 28^5 + 67^5       | c =  }} {{eqn | r = 3^5 + 54^5 + 62^5       | c =  }} {{end-eqn}}
Automorphic Numbers with 10 Digits
The only $10$-digit automorphic numbers are: :$1 \, 787 \, 109 \, 376$ :$8 \, 212 \, 890 \, 625$
Left-Truncated Automorphic Number is Automorphic
Let $n$ be an automorphic number, expressed in some conventional number base. Let any number of digits be removed from the left-hand end of $n$. Then what remains is also an automorphic number.
Square whose Sigma is Cubic
The number $1 \, 857 \, 437 \, 604$ is a square number whose $\sigma$ value is a cube.
Largest Right-Truncatable Primes allowing 1
Let $1$ be temporarily considered to be a prime number. Under that consideration, the largest right-truncatable prime numbers are: :$1 \, 979 \, 339 \, 333$ :$1 \, 979 \, 339 \, 339$
Completely Multiplicative Function is Multiplicative
Let $f: \Z \to \Z$ be a function on the integers $\Z$. Let $f$ be completely multiplicative. Then $f$ is multiplicative.
Pandigital Numbers Divisible by All Integers up to 18
The following pandigital integers are divisible by all the positive integers up to $18$: :$2 \, 438 \, 195 \, 760$ :$3 \, 785 \, 942 \, 160$ :$4 \, 753 \, 869 \, 120$ :$4 \, 876 \, 391 \, 520$
Smallest n for which 2^n-3 is Divisible by n
The smallest positive integer $n$ such that $2^n - 3$ is divisible by $n$ is $4 \, 700 \, 063 \, 497$.
Smallest Odd Abundant Number not Divisible by 3
The smallest odd abundant number not divisible by $3$ is $5 \, 391 \, 411 \, 025$.
Closure of Irreducible Subspace is Irreducible
Let $X$ be a topological space. Let $Y \subset X$ be an irreducible subspace. Then its closure $\overline Y$ is also irreducible.
Point is Contained in Irreducible Component
Let $X$ be a topological space. Let $x\in X$ be a point. Then $x$ is contained in some irreducible component of $X$.
Irreducible Subspace is Contained in Irreducible Component
Let $X$ be a topological space. Let $Y\subset X$ be an irreducible subspace. Then there exists an irreducible component of $X$ containing $Y$.
Trivial Topological Space is Irreducible
Let $X$ be a trivial topological space. Then $X$ is irreducible.
Trivial Topological Space is Indiscrete
Let $X$ be a trivial topological space. Then $X$ is indiscrete.
Irreducible Components of Hausdorff Space are Points
Let $X$ be a Hausdorff space. Then the irreducible components of $X$ are the singleton sets.
Pandigital Integer Formed by Digits in Alphabetical Order
The number $8 \, 549 \, 176 \, 320$ is the pandigital integer formed from the digits from $0$ to $9$ arranged in alphabetical order.
Pandigital Properties of 9,876,543,210
$9 \, 876 \, 543 \, 210$ has the following properties: :$9 \, 876 \, 543 \, 210 - 0 \, 123 \, 456 \, 789 = 9 \, 753 \, 086 \, 421$ all three terms of which are pandigital. {{expand|Add more}}
Union of Open Irreducible Non-Disjoint Subspaces is Irreducible
Let $T = \left({S, \tau}\right)$ be an irreducible toplogical space. Let $U$ and $V$ be open irreducible subspaces. Let their intersection $U \cap V$ be non-empty. Then their union $U \cup V$ is irreducible.
Closed Set of Ultraconnected Space is Ultraconnected
Let $T = \struct {S, \tau}$ be an ultraconnected topological space. Let $F \subset S$ be a closed set in $T$. Then $F$ is ultraconnected.
Noetherian Space is Compact
Let $X$ be a noetherian topological space. Then $X$ is compact.
Subspace of Noetherian Space is Noetherian
Let $X$ be a noetherian topological space. Let $Y\subseteq X$ be a subspace. Then $Y$ is noetherian.
Zero Locus of Larger Set is Smaller
Let $k$ be a field. Let $n \ge 1$ be a natural number. Let $A = k \sqbrk {X_1, \ldots, X_n}$ be the ring of polynomials in $n$ variables over $k$. Let $I, J \subseteq A$ be subsets, and $\map V I$ and $\map V J$ their zero loci. Let $I \subseteq J$. Then $\map V I \supseteq \map V J$.
Smallest Multiply Perfect Number of Order 5
The number $14 \, 182 \, 439 \, 040$ is multiply perfect of order $5$: :$\sigma \left({14 \, 182 \, 439 \, 040}\right) = 70 \, 912 \, 195 \, 200 = 5 \times 14 \, 182 \, 439 \, 040$ It is the smallest positive integer to be so.
Smallest Fourth Power expressible as Sum of 4 Fourth Powers
$15 \, 527 \, 402 \, 881$ is the smallest fourth power which can be expressed as the sum of $4$ fourth powers: :$15 \, 527 \, 402 \, 881 = 353^4 = 30^4 + 120^4 + 272^4 + 315^4$
Largest Known Lead by 4n+1 in Prime Number Race
In the prime number race between prime numbers of the form $4 n - 1$ and $4 n + 1$, the highest known stretch of integers where $4 n + 1$ is not less than $4 n - 1$ is between $18 \, 465 \, 126 \, 293$ and $19 \, 033 \, 524 \, 538$.
Number whose Square is in 2 Identical Halves
The number $36 \, 363 \, 636 \, 364$ has the property that its square can be split into two identical halves: :$36 \, 363 \, 636 \, 364 = 1 \, 322 \, 314 \, 049 \, 613 \, 223 \, 140 \, 496$
Fifth Power expressible as Sum of 4 Fifth Powers
$61 \, 917 \, 364 \, 224$ can be expressed as the sum of $4$ fifth powers: :$61 \, 917 \, 364 \, 224 = 144^5 = 27^5 + 84^5 + 110^5 + 133^5$
Common Sum of 3 Distinct Amicable Pairs
The integer $64 \, 795 \, 852 \, 800$ is the sum of $3$ distinct amicable pairs: :$29 \, 912 \, 035 \, 725$ and $34 \, 883 \, 817 \, 075$ :$31 \, 695 \, 652 \, 275$ and $33 \, 100 \, 200 \, 525$ :$32 \, 129 \, 958 \, 525$ and $32 \, 665 \, 894 \, 275$ all of them odd.
Probability of Receiving Complete Suit as Hand at Bridge
The probability of being dealt a complete suit in a deal at Bridge is $1$ in $158 \, 753 \, 389 \, 900$.
Polynomial Ring is Generated by Indeterminate over Ground Ring
Let $R$ be a commutative ring with unity. Let $R \sqbrk X$ be a polynomial ring over $R$. Let $\iota: R \to R \sqbrk X$ be the embedding. Then $R \sqbrk X$ is generated by $X$ over $R$.
Smallest Cunningham Chain of the First Kind of Length 12
The smallest Cunningham chain of the first kind of length $12$ is: :$554 \, 688 \, 278 \, 429$, $1 \, 109 \, 376 \, 556 \, 859$, $2 \, 218 \, 753 \, 113 \, 719$, $4 \, 437 \, 506 \, 227 \, 439$, ::$8 \, 875 \, 012 \, 454 \, 879$, $17 \, 750 \, 024 \, 909 \, 759$, $35 \, 500 \, 049 \, 819 \, 519$, $71 \, 000 \, 099 \, 639 \, 039$, ::$142 \, 000 \, 199 \, 278 \, 079$, $284 \, 000 \, 398 \, 556 \, 159$, $568 \, 000 \, 797 \, 112 \, 319$, $1 \, 136 \, 001 \, 594 \, 224 \, 639$