Center of Group is Kernel of Conjugacy Action

Let $G$ be a group. Let $Z$ be the kernel of the conjugacy action. Then $Z$ is the center of $G$.


Little-O Implies Big-O/General Result

Let $X$ be a topological space. Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\,\cdot\,}$ Let $f, g: X \to V$ be mappings. Let $x_0 \in X$. Let $f = \map o g$ as $x \to x_0$, where $o$ denotes little-O notation. Then $f = \map \OO g$ as $x \to x_0$, where $\OO$ denotes big-$\OO$ notation.


Little-O Implies Big-O/Sequences

Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers. Let $a_n = \map o {b_n}$ where $o$ denotes little-O notation. Then $a_n = \map \OO {b_n}$ where $\OO$ denotes big-$\OO$ notation.


Substitution in Big-O Estimate/Sequences

Let $(a_n)$ and $(b_n)$ be sequences of real or complex numbers. Let $a_n = O(b_n)$ where $O$ denotes big-O notation. Let $(n_k)$ be a diverging sequence of natural numbers. Then $a_{n_k} = O(b_{n_k})$.


First Sequence of Three Consecutive Strictly Decreasing Euler Phi Values

The first sequence of $3$ consecutive positive integers whose Euler $\phi$ values are strictly decreasing is: :$\map \phi {523} > \map \phi {524} > \map \phi {525}$


Bases of Finitely Generated Free Module have Equal Cardinality

Let $R$ be a commutative ring with unity. Let $M$ be a free $R$-module. Let $M$ be finitely generated. Let $B$ and $C$ be bases of $M$. Then $B$ and $C$ are finite and have the same cardinality.


Bases of Free Module have Equal Cardinality

Let $R$ be a commutative ring with unity. Let $M$ be a free $R$-module. Let $B$ and $C$ be bases of $M$. Then $B$ and $C$ are equinumerous.


Sequences of Three Consecutive Strictly Increasing Euler Phi Values

The following sequences of $3$ consecutive positive integers have the property that their Euler $\phi$ values are strictly increasing: :$\tuple {105, 106, 107}, \tuple {165, 166, 167}, \tuple {315, 316, 317}, \tuple {525, 526, 527}, \dots$ {{expand|Related sequences: A161962 (superset), A161963}}


Basis of Vector Space Injects into Generator

Let $K$ be a division ring. Let $V$ be a vector space over $K$. Let $B$ be a basis of $V$. Let $G$ be a generator of $V$. Then there exists an injection from $B$ to $G$.


Increasing Union of Ideals is Ideal/Sequence

Let $R$ be a ring. Let $S_0 \subseteq S_1 \subseteq S_2 \subseteq \dotsb \subseteq S_i \subseteq \dotsb$ be ideals of $R$. Then the increasing union $S$: :$\displaystyle S = \bigcup_{i \mathop \in \N} S_i$ is an ideal of $R$.


Increasing Union of Ideals is Ideal/Chain

Let $R$ be a ring. Let $\left({P, \subseteq}\right)$ be the ordered set consisting of all ideals of $R$, ordered by inclusion. Let $\left\{{I_\alpha}\right\}_{\alpha \in A}$ be a non-empty chain of ideals in $P$. Let $\displaystyle I = \bigcup_{\alpha \in A} I_\alpha$ be their union. Then $I$ is an ideal of $R$.


Union of Chain of Proper Ideals is Proper Ideal

Let $R$ be a ring with unity. Let $\struct {P, \subseteq}$ be the ordered set consisting of all ideals of $R$, ordered by inclusion. Let $\sequence {I_\alpha}_{\alpha \mathop \in A}$ be a non-empty chain of proper ideals in $P$. Let $\displaystyle I = \bigcup_{\alpha \mathop \in A} I_\alpha$ be their union. Then $I$ is a proper ideal of $R$.


Dimension of Free Vector Space on Set

Let $k$ be a division ring. Let $X$ be a set. Let $k^{\paren X}$ be the free vector space on $X$. The vector space $k^{\paren X}$ has dimension the cardinality of $X$.


239 is not Expressible as Sum of Fewer than 19 Fourth Powers

:$239 = 13 \times 1^4 + 4 \times 2^4 + 2 \times 3^4$


479 is not Expressible as Sum of Fewer than 19 Fourth Powers

:$479 = 13 \times 1^4 + 3 \times 2^4 + 2 \times 3^4 + 4^4$ or: :$479 = 10 \times 1^4 + 4 \times 2^4 + 5 \times 3^4$


559 is not Expressible as Sum of Fewer than 19 Fourth Powers

:$559 = 15 \times 1^4 + 2 \times 2^4 + 2 \times 4^4$ or: :$559 = 9 \times 1^4 + 4 \times 2^4 + 6 \times 3^4$


Canonical Basis of Free Module on Set is Basis

Let $R$ be a ring with unity. Let $I$ be a set. Let $R^{(I)}$ be the free $R$-module on $I$. Let $B$ be its canonical basis. Then $B$ is a basis of $R^{(I)}$.


Korselt's Theorem

Let $n \ge 2$ be an integer. Then $n$ is a '''Carmichael number''' {{iff}}: : $(1): \quad n$ is odd and the following conditions hold for every prime factor $p$ of $n$: : $(2): \quad p^2 \nmid n$ : $(3): \quad \paren {p - 1} \divides \paren {n - 1}$ where: :$\divides$ denotes divisibility :$\nmid$ denotes non-divisibility.


Intersection of Submodules is Submodule

Let $R$ be a ring. Let $M$ be an $R$-module. Let $S$ be a set of submodules of $M$. Then the intersection $\bigcap S$ is a submodule of $M$.


There are Infinitely Many Carmichael Numbers

There are infinitely many Carmichael numbers.


Product of Big-O Estimates/Sequences

Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be sequences of real or complex numbers. Let: :$a_n = \map \OO {b_n}$ :$c_n = \map \OO {d_n}$ where $\OO$ denotes big-O notation. Then: :$a_n c_n = \map \OO {b_n d_n}$


Equivalence of Definitions of Order of Entire Function

Let $f: \C \to \C$ be an entire function. Let $\ln$ denote the natural logarithm. {{TFAE| def = Order of Entire Function}}


Little-O Times Big-O is Little-O/Sequences

Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be sequences of real or complex numbers. Let: :$a_n = \map \OO {b_n}$ :$c_n = \map o {d_n}$ where: :$\OO$ denotes big-O notation :$o$ denotes little-o notation. Then: :$a_n c_n = \map o {b_n d_n}$


Field Norm of Complex Number Equals Field Norm

Let $z = a + i b$ be a complex number, where $a, b \in \R$. Then the field norm of $z$ is the field norm with respect to the field extension $\C / \R$.


Big-O Notation for Sequences Coincides with General Definition

Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ be sequences of real or complex numbers. Let $\N$ be given the discrete topology. {{TFAE}} :$(1): \quad$ $a_n = O(b_n)$, where $O$ denotes big-O notation for sequences :$(2): \quad$ $a_n = O(b_n)$, where $O$ stands for the general definition of big-O notation


Bounded iff Big-O of 1/Sequences

Let $\sequence {a_n}$ be a sequence of real or complex numbers. {{TFAE}} :$(1): \quad a_n$ is bounded :$(2): \quad a_n = \map \OO 1$, where $\OO$ denotes big-$\OO$ notation


Sum of Little-O Estimates/Sequences

Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be sequences of real or complex numbers. Let: :$a_n = \map o {b_n}$ :$c_n = \map o {d_n}$ where $o$ denotes little-o notation. Then: :$a_n + c_n = \map o {\size {b_n} + \size {d_n} }$


Equivalence of Definitions of Asymptotically Equal Sequences

Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences in $\R$. {{TFAE|def = Asymptotically Equal Sequences}}


Largest Number not Expressible as Sum of Multiples of 23 and 28

The largest integer $n$ that cannot be expressed in the form: :$n = 23 x + 28 y$ for $x, y \in \Z_{>0}$ is $593$.


Open implies There Exists Way Below Element

Let $L = \left({S, \preceq, \tau}\right)$ be a continuous topological lattice with Scott topology. Let $p \in S, A \subseteq S$ such that :$A$ is open and $p \in A$. Then :$\exists q \in A: q \ll p$


Smallest Fourth Power which is Sum of 5 Fourth Powers

$625$ is the smallest fourth power which is the sum of $5$ fourth powers: :$625 = 5^4 = 2^4 + 2^4 + 3^4 + 4^4 + 4^4$


Interior is Union of Way Above Closures

Let $\left({S, \preceq, \tau}\right)$ be a complete continuous topological lattice with Scott topology. Let $X \subseteq S$. Then $X^\circ = \bigcup \left\{ {x^\gg: x \in S \land x^\gg \subseteq X}\right\}$ where :$X^\circ$ denotes the interior of $X$, :$x^\gg$ denotes the way above closure of $x$.


Way Above Closures Form Basis

Let $L = \left({S, \preceq, \tau}\right)$ be a complete continuous topological lattice with Scott topology. Then $\left\{ {x^\gg: x \in S}\right\}$ is basis of $L$.


Record Gaps between Twin Primes

The gaps between the following pairs of twin primes are larger than those for all smaller pairs: {{begin-eqn}} {{eqn | l = \tuple {3, 5}       | o = \to       | r = \tuple {5, 7}       | c = a gap of $0$ }} {{eqn | l = \tuple {5, 7}       | o = \to       | r = \tuple {11, 13}       | c = a gap of $4$ }} {{eqn | l = \tuple {17, 19}       | o = \to       | r = \tuple {29, 31}       | c = a gap of $10$ }} {{eqn | l = \tuple {41, 43}       | o = \to       | r = \tuple {59, 61}       | c = a gap of $16$ }} {{eqn | l = \tuple {69, 73}       | o = \to       | r = \tuple {101, 103}       | c = a gap of $28$ }} {{eqn | l = \tuple {311, 313}       | o = \to       | r = \tuple {347, 349}       | c = a gap of $34$ }} {{eqn | l = \tuple {347, 349}       | o = \to       | r = \tuple {419, 421}       | c = a gap of $70$ }} {{eqn | l = \tuple {659, 661}       | o = \to       | r = \tuple {809, 811}       | c = a gap of $148$ }} {{end-eqn}} {{OEIS|A036061}} {{finish|Add this list to the Number Sequence}}


Sophie Germain Prime cannot be 6n+1

Let $p$ be a Sophie Germain prime. Then $p$ cannot be of the form $6 n + 1$, where $n$ is a positive integer.


Way Above Closure is Open

Let $L = \left({S, \preceq, \tau}\right)$ be a complete continuous topological lattice with Scott topology. Let $x \in S$. Then $x^\gg$ is open where $x^\gg$ denotes the way above closure of $x$.


Way Above Closure is Upper

Let $\left({S, \preceq}\right)$ be an ordered set. Let $x \in S$. Then $x^\gg$ is upper where $x^\gg$ denotes the way above closure of $x$.


Way Above Closures that Way Below Form Local Basis

Let $L = \left({S, \preceq, \tau}\right)$ be a complete continuous topological lattice with Scott topology. Let $p \in S$. Then $\left\{ {q^\gg: q \in S \land q \ll p}\right\}$ is local basis at $p$.


Characterization of Analytic Basis by Local Bases

Let $T = \struct {S, \tau}$ be a topological space. Let $P$ be a set of subsets of $S$ such that :$P \subseteq \tau$ and :for all $p \in S$: there exists local basis $B$ at $p: B \subseteq P$ Then $P$ is basis of $T$.


Approximation to Golden Ratio using 666

The Golden Ratio is approximated closely by the following formula: :$\phi \approx -2 \sin \left({666}\right) = 1.61803 \, 39887 \, 5 \ldots$


Euler Phi Function of 666 equals Product of Digits

The number $666$ has the following interesting property: :$\map \phi {666} = 6 \times 6 \times 6$ where $\phi$ denotes the Euler $\phi$ function.


Sum of Sequence of Squares of Primes

Let $S = \sequence {s_n}$ be the integer sequence defined as: :$\displaystyle s_n = \sum_{i \mathop = 1}^n {p_i}^2$ where $P_i$ denotes the $i$th prime number. Then $S$ begins: :$4, 13, 38, 87, 208, 377, 666, 1027, 1556, 2397, 3358, 4727, 6408, 8257, 10466, \ldots$ {{OEIS|A024450}}


Pair of Consecutive Powerful Numbers whose First is Odd

The only known pair of consecutive integers which are both powerful numbers such that the first of the pair is odd is: :$\tuple {675, 676}$


Interior is Union of Elements of Basis

Let $T = \left({S, \tau}\right)$ be a topological space. Let $B$ be a basis of $T$. Let $V$ be a subset of $S$. Then $V^\circ = \bigcup \left\{ {G \in B: G \subseteq V}\right\}$ where $V^\circ$ denotes the interior of $V$.


Palindromic Squares with Non-Palindromic Roots

The sequence of palindromic squares with non-palindromic square roots begins: :$676, 69 \, 696, 94 \, 249, 698 \, 896, 5 \, 221 \, 225, 6 \, 948 \, 496, 522 \, 808 \, 225, \ldots$ This sequence is not explicitly given in {{OEISLink}}. The sequence of those corresponding non-palindromic square roots begins: :$26, 264, 307, 836, 2285, 2636, 22 \, 865, 24 \, 846, 30 \, 693, \ldots$ {{OEIS|A251673}}


Tetrahedral Numbers which are Sum of 2 Tetrahedral Numbers

The sequence of tetrahedral numbers which are the sum of two other tetrahedral numbers begins: :$680, \ldots$ {{expand|To be expanded when we have some more.}}


Open Set is Union of Elements of Basis

Let $T = \left({S, \tau}\right)$ be a topological space. Let $B$ be a basis of $T$. Let $V$ be an open subset of $S$. Then $V = \bigcup \left\{ {G \in B: G \subseteq V}\right\}$


Equivalence of Definitions of Compatible Atlases

Let $M$ be a topological space. Let $\mathscr F, \mathscr G$ be $d$-dimensional atlases of class $C^k$ on $M$. {{TFAE|def = Compatible Atlases}}


Atlas is Contained in Unique Maximal Atlas

Let $M$ be a topological space. Let $A$ be a $d$-dimensional atlas of class $C^k$. Then $A$ is contained in a unique maximal atlas of class $C^k$.


Locally Euclidean iff has C0-Atlas

Let $M$ be a topological space. {{TFAE}} :$(1):\quad$ $M$ is locally euclidean. :$(2):\quad$ There exists a $C^0$-atlas on $M$.


Differentiable Structure Contains Unique Maximal Atlas

Let $M$ be a topological space. Let $k$ and $d$ be natural numbers. Let $S$ be a $d$-dimensional differentiable structure of class $C^k$ on $M$. Then $S$ contains a unique maximal $C^k$-atlas.


Consecutive Integers whose Product is Primorial

The following primorials can be expressed as the product of consecutive integers: :$2, 6, 30, 210, 510 \, 510$ {{OEIS|A161620}} No others are known. The corresponding indices of those primorials are: :$2, 3, 5, 7, 17$ {{OEIS|A215658}} The corresponding values of $n$ such that $p\# = \paren {n - 1} n$ are: :$2, 3, 6, 15, 715$ {{OEIS|A215659}}


Integers whose Ratio between Sigma and Phi is Square

The sequence of integers whose $\sigma$ value divided by its Euler $\phi$ value is a square begins: :$1, 14, 30, 105, 248, 264, 418, 714, 1485, 3080, \ldots$ {{OEIS|A293391}} {{finish|This sequence needs to be added to the individual integer pages.}}


Implicit Function Theorem/Real Functions

Let $n$ and $k$ be natural numbers. Let $\Omega \subset \R^{n + k}$ be open. Let $f: \Omega \to \R^k$ be continuous. Let the partial derivatives of $f$ with respect to $\R^k$ be continuous. Let $\tuple {a, b} \in \Omega$, with $a\in \R^n$ and $b\in \R^k$. Let $\map f {a, b} = 0$. For $\tuple {x_0, y_0} \in \Omega$, let $D_2 \map f {x_0, y_0}$ denote the total derivative of the function $y \mapsto \map f {x_0, y}$ at $y_0$. Let the linear map $D_2 \map f {a, b}$ be invertible. Then there exist neighborhoods $U \subset \Omega$ of $a$ and $V \subset \R^k$ of $b$ such that there exists a unique function $g: U \to V$ such that $\map f {x, \map g x} = 0$ for all $x \in U$. Moreover, $g$ is continuous.


Uniform Contraction Mapping Theorem

Let $M$ and $N$ be metric spaces. Let $M$ be complete. Let $f : M \times N \to M$ be a continuous uniform contraction. Then for all $t \in N$ there exists a unique $g \left({t}\right) \in M$ such that $f(g \left({t}\right), t) = g \left({t}\right)$, and the mapping $g: N \to M$ is continuous.


Mapping at Element is Supremum of Compact Elements implies Mapping at Element is Supremum that Way Below

Let $\left({S, \vee_1, \wedge_1, \preceq_1}\right)$ and $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be complete lattices. Let $f: S \to T$ be a mapping such that :$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \preceq_1 x \land w}\right.$ is compact$\left.{}\right\}$ Then :$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$


Inverse Function Theorem for Real Functions

Let $n\geq1$ and $k\geq1$ be natural numbers. Let $\Omega\subset \R^n$ be open. $f : \Omega \to \R^n$ be a vector-valued function of class $C^k$. Let $a\in\Omega$. Let the differential $Df(a)$ of $f$ at $a$ be invertible. Then there exist open sets $U\subset\Omega$ and $V\subset\R^n$ such that the restriction of $f$ to $U$ is a $C^k$-diffeomorphism $f:U\to V$.


720 is Product of Consecutive Numbers in Two Ways

:$720 = 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8$


Factorial which is Sum of Two Squares

The only factorial which can be expressed as the sum of two squares is: {{begin-eqn}} {{eqn | l = 6!       | r = 12^2 + 24^2       | c =  }} {{end-eqn}}


Multiplicity of 720 in 720 Factorial

The multiplicity of $720$ in $720!$ is $178$. That is: :$720^{178} \divides 720!$ but: :$720^{179} \nmid 720!$ where: :$720!$ denotes $720$ factorial :$\divides$ denotes divisibility :$\nmid$ denotes non-divisibility.


Continuous iff Mapping at Element is Supremum

Let $\left({S, \preceq_1, \tau_1}\right)$ and $\left({T, \preceq_2, \tau_2}\right)$ be complete continuous topological lattices with Scott topologies. Let $f: S \to T$ be a mapping. Then $f$ is continuous {{iff}} :$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$


Equivalence of Definitions of Generated Submodule

Let $R$ be a ring. Let $M$ be an $R$-module. Let $S\subset M$ be a subset. {{TFAE|def = Generated Submodule}}


Equivalence of Definitions of Basis of Vector Space

Let $K$ be a division ring. Let $\struct {G, +_G, \circ}_K$ be an vector space over $K$. {{TFAE| def = Basis of Vector Space}}


Equivalence of Definitions of Differentiable Real Function at Point

Let $\openint a b \subset \R$ be an open interval. Let $\xi$ be a point in $\openint a b$. {{TFAE|def = Differentiable Real Function at Point|view = differentiable real function at a point}}


Solutions to Approximate Fermat Equation x^3 = y^3 + z^3 Plus or Minus 1

The approximate Fermat equation: :$x^3 = y^3 + z^3 \pm 1$ has the solutions: {{begin-eqn}} {{eqn | l = 9^3       | r = 6^3 + 8^3 + 1 }} {{eqn | l = 103^3       | r = 64^3 + 94^3 - 1       | c =  }} {{end-eqn}}


Cubes which are Sum of Five Cubes

The following cube numbers can be expressed as the sum of $5$ positive cube numbers: :$9^3, \ldots$ {{expand|More terms needed. It seems that: </br> $4$ and all numbers $> 8$ can be so expressed </br> only $4, 8, 10, 11, 13$ require repeated cubes}}


Period of Reciprocal of 729 is 81

The decimal expansion of the reciprocal of $729$ has $\dfrac 1 9$ the maximum period, that is, $81$: :$\dfrac 1 {729} = 0 \cdotp \dot 00137 \, 17421 \, 12482 \, 85322 \, 35939 \, 64334 \, 70507 \, 54458 \, 16186 \, 55692 \, 72976 \, 68038 \, 40877 \, 91495 \, 19890 \, 26063 \, \dot 1$ The recurring part can be arranged in groups of $9$ digits each, revealing an interesting pattern: {{begin-eqn}} {{eqn | l = 001 \, 371 \, 742       | o =  }} {{eqn | l = 112 \, 482 \, 853       | o =  }} {{eqn | l = 223 \, 593 \, 964       | o =  }} {{eqn | l = 334 \, 705 \, 075       | o =  }} {{eqn | l = 445 \, 816 \, 186       | o =  }} {{eqn | l = 556 \, 927 \, 297       | o =  }} {{eqn | l = 668 \, 038 \, 408       | o =  }} {{eqn | l = 779 \, 149 \, 519       | o =  }} {{eqn | l = 890 \, 260 \, 631       | o =  }} {{end-eqn}} that is, each row (apart from the last) can be obtained from the previous one by adding $111 \, 111 \, 111$ to it. {{OEIS|A021733}}


Implicit Function Theorem for Differentiable Real Functions

Let $\Omega \subset \R^{n+k}$ be open. Let $f : \Omega \to \R^k$ be differentiable. Let the $i$th partial derivatives of $f$ be continuous in $\Omega$ for $n+1 \leq i \leq n+k$. Let $(a,b) \in \Omega$, with $a\in \R^n$ and $b\in \R^k$. Let $f(a,b) = 0$. For $(x_0,y_0)\in\Omega$, let $D_2 f(x_0,y_0)$ denote the differential of the function $y\mapsto f(x_0, y)$ at $y_0$. Let the linear map $D_2 f(a,b)$ be invertible. Then there exist neighborhoods $U\subset\Omega$ of $a$ and $V\subset\R^k$ of $b$ such that there exists a unique function $g : U \to V$ such that $f(x, g(x)) = 0$ for all $x\in U$. Moreover, $g$ is differentiable, and its differential satisfies: :$dg (x) = - \left( (D_2f)(x, g(x)) \right)^{-1} \circ (D_1 f)(x, g(x))$ for all $x\in U$.


Implicit Function Theorem for Smooth Real Functions

Let $\Omega \subset \R^{n+k}$ be open. Let $f : \Omega \to \R^k$ be smooth. Let $(a,b) \in \Omega$, with $a\in \R^n$ and $b\in \R^k$. Let $f(a,b) = 0$. For $(x_0,y_0)\in\Omega$, let $D_2 f(x_0,y_0)$ denote the differential of the function $y\mapsto f(x_0, y)$ at $y_0$. Let the linear map $D_2 f(a,b)$ be invertible. Then there exist neighborhoods $U\subset\Omega$ of $a$ and $V\subset\R^k$ of $b$ such that there exists a unique function $g : U \to V$ such that $f(x, g(x)) = 0$ for all $x\in U$. Moreover, $g$ is smooth, and its differential satisfies: :$dg (x) = - \left( (D_2f)(x, g(x)) \right)^{-1} \circ (D_1 f)(x, g(x))$ for all $x\in U$.


Sum of 4 Consecutive Binomial Coefficients forming Square

Consider the Diophantine equation: :$\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 = m^2$ where: :$\dbinom a b$ denotes a binomial coefficient :$n$ is an integer :$m$ is a non-negative integer. Then $n$ has one of the following values: :$-1, 0, 2, 7, 15, 74, 767$ {{OEIS|A047694}} The corresponding values of $m$ are: :$0, 1, 2, 8, 24, 260, 8672$ {{OEIS|A047695}}


Implicit Function Theorem for Lipschitz Contractions

Let $M$ and $N$ be metric spaces. Let $M$ be complete. Let $f : M \times N \to M$ be a Lipschitz continuous uniform contraction. Then for all $t\in N$ there exists a unique $g(t) \in M$ such that $f(g(t), t) = g(t)$, and the mapping $g : N \to M$ is Lipschitz continuous.


Local Normal Form for Immersions

Let $\Omega\subset\R^k$ be open. Let $f: \Omega \to \R^n$ be an immersion. Let $p \in \Omega$. Then: :$k \le n$ and there exists a local diffeomorphism $\phi$ around $\map f p$ such that: :$\phi \circ \map f x = \tuple {x, 0}$ for all $x$ in a neighborhood of $p$.


Local Normal Form for Submersions

Let $\Omega\subset\R^n$ be open. Let $f : \Omega \to \R^k$ be an submersion. Let $p\in\Omega$. Then $n\geq k$, and there exists a local diffeomorphism $\phi$ around $f(p)$ such that :$\phi\circ f (x, y) = x$ for all $(x, y)$ in a neighborhood of $p$.


Smallest Square Inscribed in Two Pythagorean Triangles

The smallest square with integer sides that can be inscribed within two different Pythagorean triangles so that one side of the square lies on the hypotenuse has side length $780$. The two Pythagorean triangles in question have side lengths $\tuple {1443, 1924, 2405}$ and $\tuple {1145, 2748, 2977}$.


Sequence of Numbers Divisible by Sequence of Primes

The integers in this sequence: :$788, 789, 790, 791, 792, 793$ are divisible by: :$2, 3, 5, 7, 11, 13$ respectively.


Numbers whose Square is Palindromic with Even Number of Digits

The sequence of positive integers whose square is a palindromic number with an even number of digits begins: :$836, 798 \, 644, 64 \, 030 \, 648, 83 \, 163 \, 115 \, 486, 6 \, 360 \, 832 \, 925 \, 898, \ldots$ {{OEIS|A016113}}


Sum of Sequence of Factorials

The sequence $S = \sequence {s_n}$ defined as: :$\displaystyle s_n = \sum_{k \mathop = 1}^n k!$ begins: :$1, 3, 9, 33, 153, 873, 5913, 46 \, 233, 409 \, 113, 4 \, 037 \, 913, \ldots$ {{OEIS|A007489}}


Number of Magic Squares of Order 4

The number of different magic squares of order $4$, up to rotation and reflection, is $880$.


Primitive Semiperfect Numbers which are not Primitive Abundant

The sequence of primitive semiperfect numbers which are not also primitive abundant starts: :$6, 28, 350, 490, 496, 770, 910, 1190, \ldots$ These are semiperfect numbers which are either: : perfect or: : whose only abundant aliquot parts are weird.


Sequence of Odd Abundant Numbers

The sequence of odd abundant numbers begins: :$945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, \ldots$ {{OEIS|A005231}}


Set of 5 Triplets whose Sums and Products are Equal

The following set of $5$ triplets of integers have the property that: :the sum of the integers in each triplet are equal and: :the product of the integers in each triplet are equal: :$\tuple {6, 480, 495}$, $\tuple {11, 160, 810}$, $\tuple {12, 144, 825}$, $\tuple {20, 81, 880}$, $\tuple {33, 48, 900}$ The sum is $981$, and the product is $1 \, 425 \, 600$. This is the only known such set of $5$ triplets of integers with this property.


Divisibility Test for 7, 11 and 13

Mark off the integer $N$ being tested into groups of $3$ digits. Because of the standard way of presenting integers, this may already be done, for example: :$N = 22 \, 846 \, 293 \, 462 \, 733 \, 356$ Number the groups of $3$ from the right:  :$N = \underbrace{22}_6 \, \underbrace{846}_5 \, \underbrace{293}_4 \, \underbrace{462}_3 \, \underbrace{733}_2 \, \underbrace{356}_1$ Considering each group a $3$-digit integer, add the even numbered groups together, and subtract the odd numbered groups: :$22 - 846 + 293 - 462 + 733 - 356 = -616$ where the sign is irrelevant. If the result is divisible by $7$, $11$ or $13$, then so is $N$. In this case: :$616 = 2^3 \times 7 \times 11$ and so $N$ is divisible by $7$ and $11$ but not $13$.


Solutions to Diophantine Equation x (x + 1) = y (y + 5) (y + 10) (y + 15)

The Diophantine equation : $n = x \left({x + 1}\right) = y \left({y + 5}\right) \left({y + 10}\right) \left({y + 15}\right)$ has exactly $2$ solutions: {{begin-eqn}} {{eqn | l = 1056       | r = 32 \times 33 = 1 \times 6 \times 11 \times 16 }} {{eqn | l = 43 \, 056       | r = 207 \times 208 = 8 \times 13 \times 18 \times 23 }} {{end-eqn}}


Numbers Reversed when Multiplying by 9

Numbers of the form $\sqbrk {10 (9) 89}_{10}$ are reversed when they are multiplied by $9$: {{begin-eqn}} {{eqn | l = 1089 \times 9       | r = 9801 }} {{eqn | l = 10 \, 989 \times 9       | r = 98 \, 901 }} {{eqn | l = 109 \, 989 \times 9       | r = 989 \, 901 }} {{end-eqn}} and so on.


Square which is Difference between Square and Square of Reversal

$33^2 = 65^2 - 56^2$ This is the only square of a $2$-digit number which has this property.


Numbers for which Sixth Power plus 1091 is Composite

The number $1091$ has the property that: :$x^6 + 1091$ is composite for all integer values of $x$ from $1$ to $3905$.


Wieferich's Criterion

Suppose Fermat's equation: :$x^p + y^p = z^p$ has a solution in which $p$ is an odd prime that does not divide any of $x$, $y$ or $z$. Then $2^{p - 1} - 1$ is divisible by $p^2$.


1105 as Sum of Two Squares

$1105$ can be expressed as the sum of two squares in more ways than any smaller integer: {{begin-eqn}} {{eqn | l = 1105       | m = 1089 + 16       | mo= =       | r = 33^2 + 4^2       | c =  }} {{eqn | m = 1024 + 81       | mo= =       | r = 32^2 + 9^2       | c =  }} {{eqn | m = 961 + 144       | mo= =       | r = 31^2 + 12^2       | c =  }} {{eqn | m = 625 + 529       | mo= =       | r = 24^2 + 23^2       | c =  }} {{end-eqn}}


Integer as Difference between Two Squares

Let $n$ be a positive integer. Then $n$ can be expressed as: :$n = a^2 - b^2$ {{iff}} $n$ has at least two distinct divisors of the same parity.


Difference between Two Squares equal to Repunit/Corollary 1

{{begin-eqn}} {{eqn | l = 6^2 - 5^2       | r = 11       | c =  }} {{eqn | l = 56^2 - 45^2       | r = 1111       | c =  }} {{eqn | l = 556^2 - 445^2       | r = 111 \, 111       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}} and in general for integer $n$: :$R_{2 n} = {\underbrace {55 \ldots 56}_{n - 1 \ 5 \text{'s} } }^2 - {\underbrace {44 \ldots 45}_{n - 1 \ 4 \text{'s} } }^2$ that is: :$\displaystyle \sum_{k \mathop = 0}^{2 n - 1} 10^k = \paren {\sum_{k \mathop = 1}^{n - 1} 5 \times 10^k + 6}^2 - \paren {\sum_{k \mathop = 1}^{n - 1} 4 \times 10^k + 5}^2$


Difference between Two Squares equal to Repunit/Corollary 2

{{begin-eqn}} {{eqn | l = 6^2 - 5^2       | r = 11       | c =  }} {{eqn | l = 56^2 - 45^2       | r = 1111       | c =  }} {{eqn | l = 5056^2 - 5045^2       | r = 111 \, 111       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}} and in general for integer $n$: :$R_{2 n} = {\underbrace{5050 \ldots 56}_{n - 1 \ 5 \text{'s} } }^2 - {\underbrace{5050 \ldots 45}_{n - 1 \ 5 \text{'s} } }^2$ that is: :$\displaystyle \sum_{k \mathop = 0}^{2 n - 1} 10^k = \left({\sum_{k \mathop = 1}^{n - 1} 5 \times 10^{2 k - 1} + 6}\right)^2 - \left({\sum_{k \mathop = 1}^{n - 1} 5 \times 10^{2 k - 1} - 5}\right)^2$


Numbers whose Squares are Consecutive Odd or Even Integers Juxtaposed

Integers whose squares consists of $2$ consecutive odd or even integers juxtaposed include: :$1127^2 = 01 \, 270 \, 129$ :$8874^2 = 78 \, 747 \, 876$ Such integers come in pairs which add to $1$ more than a power of $10$: :$1127 + 8874 = 10 \, 001$ {{expand|A great deal more to be done here}}


Sixth Power as Sum of 7 Sixth Powers

The smallest known integer whose $6$th power can be expressed as the sum of $7$ smaller $6$th powers is $1141$: :$1141^6 = 74^6 + 234^6 + 402^6 + 474^6 + 702^6 + 894^6 + 1077^6$


Numbers Not Expressible as Sum of no more than 5 Squares of Composite Numbers

There are $256$ integers which cannot be expressed as the sum of no more than $5$ squares of composite numbers: :$1, 2, 3, \ldots, 1167$ {{finish}}


Product of Injective Spaces is Injective

Let $I$ be a non-empty set. Let $\left({\left({S_i, \tau_i}\right)}\right)_{i \in I}$ be an indexed family of injective topological spaces. Then $\displaystyle \prod_{i \mathop \in I} \left({S_i, \tau_i}\right)$ is injective space.


Retract of Injective Space is Injective

Let $T = \left({S, \tau}\right)$ be an injective topological space. Let $R = \left({Z, \tau'}\right)$ be a retract of $T$. Then $R$ is injective.


Numbers equal to Sum of Squares of two Parts

Integers that can be split into two parts whose squares add up to it include: :$1233 = 12^2 + 33^2$ :$8833 = 88^2 + 33^2$ {{expand|Need to establish the parameters of this}}


Triples of Consecutive Sphenic Numbers

The sequence of triplets of consecutive sphenic numbers starts: :$\tuple {1309, 1310, 1311}, \tuple {1885, 1886, 1887}, \tuple {2013, 2014, 2015}, \ldots$ {{OEIS|A066509|order = first}} {{OEIS|A248202|order = middle}}


Smallest Quadruplet of Consecutive Integers Divisible by Cube

The smallest sequence of quadruplets of consecutive integers each of which is divisible by a cube greater than $1$ is: :$\tuple {22 \, 624, 22 \, 625, 22 \, 626, 22 \, 627}$


Riemann Zeta Function as a Multiple Integral

For $n \in \Z_{> 0}$, the Riemann zeta function is given by: :$\displaystyle \map \zeta n = \int_{\closedint 0 1^n} \frac 1 {1 - \prod_{i \mathop = 1}^n x_i} \prod_{i \mathop = 1}^n \rd x_i$ where $\closedint 0 1^n$ denotes the Cartesian $n$th power of the closed real interval $\closedint 0 1$.


Closed Form for Hexagonal Pyramidal Numbers

The closed-form expression for the $n$th hexagonal pyramidal number is: :$S_n = \dfrac {n \paren {n + 1} \paren {4 n - 1} } 6$


Tetrahedral and Triangular Numbers

The only positive integers which are simultaneously tetrahedral and triangular are: :$1, 10, 120, 1540, 7140$


Restriction of Composition is Composition of Restriction

Let $X, Y, Z$ be sets. Let $f: X \to Y$ and $g: Y \to Z$ be mappings. Let $S \subseteq X$. Then: : $\left({g \circ f}\right) \restriction S = g \circ \left({f \restriction S}\right)$


Smallest Fermat Pseudoprime to Bases 2, 3 and 5

The smallest Fermat pseudoprime to bases $2$, $3$ and $5$ is $1729$.


Numbers that Factorise into Sum of Digits and Reversal

The following positive integers can each be expressed as the product of the sum of its digits and the reversal of the sum of its digits: :$1, 81, 1458, 1729$ {{OEIS|A110921}}


1782 is 3 Times Sum of all 2-Digit Numbers from its Digits

$1782$ equals $3$ multiplied by the sum of all the $2$-digit integers that can be formed from its digits.


Triple of Consecutive Happy Numbers

The smallest triple of consecutive integers all of which are happy is: :$\left({1880, 1881, 1882}\right)$


Numbers whose Digits are Unchanged when Subtracting Reversal

The following sequence consists of the integers which have the property that subtraction of their reversals results in anagrams of them: :$954, 1980, 2961, 3870, 5823, 7641, 9108, 19980, 29880, 29961, 32760, \ldots$ {{OEIS|A121969}}


17 Consecutive Integers each with Common Factor with Product of other 16

The $17$ consecutive integers from $2184$ to $2200$ have the property that each one is not coprime with the product of the other $16$.


Lower and Upper Bounds for Sequences/Warning

Let $\sequence {x_n}$ be a sequence in $\R$. Let $x_n \to l$ as $n \to \infty$. Then it is '''not''' the case that: :$(1): \quad \forall n \in \N: x_n > a \implies l > a$ :$(2): \quad \forall n \in \N: x_n < b \implies l < b$


Smallest Fourth Power as Sum and Difference of Fourth Powers

The smallest $4$th power that can be expressed as the sum of $2$ $4$th powers minus a $3$rd is: :$2401 = 7^4 = 227^4 + 157^4 - 239^4$ with all numbers less than $10^4$.


Zsigmondy's Theorem for Sums

Let $a > b > 0$ be coprime positive integers. Let $n \ge 1$ be a (strictly) positive integer. Then there is a prime number $p$ such that :$p$ divides $a^n + b^n$ :$p$ does not divide $a^k + b^k$ for all $k < n$ with the following exception: :$n = 3$, $a = 2$, $b = 1$


Cyclotomic Polynomial of Index times Prime Power

Let $n, k \ge 1$ be natural numbers. Let $p$ be a prime number. Let $\Phi_n$ denote the $n$th cyclotomic polynomial. Then $\map {\Phi_{p^k n}} x = \begin{cases} \map {\Phi_n} {x^{p^k}} & \text{if } p \divides n\\ \dfrac {\map {\Phi_n} {x^{p^k}}} {\map {\Phi_n} {x^{p^{k - 1}}}} & \text{if } p \nmid n \end{cases}$


Multiplicative Order of Roots of Cyclotomic Polynomial Modulo Prime

Let $n\geq1$ be a natural number. Let $p$ be a prime number Let $n=p^\alpha q$ where $\alpha = \nu_p(n)$ is the valuation of $p$ in $n$. Let $a\in\Z$ with $\Phi_n(a)\equiv0\pmod p$. Then the order of $a$ modulo $p$ is $q$: :$\operatorname{ord}_p(a) = q$.


Lower Topology is Unique

Let $T_1 = \left({S, \preceq, \tau_1}\right)$ and $T_2 = \left({S, \preceq, \tau_2}\right)$ be relational structures with lower topologies. Then: : $\tau_1 = \tau_2$


Homogeneous Cyclotomic Polynomial is Symmetric

Let $n>1$ be a natural number. Let $\Phi_n(x,y)$ be the $n$th homogeneous cyclotomic polynomial. Then $\Phi_n(x,y) = \Phi_n(y,x)$, that is, $\Phi_n(x,y)$ is symmetric.


Cyclotomic Polynomial of Index Power of Two

Let $n \ge 1$ be a natural number. Then the $2^n$th cyclotomic polynomial is: :$\map {\Phi_{2^n} } x = x^{2^{n - 1} } + 1$


Trivial Estimate for Cyclotomic Polynomials

Let $n \ge 1$ be a natural number. Let $\Phi_n$ be the $n$th cyclotomic polynomial. Let $\phi$ be the Euler totient function. Let $z \in \C$ be a complex number. Then: :$\size {\size z - 1}^{\map \phi n} \le \size {\map {\Phi_n} z} \le \paren {\size z + 1}^{\map \phi n}$ where: :the first inequality becomes an equality only if: :::$n = 1$ and $z \in \R_{\ge 0}$ ::or: :::$n = 2$ and $z \in \R_{\le 0}$ :the second inequality becomes an equality only if: :::$n = 1$ and $z \in \R_{\le 0}$ ::or: :::$n = 2$ and $z \in \R_{\ge 0}$


Complete List of Special Highly Composite Numbers

There are exactly $6$ special highly composite numbers: :$1, 2, 6, 12, 60, 2520$ {{OEIS|A106037}}


Prime Decomposition of Highly Composite Number

Let $n$ be a highly composite number. Let the prime decomposition of $n$ be expressed as: :$n = \displaystyle \prod_{k \mathop \in \N} {p_k}^{r_k}$ where $p_k$ denotes the $k$th prime. Then the sequence $\left\langle{r_k}\right\rangle$ is decreasing. That is: :$\forall k \in \N: r_k \ge r_{k + 1}$


Complement of Upper Closure of Element is Open in Lower Topology

Let $T = \left({S, \preceq, \tau}\right)$ be a relational structure with lower topology. Let $x \in S$. Then $\complement_S\left({x^\succeq}\right)$ is open and $x^\succeq$ is closed.


Open Subset is Lower in Lower Topology

Let $T = \left({S, \preceq, \tau}\right)$ be a transitive relational structure with lower topology. Let $A \subseteq S$ such that :$A$ is open. Then $A$ is lower.


Ratio between Consecutive Highly Composite Numbers Greater than 2520 is Less than 2

The ratio between $2$ consecutive highly composite numbers both greater than $2520$ is less than $2$.


Modified Kaprekar Process on 4-Digit Number terminates in 2538

Let $n$ be a $4$-digit number. Let $n$ be operated on by the modified Kaprekar process. The eventual result is always $2538$.


Fibonacci Number equal to Sum of Sequence of Cubes

The following Fibonacci number can be expressed as the sum of a sequence of cubes: :$F_{18} = 2584 = 7^3 + 8^3 + 9^3 + 10^3$ {{expand|Any more?}}


Dudeney's Property of 2592

:$2592 = 2^5 \times 9^2$ It is the only number $n$ that has the property that: :$n = \sqbrk {abcd} = a^b \times c^d$ where $\sqbrk {abcd}$ denotes the decimal representation of $n$.


2601 as Sum of 3 Squares in 12 Different Ways

$2601$ can be expressed as the sum of $3$ squares in $12$ different ways.


Reversal of Number Multiplied by 11

Let $n \in \N$ be a number for which, when written in decimal notation, no two adjacent digits total to more than $9$. Let $n'$ denote the reversal of $n$. Then $n \times 11$ is the reversal of $n' \times 11$.


Reduction Formula for Integral of Power of Tangent

For all $n \in \Z_{> 1}$:   :$\displaystyle \int \map {\tan^n} x \rd x = \frac {\map {\tan^{n - 1} } x} {n - 1} - \int \map {\tan^{n - 2} } x \rd x$


Numbers Appearing 8 Times in Pascal's Triangle

The number $3003$ is the smallest integer to appear $8$ times in Pascal's triangle. No other number below $2^{23}$ appears as often.


3367 Multiplied by 2-Digit Number

In order to multiply $3367$ by a $2$-digit integer $\sqbrk {xy}$: :divide the $6$-digit integer $\sqbrk {xyxyxy}$ by $3$.


Integer as Sum of 2 Cubes in 3 Ways

$4104$ is the smallest natural number which can be expressed as the sum of $2$ cubes in $3$ different ways: {{begin-eqn}} {{eqn | l = 4104       | r = 16^3 + 2^3       | c =  }} {{eqn | r = 15^3 + 9^3       | c =  }} {{eqn | r = \paren {-12}^3 + 18^3       | c =  }} {{end-eqn}}


Composite Fibonacci Numbers with Prime Index

The sequence of composite Fibonacci numbers with a prime index begins: :$4181, 1 \, 346 \, 269, 24 \, 157 \, 817, 165 \, 580 \, 141, \ldots$ {{OEIS|A050937}} The corresponding sequence of prime indices begins: :$19, 31, 37, 41, 53, 59, 61, 67, 71, 73, 79, \ldots$ {{OEIS|A038672}}


Closed Subset is Upper in Lower Topology

Let $T = \left({S, \preceq, \tau}\right)$ be a transitive relational structure with lower topology. Let $A \subseteq S$ such that :$A$ is closed. Then $A$ is upper.


Square Pyramidal Number also Square

$4900$ is the only square pyramidal number which is also square: :$4900 = 70^2 = \displaystyle \sum_{k \mathop = 1}^{24} k^2 = \dfrac {24 \paren {24 + 1} \paren {2 \times 24 + 1} } 6$


Mapping Preserves Non-Empty Infima implies Mapping is Continuous in Lower Topological Lattice

Let $T = \left({S, \preceq, \tau}\right)$ and $Q = \left({X, \preceq', \tau'}\right)$ be complete topological lattices with lower topologies. Let $f: S \to X$ be a mapping such that :for all non-empty subsets $Y$ of $S$: $f$ preserves the infimum of $Y$. Then $f$ is continuous mapping.


5040 is Product of Consecutive Numbers in Two Ways

:$5040 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8 \times 7$


Products of Consecutive Integers in 2 Ways

The following integers are the product of consecutive integers in $2$ ways: :$-720, 720, 5040$


8 Mutually Non-Attacking Rooks on Chessboard

On a standard chessboard, it is possible to arrange a maximum of $8$ rooks so that no rook is attacking any other rook. There are $5282$ such arrangements, up to rotation and reflection.


Triangular Lucas Numbers

The only Lucas numbers which are also triangular are: :$1, 3, 5778$ {{OEIS|A248506}}


Subspace of Subspace is Subspace

Let $T = \struct{S, \tau}$ be a topological space. Let $H \subseteq S$ and $\tau_H$ be the subspace topology on $H$. Let $K\subseteq H$. Then the subspace topology on $K$ induced by $\tau$ equals the subspace topology on $K$ induced by $\tau_H$.


Positive Integer Sum of 3 Fourth Powers in 2 Ways

The smallest positive integer which can be expressed as the sum of $3$ fourth powers in $2$ different ways is $6578$: {{begin-eqn}} {{eqn | l = 1^4 + 2^4 + 9^4       | r = 1 + 16 + 6561       | c =  }} {{eqn | r = 6578       | c =  }} {{eqn | r = 81 + 2401 + 4096       | c =  }} {{eqn | r = 3^4 + 7^4 + 8^4       | c =  }} {{end-eqn}}


Mapping Preserves Infima implies Mapping is Continuous in Lower Topological Lattice

Let $T = \struct {S, \preceq, \tau}$ and $Q = \struct {X, \preceq', \tau'}$ be complete topological lattices with lower topologies. Let $f: S \to X$ be a mapping such that :$f$ preserves all infima. Then $f$ is continuous mapping.


Compact in Subspace is Compact in Topological Space

Let $T = \struct {S, \tau}$ be a topological space. Let $K \subseteq S$ be a subset. Let $\tau_K$ be the subspace topology on $K$. Let $T' = \struct {K, \tau_K}$ be the topological subspace of $T$ determined by $K$. Let $H \subseteq K$ be compact in $T'$. Then $H$ is compact in $T$.


If Infimum of Filtered Subset belongs to Element of Sub-Basis then Subset and Element Intersect implies Infimum of Subset belongs to Closure of Subset

Let $T = \left({S, \preceq, \tau}\right)$ be a complete topological lattice with lower topology. Let $B$ be an analytic sub-basis of $T$. Let $F$ be a filtered subset of $S$ such that :$\forall A \in B: \inf F \in A \implies F \cap A \ne \varnothing$ Then $\inf F \in F^-$ where $F^-$ denotes the topological closure of $F$.


Product with Repdigit can be Split into Parts which Add to Repdigit

Let $n$ be a positive integer with $d_1$ digits. Let $m$ be a repdigit number with $d_2$ digits such that $d_2 > d_1$. Let $r$ consist of the result when the rightmost $d_2$ digits of $m n$ is cut off and added to the remaining left hand portion. Then $r$ is a repdigit number.


Sum of Reciprocals of Squares of Odd Integers as Double Integral

:$\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^2} = \int_0^1 \int_0^1 \frac 1 {1 - x^2 y^2} \rd x \rd y$


Three Tri-Automorphic Numbers for each Number of Digits

Let $d \in \Z_{>0}$ be a (strictly) positive integer. Then there exist exactly $3$ tri-automorphic numbers with exactly $d$ digits. These tri-automorphic numbers all end in $2$, $5$ or $7$.


Fourier Series/x squared over Minus Pi to Pi

:$\displaystyle x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos n x}$


Integer and its Double forming Pandigital Pair

$6729$ and its double contain all the digits from $1$ to $9$ between them.


4-Digit Numbers forming Longest Reverse-and-Add Sequence

Let $m \in \Z_{>0}$ be a positive integer expressed in decimal notation. Let $r \left({m}\right)$ be the reverse-and-add process on $m$. Let $r$ be applied iteratively to $m$. The $4$-digit integers $m$ which need the largest number of iterations before reaching a palindromic number are: :$6999, 7998, 8997, 9996$ all of which need $20$ iterations.


Two Consecutive Integers each Product of Four Distinct Primes

The sequence of pairs of consecutive positive integers which are each the product of exactly $4$ distinct prime numbers begins: :$\tuple {7314, 7315}, \tuple {8294, 8295}, \tuple {8645, 8646}, \tuple {9009, 9010}, \ldots$ {{OEIS|A140078|order = first}}


Fourier Series/Identity Function over Minus Pi to Pi

For $x \in \openint {-\pi} \pi$: :$\displaystyle x = 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin n x$


Fourier Series/Fourth Power of x over Minus Pi to Pi

:$\displaystyle x^4 = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \cos n \pi \cos n x$


Fourier Series/Absolute Value of x over Minus Pi to Pi

For $x \in \openint {-\pi} \pi$: :$\displaystyle \size x = \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\map \cos {2 n - 1} x} {\paren {2 n - 1}^2}$


Fourier Series/Pi minus x over 0 to 2 Pi

:$\displaystyle \pi - x = 2 \sum_{n \mathop = 1}^\infty \frac {\sin n x} n$


Mapping is Continuous implies Mapping Preserves Filtered Infima in Lower Topological Lattice

Let $T = \left({S, \preceq, \tau}\right)$ and $Q = \left({X, \preceq', \tau'}\right)$ be complete topological lattices with lower topologies. Let $f: S \to X$ be a mapping such that :$f$ is a continuous mapping. Then $f$ preserves filtered infima.


3 Numbers in A.P. whose 4th Powers are Sum of Four 4th Powers

The following triplets of integers in arithmetic sequence with common difference of $60$ can all be expressed as the sum of four $4$th powers: :$\tuple {8373, 8433, 8493}, \tuple {8517, 8577, 8637}, \ldots$


Intersection of Chain of Prime Ideals of Commutative Ring is Prime Ideal

Let $R$ be a commutative ring. Let $\Spec R$ be the spectrum of $R$, ordered by inclusion. Let $\set {P_\alpha}_{\alpha \mathop \in A}$ be a non-empty chain of prime ideals of $\Spec R$. Let $\displaystyle P = \bigcap_{\alpha \mathop \in A} P_\alpha$ be their intersection. Then $P$ is a prime ideal of $R$.


Number of Different Ways to play First n Moves in Chess

The sequence formed from the number of ways to play the first $n$ moves in chess begins: :$20, 400, 8902, 197 \, 742, \ldots$ {{OEIS|A007545}} The count for the fourth move is already ambiguous, as it depends on whether only legal moves count, or whether all moves, legal or illegal, are included. The count as given here does include illegal moves in addition to legal ones.


Largest Product of Pandigital Factors

The largest integer that can be obtained by multiplying $2$ integers which between them use all the digits from $1$ to $9$ is: :$843 \, 973 \, 902 = 9642 \times 87531$


Smallest Pandigital Square

The smallest square number which contains all the digits from $1$ to $9$ is: :$11 \, 826^2 = 139 \, 854 \, 276$


Locally Compact Space is Weakly Locally Compact

Let $T = \struct{S, \tau}$ be a locally compact topological space. Then $T$ is weakly locally compact.


Space is Neighborhood of all its Points

Let $T = \struct {X, \tau}$ be a topological space. Let $x\in X$. Then $X$ is a neighborhood of $x$.


Local Compactness is Preserved under Open Continuous Surjection

Let $T_A = \left({S_A, \tau_A}\right)$ and $T_B = \left({S_B, \tau_B}\right)$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous mapping which is also an open mapping and a surjection. If $T_A$ is locally compact, then $T_B$ is also locally compact.


Equivalence of Definitions of Noetherian Module

Let $A$ be a commutative ring with unity. Let $M$ be an $A$-module. {{TFAE|def = Noetherian Module}}


Largest Positive Integer not Sum of Distinct Cubes

$12 \, 758$ is the largest positive integer that cannot be expressed as the sum of distinct cubes.


Particular Point Space is Locally Compact

Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is locally compact.


Product of Summations is Summation Over Cartesian Product of Products

This is a generalization of the distributive law: :$\displaystyle \prod_{a \mathop \in A} \sum_{b \mathop \in B_a} t_{a, b} = \sum_{c \mathop \in \prod_{a \mathop \in A} B_a} \prod_{a \mathop \in A} t_{a, c_a}$ where the product of sets $\prod_{a \mathop \in A} B_a$ is taken to be a cartesian product. {{explain|In order to make this comprehensible, the precise nature of $A$ and $B$ need to be defined. Presumably $A$ is a set (of numbers?), and $B$ is a family of sets of numbers indexed by $A$? And the $t$s are elements of $B$ and are numbers? I'm having difficulty.}}


Largest Number not Sum of Squares of Distinct Primes

The largest positive integer which cannot be expressed as the sum of the squares of distinct prime numbers is $17 \, 163$.


Volume of Smallest Rational Tetrahedron

The only rational tetrahedron whose edge lengths are less than $157$ has: : edges of length $117$, $80$, $53$, $52$, $51$, $84$ : faces of area $1800$, $1890$, $2016$, $1170$ : volume of $18 \, 144$.


Square and Tetrahedral Numbers

The only positive integers which are simultaneously tetrahedral and square are: :$1, 4, 19 \, 600$


Smallest Pythagorean Quadrilateral with Integer Sides

The smallest Pythagorean quadrilateral in which the sides of the $4$ right triangles formed by its sides and perpendicular diagonals are all integers has an area of $21 \, 576$. The sides of the right triangles in question are: :$25, 60, 65$ :$91, 60, 109$ :$91, 312, 325$ :$25, 312, 313$


Integers whose Tau value equals Cube Root

There are $3$ positive integers whose $\tau$ value equals its cube root: {{begin-eqn}} {{eqn | l = 1 = 1^3       | o = :       | r = \map \tau 1 = 1       | c = {{TauLink|1}} }} {{eqn | l = 21 \, 952 = 28^3       | o = :       | r = \map \tau {21 \, 952} = 28       | c = {{TauLink|21,952|21 \, 952}} }} {{eqn | l = 64 \, 000 = 40^3       | o = :       | r = \map \tau {64 \, 000} = 40       | c = {{TauLink|64,000|64 \, 000}} }} {{end-eqn}} {{OEIS|A066693}}


Smallest Integer which is Product of 4 Triples all with Same Sum

The smallest integer which can be expressed as the product of $4$ different triplets of integers each of which has the same sum is: {{begin-eqn}} {{eqn | l = 25 \, 200       | r = 6 \times 56 \times 75 }} {{eqn | r = 7 \times 40 \times 90 }} {{eqn | r = 9 \times 28 \times 100 }} {{eqn | r = 12 \times 20 \times 105 }} {{end-eqn}}


4 Integers whose Euler Phi Value is 10,368

:$\map \phi {25 \, 930} = \map \phi {25 \, 935} = \map \phi {25 \, 940} = \map \phi {25 \, 942} = 10 \, 368 = 2^7 \times 3^4$ where $\phi$ denotes the Euler $\phi$ function.


Change of Lead in Prime Number Race 4n+1 vs. 4n-1

Consider the prime number race between $4 n + 1$ and $4 n - 1$. While the prime numbers of the form $4 n - 1$ appear usually to be in the majority, the lead changes from one to the other an infinite number of times.


Fourth Power expressible as Sum of 6 Fourth Powers

$28 \, 561$ can be expressed as the sum of $6$ fourth powers: :$28 \, 561 = 13^4 = 12^4 + 8^4 + 7^4 + 6^4 + 2^4 + 2^4$


Smallest Differences between Fractional Parts of Square and Cube Roots

Apart from $6$th powers, the value of $n$ less than $50 \, 000$ for which the difference between the fractional parts of $\sqrt n$ and $\sqrt [3] n$ is smallest is $30 \, 739$. The next integer to produce a smaller difference above that is $62 \, 324$.


Smallest Triplet of Consecutive Integers each Divisible by Fourth Power

This triplet of consecutive integers has the property that each of them is divisible by a fourth power: :$33 \, 614, 33 \, 615, 33 \, 616$ This is the smallest such triplet.


Abundancy of Integers in form 945 + 630n

A large number of integers of the form $945 + 630 n$, for $n \in \Z_{\ge 0}$, are abundant. The first counterexample is for $n = 52$.


Sigma Function of Integer/Corollary

:$\displaystyle \map \sigma n = \prod_{\substack {1 \mathop \le i \mathop \le r \\ k_i \mathop > 1} } \frac {p_i^{k_i + 1} - 1} {p_i - 1} \prod_{\substack {1 \mathop \le i \mathop \le r \\ k_i \mathop = 1} } \paren {p_i + 1}$


Affine Group of One Dimension as Semidirect Product

Let $\map {\operatorname{Af}_1} \R$ be the $1$-dimensional affine group on $\R$. Let $\R^+$ be the additive group of real numbers. Let $\R^\times$ be the multiplicative group of real numbers. Let $\phi: \R^\times \to \Aut {\R^+}$ be defined as: :$\forall b \in \R^\times: \map \phi b = \paren {a \mapsto a b}$ Let $\R^+ \rtimes_\phi \R^\times$ be the corresponding semidirect product. Then: :$\map {\operatorname {Af}_1} \R \cong \R^+ \rtimes_\phi \R^\times$ where $\cong$ denotes (group) isomorphism.


Sequence of Consecutive Integers with Same Number of Divisors

The following sequence of consecutive integers all have the same number of divisors, that is, $8$: :$40 \, 311, 40 \, 312, 40 \, 313, 40 \, 314, 40 \, 315$ This is the longest such sequence known.


40,585/Historical Note

The fact that $40 \, 585$ is a factorion base $10$  was discovered as late as $1964$ by {{AuthorRef|Leigh Janes}}.


Smallest Sequence of 5 Consecutive Numbers which are Happy

The smallest sequence of $5$ consecutive integers all of which are happy numbers is: :$44 \, 488, 44 \, 489, 44 \, 490, 44 \, 491, 44 \, 492$


Reciprocal of 21

The decimal expansion of the reciprocal of $21$ has period $6$: :$\dfrac 1 {21} = 0 \cdotp \dot 04761 \, \dot 9$


Properties of 47,619

If you split $047 \, 619$ into two halves, they add up to $666$: :$047 + 619 = 666$ which is a multiple of $333$. and: : $047 \, 619 = 143 \times 333$ Similarly, you can split $047 \, 619$ into three thirds, and these add up to $99$: :$04 + 76 + 19 = 99$ and: : $047 \, 619 = 481 \times 99$ The square of $47 \, 619$: :$47 \, 619^2 = 2 \, 267 \, 569 \, 161$ can itself be split into two $6$-digit halves which together add to the recurring part of $\dfrac 4 7$: :$2267 + 569 \, 161 = 571 \, 428$ This is caused by the fact that $047 \, 619$ is the recurring part of the Reciprocal of 21: :$\dfrac 1 {21} = 0 \cdotp \dot 04761 \, \dot 9$ where $21$ is the product of (the smallest) $2$ distinct primes which do not divide $10$. {{finish|for whatever value of "finish off" is determined}}


Smallest Fourth Power as Sum of 5 Fourth Powers

The smallest $4$th power which can be expressed as the sum of $5$ $4$th powers is: :$15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4$


Boolean Interpretation is Well-Defined

Let $\LL_0$ be the language of propositional logic. Let $v: \LL_0 \to \set {\T, \F}$ be a boolean interpretation. Then $v$ is well-defined.


Pandigital Pairs whose Squares are Pandigital

The elements of the following pandigital pairs of integers each have squares which are themselves pandigital: :$\left({35 \, 172, 60 \, 984}\right), \left({57 \, 321, 60 \, 984}\right), \left({58 \, 413, 96 \, 702}\right), \left({59 \, 403, 76 \, 182}\right)$ {{OEIS|A085545|order = first}}


Language of Propositional Logic has Unique Parsability

The language of propositional logic $\mathcal L_0$ has unique parsability.


Powers of 2 not containing Digit Power of 2

$2^{16} = 65 \, 536$ is the only known power of $2$, up to $2^{31 \, 000}$, whose digits do not contain $1$, $2$, $4$ or $8$.


Construction of Regular 65,537-Gon

It is possible to construct a regular polygon with $65 \, 537$ sides) using a compass and straightedge construction.


There are no Odd Unitary Perfect Numbers

No unitary perfect numbers exist which are odd.


Automorphic Numbers with 5 Digits

The only $5$-digit automorphic number which does not begin with a zero is $90 \, 625$.


Kaprekar's Process on 5 Digit Number

Let $n$ be a $5$-digit integer whose digits are not all the same. Kaprekar's process, when applied to $n$, results in one of the following $3$ cycles: :$53 \, 955 \to 59 \, 994 \to 53 \, 955$ :$61 \, 974 \to 82 \, 962 \to 75 \, 933 \to 63 \, 954 \to 61 \, 974$ :$62 \, 964 \to 71 \, 973 \to 83 \, 952 \to 74 \, 943 \to 62 \, 964$


Tableau Extension Lemma/General Statement

Let $\mathbf H'$ be another finite set of WFFs. Then there exists a finished finite propositional tableau $T'$ such that: $(1):\quad$ the root of $T'$ is $\mathbf H \cup \mathbf H'$; $(2):\quad$ $T$ is a rooted subtree of $T'$.


Tableau Extension Lemma/General Statement/Proof 2

Let $T$ be a finite propositional tableau. Let its hypothesis set $\mathbf H$ be finite. {{:Tableau Extension Lemma/General Statement}}


Factors of Sums of Powers of 100,000

All integers $n$ of the form: :$n = \displaystyle \sum_{k \mathop = 0}^m 10^{5 k}$ for $m \in \Z_{> 0}$ are composite.


Points Defined by Adjacent Pairs of Digits of Reciprocal of 7 lie on Ellipse

Consider the digits that form the recurring part of the reciprocal of $7$: :$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$ Take the digits in ordered pairs, and treat them as coordinates of a Cartesian plane. It will be found that they all lie on an ellipse: :400px


Continuous Function on Compact Subspace of Euclidean Space is Bounded

Let $\R^n$ be the $n$-dimensional Euclidean space. Let $S \subseteq \R^n$ be a compact subspace of $\R^n$. Let $f: S \to \R$ be a continuous function. Then $f$ is bounded in $\R$.


Points Defined by Adjacent Pairs of Digits of Reciprocal of 13 lie on Hyperbola

Consider the digits that form the recurring part of the reciprocal of $13$: :$\dfrac 1 {13} = 0 \cdotp \dot 07692 \dot 3$ Take the digits in ordered pairs, and treat them as coordinates of a Cartesian plane. It will be found that they all lie on a hyperbola: :600px


Recurring Part of Fraction times Period gives 9-Repdigit

Let a (strictly) positive integer $n$ be such that the decimal expansion of its reciprocal has a recurring part of period $d$. Let $m$ be the integer formed from the $d$ digits of the recurring part. Then $m \times n$ is a $d$-digit repdigit number consisting of $9$s. {{mistake|Counterexample:1/6. Possibly true if n is not a multiple of 2 or 5}}


Properties of 142,857

This page gathers together some properties of $142 \, 857$ which arise through its being the digits of the recurring part of the reciprocal of $7$. Multiplication of $142 \, 857$ by numbers higher than $7$ produces a similar pattern to when you multiply it by a single digit, but with added complications. For example: {{begin-eqn}} {{eqn | l = 142 \, 857 \times 12       | r = 1 \, 714 \, 284       | c =  }} {{end-eqn}} This becomes $714 \, 285$ when you take the $1$ off the front and add it to the back. The exception is when you multiply it by $7$ or a multiple of $7$: {{begin-eqn}} {{eqn | l = 142 \, 857 \times 7       | r = 999 \, 999       | c =  }} {{eqn | l = 142 \, 857 \times 14       | r = 1 \, 999 \, 998       | c =  }} {{end-eqn}} From Recurring Part of Fraction times Period gives 9-Repdigit, it is seen that this property is shared of all numbers formed from the digits of the recurring part of a recurring fraction. If you divide $142 \, 857$ into two equal parts and add them, you get $999$: :$142 + 857 = 999$ Thus by Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999, $142 \, 857$ is divisible by $999$: :$142 \, 857 = 143 \times 999$ Also, we have: :$999 \, 999 = 1001 \times 999$ and so $999 \, 999$ is divisible by $999$. But as $999 \, 999 = 7 \times 142 \, 857$ we have that $999 \, 999$ is divisible by $7$. Thus it follows from Euclid's Lemma that $142 \, 857$ is divisible by $999$. {{finish|A lot of material from this chapter of Wells has been skipped, because it's just not very interesting. I leave it open for someone else to complete, if they want to.}}


Quotient of Group by Itself

Let $G$ be a group. Let $G / G$ be the quotient group of $G$ by itself. Then: :$G / G \cong \set e$ That is, the quotient of a group by itself is isomorphic to the trivial group.


Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999

Let $n$ be an integer which has at least $3$ digits when expressed in decimal notation. Let the digits of $n$ be divided into groups of $3$, counting from the right, and those groups added. Then the result is equal to a multiple of $999$ {{iff}} $n$ is divisible by $999$.


Number which is Sum of Subfactorials of Digits

The only integer which is the sum of the subfactorials of its digits is $148 \, 349$: :$148 \, 349 = \mathop !1 + \mathop !4 + \mathop !8 + \mathop !3 \mathop + \mathop !4 \mathop + \mathop !9$


Integers Representable as Product of both 3 and 4 Consecutive Integers

There are $3$ integers which can be expressed as both $x \paren {x + 1} \paren {x + 2} \paren {x + 3}$ for some $x$, and $y \paren {y + 1} \paren {y + 2}$ for some $y$: :$24, 120, 175 \, 560$


Squares whose Digits form Consecutive Increasing Integers

The sequence of integers whose squares have a decimal representation consisting of the concatenation of $2$ consecutive increasing integers begins: :$428, 573, 727, 846, 7810, 36 \, 365, 63 \, 636, 326 \, 734, \ldots$ {{OEIS|A030467}}


Smallest Fifth Power which is Sum of 6 Fifth Powers

The smallest fifth power which is the sum of $6$ fifth powers is $12^5 = 248 \, 832$: :$12^5 = 4^5 + 5^5 + 6^5 + 7^5 + 9^5 + 11^5$


Prime Numbers Embedded in Digits of Pi

The sequence of prime numbers that can be found starting from the beginning of the decimal expansion of $\pi$ (pi) begins: :$3, 31, 314 \, 159, 31 \, 415 \, 926 \, 535 \, 897 \, 932 \, 384 \, 626 \, 433 \, 832 \, 795 \, 028 \, 841, \ldots$ {{OEIS|A005042}}


Cube which can be Represented as Sum of 3, 4, 5, 6, 7 or 8 Cubes

:$351 \, 120^3$ can be represented as the sum of $3$, $4$, $5$, $6$, $7$ or $8$ cubes.


Prime Gaps of 100

The following pairs of consecutive prime numbers are those whose difference is $100$: :$\tuple {396 \, 733, 396 \, 833}, \ldots$ {{expand|Only know the first pair so far. Research needed to find the next one(s).}}


Property of 490,689

The number $490 \, 689$ can be expressed as the sum of $3$ cubes in $2$ different ways: :$490 \, 689 = 4^3 + 60^3 + 65^3 = 8^3 + 25^3 \times 78^3$ while at the same time the products of the contributory cube roots of each of those $2$ ways are equal: :$4 \times 60 \times 65 = 8 \times 25 \times 78$


510,510 is Product of 4 Consecutive Fibonacci Numbers

$510 \, 510$ Can be expressed as the product of $4$ distinct Fibonacci numbers: :$510 \, 510 = 13 \times 21 \times 34 \times 55$ and is also the $7$th primorial: :$510 \, 510 = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17$


Tableau Confutation contains Finite Tableau Confutation

Let $\mathbf H$ be a countable set of WFFs of propositional logic. Let $T$ be a tableau confutation of $\mathbf H$. Then there exists a finite rooted subtree of $T'$ that is also a tableau confutation of $\mathbf H'$.


Set of 7 Anagrams which are Square

The following integers are all anagrams, and all square: :$1 \, 048 \, 576, 1 \, 056 \, 784, 1 \, 085 \, 764, 5 \, 740 \, 816, 5 \, 764 \, 801, 6 \, 754 \, 801, 7 \, 845 \, 601$


Burnside's Lemma

Let $G$ be a finite group acting on a set $X$. Let $X / G$ be the set of orbits under this action. For $x \in X$, let $\Stab x$ be the stabilizer of $x$ by $G$. For $g \in G$, let $X^g$ denotes the set of all elements in $X$ which is fixed by $g$, that is: :$X^g := \set {x \in X: g x = x}$ Then: :$\displaystyle \size {X / G} = \frac 1 {\order G} \sum_{g \mathop \in G} \size {X^g}$ In words, the number of orbits equals the average number of fixed elements.


Smallest Cunningham Chain of the First Kind of Length 7

The smallest Cunningham chain of the first kind of length $7$ is: :$\left({1 \, 122 \, 659, 2 \, 245 \, 319, 4 \, 490 \, 639, 8 \, 981 \, 279, 17 \, 962 \, 559, 35 \, 925 \, 119, 71 \, 850 \, 239}\right)$


Factorisation of Quintic x^5 - x + n into Irreducible Quadratic and Irreducible Cubic

The quintic $x^5 - x + n$ can be factorized into the product of an irreducible quadratic and an an irreducible cubic {{iff}} $n$ is in the set: :$\set {\pm 15, \pm 22 \, 440, \pm 2 \, 759 \, 640}$


Factorial as Product of Consecutive Factorials

The only factorials which are the product of consecutive factorials are: {{begin-eqn}} {{eqn | l = 0!       | r = 0! \times 1!       | c =  }} {{eqn | l = 1!       | r = 0! \times 1!       | c =  }} {{eqn | l = 2!       | r = 1! \times 2!       | c =  }} {{eqn | r = 0! \times 1! \times 2!       | c =  }} {{eqn | l = 10!       | r = 6! \times 7!       | c =  }} {{end-eqn}}


Smallest Solution to Equation p^p times q^q = r^r

Consider the Diophantine equation: :$p^p \times q^q = r^r$ Its smallest solution is: {{begin-eqn}} {{eqn | l = p       | r = 12^6       | rr=  = 2 \, 985 \, 984 }} {{eqn | l = q       | r = 6^8       | rr=  = 1 \, 679 \, 616 }} {{eqn | l = r       | r = 2^{11} \times 3^7       | rr=  = 4 \, 478 \, 976 }} {{end-eqn}}


Existence of Matrix Logarithm

Let $T$ be a square matrix of order $n$. Then there exists a real matrix $S$ such that $e^S = T$ {{iff}}: :$(1): \quad  T$ is not a singular matrix and: :$(2): \quad $for every negative eigenvalue $\lambda$ of $T$ and for every positive integer $k$, the Jordan form of $T$ has an even number of $k \times k$ blocks associated with $\lambda$. pcu6qfvqi59x43t8w6ubk68j3p2rko5


Weak Existence of Matrix Logarithm

Let $T$ be a square matrix of order $n$. Let $\norm {T - I} < 1$ in the norm on bounded linear operators, where $I$ the identity matrix. Then there is a square matrix $S$ such that: :$e^S = T$ where $e^S$ is the matrix exponential.


Norm on Vector Space is Continuous Function

Let $V$ be a vector space with norm $\norm {\, \cdot \,}$. The function $\norm {\, \cdot \,}: V \to \R$ is continuous.


Largest Integer Not Expressible as Sum of Distinct 4th Powers

The largest integer which cannot be expressed as the sum of distinct $4$th powers is $5 \, 134 \, 240$.


Properties of 12,345,679

$12 \, 345 \, 679$ has the following properties: {{begin-eqn}} {{eqn | l = 12 \, 345 \, 679 \times 1       | r = 12 \, 345 \, 679       | c = digit $8$ is missing }} {{eqn | l = 12 \, 345 \, 679 \times 2       | r = 24 \, 691 \, 358       | c = digit $7$ is missing }} {{eqn | l = 12 \, 345 \, 679 \times 3       | r = 37 \, 037 \, 037       | c =  }} {{eqn | l = 12 \, 345 \, 679 \times 4       | r = 49 \, 382 \, 716       | c = digit $5$ is missing }} {{eqn | l = 12 \, 345 \, 679 \times 5       | r = 61 \, 728 \, 395       | c = digit $4$ is missing }} {{eqn | l = 12 \, 345 \, 679 \times 6       | r = 74 \, 074 \, 074       | c =  }} {{eqn | l = 12 \, 345 \, 679 \times 7       | r = 86 \, 419 \, 753       | c = digit $2$ is missing }} {{eqn | l = 12 \, 345 \, 679 \times 8       | r = 98 \, 765 \, 432       | c = digit $1$ is missing }} {{eqn | l = 12 \, 345 \, 679 \times 9       | r = 111 \, 111 \, 111       | c =  }} {{end-eqn}} In each product, the sequence $1$ to $9$, with the one given digit missing, can be read in order by cycling round it, skipping a fixed number of digits (counting an extra one when going from start to end), for example: :$2 \ (4691) \ 3 \ (58?2) \ 4 \ (6913) \ 5 \ (8?24) \ 6 \ (9135) \ 8 (?246) \ 9$ {{expand|Add some mathematical analysis explaining this phenomenon}}


Number of Ways to Tile Standard Chessboard with Dominoes

The number of ways to tile a standard chessboard with dominoes is $12 \, 988 \, 816$


Sequence of 8 Consecutive Primes with Same Pattern of Differences as from 11

The $8$ prime numbers starting at $15 \, 760 \, 091$ have the same prime gaps as the $8$ primes starting at $11$. Those $8$ primes are: :$15 \, 760 \, 091, 15 \, 760 \, 093, 15 \, 760 \, 097, 15 \, 760 \, 099, 15 \, 760 \, 103, 15 \, 760 \, 109, 15 \, 760 \, 111, 15 \, 760 \, 117$ Their prime gaps are: :$2, 4, 2, 4, 6, 2, 6$ The $8$ primes starting at $11$: :$11, 13, 17, 19, 23, 29, 31, 37$ Their prime gaps are: :$2, 4, 2, 4, 6, 2, 6$


Smallest Even Integer whose Euler Phi Value is not the Euler Phi Value of an Odd Integer

The smallest even integer whose Euler $\phi$ value is shared by no odd integer is $33 \, 817 \, 088$.


Smallest Integer which is Sum of 2 Cubes in 4 Ways

The smallest positive integer which can be expressed as the sum of $2$ cubes in $4$ different ways is: {{begin-eqn}} {{eqn | l = 42 \, 549 \, 416       | r = 348^3 + 74^3       | c =  }} {{eqn | r = 282^3 + 272^3       | c =  }} {{eqn | r = \left({-2662}\right)^3 + 2664^3       | c =  }} {{eqn | r = \left({-475}\right)^3 + 531^3       | c =  }} {{end-eqn}}


Squares whose Digits form Consecutive Integers

The sequence of integers whose squares have a decimal representation consisting of the concatenation of $2$ consecutive integers, either increasing or decreasing begins: :$91, 428, 573, 727, 846, 7810, 9079, 9901, 36 \, 365, 63 \, 636, 326 \, 734, 673 \, 267, 733 \, 674, \ldots$ This sequence can be divided into two subsequences:  Those where the consecutive integers are increasing: :$428, 573, 727, 846, 7810, 36 \, 365, 63 \, 636, 326 \, 734, 673 \, 267, \ldots$ {{OEIS|A030467}} Those where the consecutive integers are decreasing: :$91, 9079, 9901, 733 \, 674, 999 \, 001, 88 \, 225 \, 295, \ldots$ {{OEIS|A054216}}


Squares whose Digits form Consecutive Decreasing Integers

The sequence of integers whose squares have a decimal representation consisting of the concatenation of $2$ consecutive decreasing integers begins: :$91, 9079, 9901, 733 \, 674, 999 \, 001, 88 \, 225 \, 295, 99 \, 990 \, 001, \ldots$ {{OEIS|A030467}}


Numbers n whose Euler Phi value Divides n + 1

The following integers $n$ satisfy the equation: :$\exists k \in \Z: k \, \map \phi n = n + 1$ where $\phi$ denotes the Euler $\phi$ function: :$83 \, 623 \, 935, 83 \, 623 \, 935 \times 83 \, 623 \, 937$


Hardy-Ramanujan Number/Examples/87,539,319

The $3$rd Hardy-Ramanujan number $\operatorname {Ta} \left({3}\right)$ is $87 \, 539 \, 319$: {{begin-eqn}} {{eqn | l = 87 \, 539 \, 319       | r = 167^3 + 436^3       | c =  }} {{eqn | r = 228^3 + 423^3       | c =  }} {{eqn | r = 255^3 + 414^3       | c =  }} {{end-eqn}}


Polynomial is Linear Combination of Monomials

Let $R$ be a commutative ring with unity. Let $R \sqbrk X$ be a polynomial ring over $R$ in the variable $X$. Let $P \in R \sqbrk X$. Then $P$ is a linear combination of the monomials of $R \sqbrk X$, with coefficients in $R$. {{explain|this needs to be made more precise}}


Sum over Disjoint Union of Finite Sets

Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S$ and $T$ be finite disjoint sets. Let $S \cup T$ be their union. Let $f: S \cup T \to \mathbb A$ be a mapping. Then we have the equality of summations over finite sets: :$\displaystyle \sum_{u \mathop \in S \mathop \cup T} f \left({u}\right) = \sum_{s \mathop \in S} f \left({s}\right) + \sum_{t \mathop \in T} f \left({t}\right)$


Finite Summation does not Change under Permutation

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 $\sigma : S\to S$ be a permutation. Then we have the equality of summations over finite sets: :$\displaystyle \sum_{s \mathop \in S} \map f s = \sum_{s \mathop \in S} \map f {\map \sigma s}$


Change of Variables in Summation over Finite Set

Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S$ and $T$ be finite sets. Let $f: S \to \mathbb A$ be a mapping. Let $g: T \to S$ be a bijection. Then we have an equality of summations over finite sets: :$\displaystyle \sum_{s \mathop \in S} f \left({s}\right) = \sum_{t \mathop \in T} f \left({g \left({t}\right)}\right)$


Indexed Summation does not Change under Permutation

Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $a$ and $b$ be integers. Let $\closedint a b$ be the integer interval between $a$ and $b$. Let $f: \closedint a b \to \mathbb A$ be a mapping. Let $\sigma: \closedint a b \to \closedint a b$ be a permutation. Then we have an equality of indexed summations: :$\displaystyle \sum_{i \mathop = a}^b \map f i = \sum_{i \mathop = a}^b \map f {\map \sigma i}$


Indexed Summation over Translated Interval

Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $a$ and $b$ be integers. Let $\closedint a b$ be the integer interval between $a$ and $b$. Let $f: \closedint a b \to \mathbb A$ be a mapping. Let $c\in\Z$ be an integer. Then we have an equality of indexed summations: :$\displaystyle \sum_{i \mathop = a}^b f(i) = \sum_{i \mathop = a + c}^{b + c} \map f {i - c}$


Indexed Summation over Adjacent Intervals

Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$. Let $a,b,c$ be integers. Let $\left[{a \,.\,.\, c}\right]$ denote the integer interval between $a$ and $c$. Let $b \in \left[{a-1 \,.\,.\, c}\right]$. Let $f : \left[{a \,.\,.\, c}\right] \to \mathbb A$ be a mapping. Then we have an equality of indexed summations: :$\displaystyle \sum_{i \mathop = a}^c f(i) = \sum_{i \mathop = a}^b f(i) + \sum_{i \mathop = b+1}^c f(i)$


Indexed Summation over Interval of Length Two

Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$. Let $a\in\Z$ be an integer. Let $f : \{a, a+1\} \to \mathbb A$ be a real-valued function. Then the indexed summation: :$\displaystyle \sum_{i \mathop = a}^{a+1} f(i) = f(a) + f(a+1)$.


Indexed Summation over Interval of Length One

Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$. Let $a\in\Z$ be an integer. Let $f : \{a\} \to \mathbb A$ be a mapping on the singleton $\{a \}$. Then the indexed summation: :$\displaystyle \sum_{i \mathop = a}^{a} f(i) = f(a)$


Change of Variables in Indexed Summation

Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $a, b, c, d$ be integers. Let $\closedint a b$ denote the integer interval between $a$ and $b$. Let $f: \closedint a b \to \mathbb A$ be a mapping. Let $g: \closedint c d \to \closedint a b$ be a bijection. Then we have an equality of indexed summations: :$\displaystyle \sum_{i \mathop = a}^b \map f i = \sum_{i \mathop = c}^d \map f {\map g i}$


Translation of Integer Interval is Bijection

Let $a, b, c \in \Z$ be integers. Let $\closedint a b$ denote the integer interval between $a$ and $b$. Then the mapping $T: \closedint a b \to \closedint {a + c} {b + c}$ defined as: :$\map T k = k + c$ is a bijection.


Indexed Summation without First Term

Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$. Let $a$ and $b$ be integers with $a\leq b$. 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 we have an equality of indexed summations: :$\displaystyle \sum_{i \mathop = a}^b f(i) = f(a) + \sum_{i \mathop = a+1}^b f(\sigma(i))$


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: :$554 \, 688 \, 278 \, 429$, $1 \, 109 \, 376 \, 556 \, 859$, $2 \, 218 \, 753 \, 113 \, 719$, $4 \, 437 \, 506 \, 227 \, 439$, ::$8 \, 875 \, 012 \, 454 \, 879$, $17 \, 750 \, 024 \, 909 \, 759$, $35 \, 500 \, 049 \, 819 \, 519$, $71 \, 000 \, 099 \, 639 \, 039$, ::$142 \, 000 \, 199 \, 278 \, 079$, $284 \, 000 \, 398 \, 556 \, 159$, $568 \, 000 \, 797 \, 112 \, 319$, $1 \, 136 \, 001 \, 594 \, 224 \, 639$


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.