Square Pyramidal and Triangular Numbers

The only positive integers which are simultaneously square pyramidal and triangular are: :$1, 55, 91, 208 \, 335$ {{OEIS|A039596}}


Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes

The following positive integers cannot be expressed as the sum of distinct non-pythagorean primes: :$1, 2, 4, 5, 6, 8, 9, 12, 13, 15, 16, 17, 20, 24, 25, 27, 28, 32, 35, 36, 39, 48, 51, 55$ {{OEIS|A048262}} All positive integers greater than $55$ can be so expressed.


Sets of 4 Integers a, b, c, d for which Every Integer is in form ax^2 + by^2 + cz^2 + du^2

There are exactly $55$ sets of $4$ integers $\left\{ {a, b, c, d}\right\}$ such that all integers can be written in the form: :$n = a x^2 + b y^2 + c z^2 + d w^2$ for integer $x, y, z, w$.


Palindromes Formed by Multiplying by 55

$55$ multiplied by any of the odd integers between $91$ and $109$ inclusive produces a palindromic number.


Upper Adjoint of Galois Connection is Surjection implies Lower Adjoint at Element is Minimum of Preimage of Singleton of Element

Let $L = \struct {S, \preceq}, R = \paren {T, \precsim}$ be ordered sets. Let $g: S \to T, d:T \to S$ be mappings such that: :$\tuple {g, d}$ is a Galois connection and :$g$ is a surjection. Then :$\forall t \in T: \map d t = \min \set {g^{-1} \sqbrk {\set t} }$


Dirichlet Convolution Preserves Multiplicativity/General Result

Let $S \subset \N$ be a set of natural numbers with the property: :$m n \in S, \; \gcd \left({m, n}\right) = 1 \implies m, n \in S$ Define: :$\left({f*_S g}\right) \left({n}\right) = \displaystyle \sum_{\substack {d \mathop \backslash n \\ d \mathop \in S} } f \left({d}\right) g \left({n / d}\right)$ Then $f*_S g$ is multiplicative.


Even Integer with Abundancy Index greater than 9

Let $n \in \Z_{>0}$ have an abundancy index greater than $9$. Then $n$ has at least $35$ distinct prime factors.


Relation Between Rank and Exponent of Convergence

Let $f: \C \to \C$ be an entire function. Let $k$ be its rank and $\tau$ be its exponent of convergence. Then: * $k=\tau=0$ if $f$ has finitely many zeroes. * $k<\tau\leq k+1$ otherwise.


Sum of Sequence of Odd Squares/Formulation 1

:$\displaystyle \forall n \in \N: \sum_{i \mathop = 0}^n \paren {2 i + 1}^2 = \frac {\paren {n + 1} \paren {2 n + 1} \paren {2 n + 3} } 3$


Sum of Sequence of Even Squares

:$\displaystyle \forall n \in \N: \sum_{i \mathop = 0}^n \left({2 i}\right)^2 = \frac {2 n \left({n + 1}\right) \left({2 n + 1}\right)} 3$


Killing Form of Orthogonal Lie Algebra

Let $\mathbb K \in \left\{ {\C, \R}\right\}$. Let $n$ be a positive integer. Let $\mathfrak{so}_n \left({\mathbb K}\right)$ be the Lie algebra of the special orthogonal group $\operatorname{SO}_n \left({\mathbb K}\right)$. Then its Killing form is $B: \left({X, Y}\right) \mapsto \left({n - 2}\right) \operatorname {tr} \left({X Y}\right)$.


Trace of Alternating Product of Matrices and Almost Zero Matrices

Let $R$ be a ring with unity. Let $n, m$ be positive integers. Let $E_{ij}$ denote the $n \times n$ matrix with only zeroes except a $1$ at the $\tuple {i, j}$th element. Let $A_1, \ldots, A_m \in R^{n \times n}$. Let $i_k, j_k \in \set {1, \ldots, n}$ for $k \in \set {1, \ldots, m}$. Let $i_0 = i_m$ and $j_0 = j_m$. Then: :$\map \tr {A_1 E_{i_1, j_1} A_2 E_{i_2, j_2} \cdots A_m E_{i_m, j_m} } = \displaystyle \prod_{k \mathop = 1}^m \sqbrk {A_k}_{j_{k - 1} i_k}$


Trace in Terms of Orthonormal Basis

Let $\mathbb K \subset \C$ be a field. Let $\struct {V, \innerprod {\, \cdot \,} {\, \cdot \,} }$ be an inner product space over $\mathbb K$ of dimension $n$. Let $\tuple {e_1, \ldots, e_n}$ be an orthonormal basis of $V$. Let $f: V \to V$ be a linear operator. Then its trace equals: :$\map \tr f = \displaystyle \sum_{i \mathop = 1}^n \innerprod {\map f {e_i} } {e_i}$


Galois Connection with Upper Adjoint Surjective implies Scond Ordered Set and Image of Lower Adjoint are Isomorphic

Let $L = \left({S, \preceq}\right), R = \left({T, \precsim}\right)$ be ordered sets. Ley $g:S \to T, d:T \to S$ be mappings such that :$\left({g, d}\right)$ is Galois connection and :$g$ is a surjection/ Let $N = \left({d\left[{T}\right], \preceq'}\right)$ be an ordered subset of $L$. Then $R$ and $N$ are order isomorphic.


Positive Integer 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 $635 \, 318 \, 657$: {{begin-eqn}} {{eqn | l = 59^4 + 158^4       | r = 12 \, 117 \, 361 + 62 \, 201 \, 296       | c =  }} {{eqn | r = 635 \, 318 \, 657       | c =  }} {{eqn | r = 312 \, 900 \, 721 + 322 \, 417 \, 936       | c =  }} {{eqn | r = 133^4 + 134^4       | c =  }} {{end-eqn}}


Trace in Terms of Dual Basis

Let $R$ be a ring with unity. Let $M$ be a free $R$-module of dimension $n$. Let $\tuple {e_1, \ldots, e_n}$ be a basis of $M$. Let $\tuple {e_1^*,\ldots, e_n^*}$ be its dual basis Let $f: M \to M$ be a linear operator. Then its trace equals: :$\map \tr f = \displaystyle \sum_{i \mathop = 1}^n e_i^* \paren {\map f {e_i} }$


Internal Angle of Equilateral Triangle

The internal angles of an equilateral triangle measure $60^\circ$ or $\dfrac \pi 3$ radians.


Smallest Positive Integer which is Sum of 2 Odd Primes in 6 Ways

The smallest positive integer which is the sum of $2$ odd primes in $6$ different ways is $60$.


Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways

The sequence of positive integers $n$ which are the smallest such that they are the sum of $2$ odd primes in $k$ different ways begins as follows: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $k$  ! align="right" style = "padding: 2px 10px" | $n$  |- | align="right" style = "padding: 2px 10px" | $1$ | align="right" style = "padding: 2px 10px" | $6$ |- | align="right" style = "padding: 2px 10px" | $2$ | align="right" style = "padding: 2px 10px" | $10$ |- | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $22$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $34$ |- | align="right" style = "padding: 2px 10px" | $5$ | align="right" style = "padding: 2px 10px" | $48$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="right" style = "padding: 2px 10px" | $60$ |} {{OEIS|A001172}}


Change of Basis Matrix from Basis to Itself is Identity

Let $R$ be a ring with unity. Let $M$ be a free $R$-module of finite dimension $n>0$. Let $\mathcal B$ be an ordered basis of $M$. Then the change of basis matrix from $\mathcal B$ to $\mathcal B$ is the $n\times n$ identity matrix: :$\mathbf M_{\mathcal B, \mathcal B} = \mathbf I$


Sum of Sequence of Seventh Powers

:$\displaystyle \sum_{j \mathop = 0}^n j^7 = \dfrac {n^2 \paren {n + 1}^2 \paren {3 n^4 + 6 n^3 - n^2 - 4 n + 2} } {24}$


Change of Coordinate Vector Under Change of Basis

Let $R$ be a ring with unity. Let $M$ be a free $R$-module of finite dimension $n>0$. Let $\BB$ and $\CC$ be bases of $M$. Let $\mathbf M_{\BB, \CC}$ be the change of basis matrix from $\BB$ to $\CC$. Let $m\in M$. Let $\sqbrk m_\BB$ and $\sqbrk m_\CC$ be its coordinate vectors relative to $\BB$ and $\CC$ respectively. Then $\sqbrk m_\BB = \mathbf M_{\BB, \CC} \cdot \sqbrk m_\CC$.


Necessary and Sufficient Condition for Boundary Conditions to be Self-adjoint

Let $\mathbf p$ be continuously differentiable. The boundary conditions :$\bigvalueat {\map {\mathbf y} a} {x \mathop = a} = \bigvalueat {\map {\boldsymbol \psi} {\mathbf y} } {x \mathop = a}$ are self-adjoint {{iff}}: :$\forall i, k \in \N: 1 \le i, k \le N: \valueat {\dfrac {\partial p_i \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } } {\partial y_k} } {x \mathop = a} = \valueat {\dfrac {\partial p_k \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } } {\partial y_i} } {x \mathop = a}$


Reciprocal of 61

The decimal expansion of the reciprocal of $61$ has the maximum period, that is: $60$: :$\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$ {{OEIS|A007450}} It also contains an equal number ($6$) of each of the digits from $0$ to $9$.


Composition of Galois Connections is Galois Connection

Let $L_1 = \left({S_1, \preceq_1}\right), L_2 = \left({S_2, \preceq_2}\right), L_3 = \left({S_3, \preceq_3}\right)$ be ordered sets. Let $g_1:S_1 \to S_2, g_2:S_2 \to S_3, d_1:S_2 \to S_1, d_2:S_3 \to S_2$ be mappings such that :$\left({g_1, d_1}\right)$ and $\left({g_2, d_2}\right)$ are Galois connections. Then $\left({g_2 \circ g_1, d_1 \circ d_2}\right)$ is also Galois connection.


Vector Space has Basis

Let $K$ be a division ring. Let $V$ be a vector space over $K$. Then $V$ has a basis.


Product of Change of Basis Matrices

Let $R$ be a ring with unity. Let $M$ be a free $R$-module of finite dimension $n>0$. Let $\mathcal A$, $\mathcal B$ and $\mathcal C$ be ordered bases of $M$. Let $\mathbf M_{\mathcal A,\mathcal B}$, $\mathbf M_{\mathcal B,\mathcal C}$ and $\mathbf M_{\mathcal A,\mathcal C}$ be the change of basis matrices from $\mathcal A$ to $\mathcal B$, $\mathcal B$ to $\mathcal C$ and $\mathcal A$ to $\mathcal C$ respectively. Then $\mathbf M_{\mathcal A,\mathcal C} = \mathbf M_{\mathcal A,\mathcal B} \cdot \mathbf M_{\mathcal B,\mathcal C}$


Relative Matrix of Composition of Linear Transformations

Let $R$ be a ring with unity. Let $M, N, P$ be free $R$-modules of finite dimension $m, n, p > 0$ respectively. Let $\mathcal A,\mathcal B,\mathcal C$ be ordered bases of $M, N, P$. Let $f: M \to N$ and $g : N \to P$ be linear transformations, and $g \circ f$ be their composition. Let $\mathbf M_{f, \mathcal B, \mathcal A}$ and $\mathbf M_{g, \mathcal C, \mathcal B}$ be their matrices relative to $\mathcal A, \mathcal B$ and $\mathcal B, \mathcal C$ respectively. Then the matrix of $g \circ f$ relative to $\mathcal A$ and $\mathcal C$ is: :$\mathbf M_{g \mathop \circ f, \mathcal C, \mathcal A} = \mathbf M_{g, \mathcal C, \mathcal B}\cdot \mathbf M_{f, \mathcal B, \mathcal A}$


Change of Coordinate Vectors Under Linear Transformation

Let $R$ be a ring with unity. Let $M, N$ be free $R$-modules of finite dimension $m, n > 0$ respectively. Let $\mathcal A, \mathcal B$ be ordered bases of $M$ and $N$ respectively. Let $f: M \to N$ be a linear transformation. Let $\mathbf M_{f, \mathcal B, \mathcal A}$ be its matrix relative to $\mathcal A$ and $\mathcal B$. Then for all $m \in M$: :$\left[{f \left({m}\right)}\right]_{\mathcal B} = \mathbf M_{f, \mathcal B, \mathcal A} \cdot \left[{m}\right]_{\mathcal A}$ where $\left[{\, \cdot \,}\right]_{-}$ denotes the coordinate vector with respect to a basis.


Matrix of Bilinear Form Under Change of Basis

Let $R$ be a ring with unity. Let $M$ be a free $R$-module of finite dimension $n>0$. Let $\mathcal A$ and $\mathcal B$ be ordered bases of $M$. Let $\mathbf M_{\mathcal A, \mathcal B}$ be the change of basis matrix from $\mathcal A$ to $\mathcal B$. Let $f : M\times M \to R$ be a bilinear form. Let $\mathbf M_{f, \mathcal A}$ be its matrix relative to $\mathcal A$. Then its matrix relative to $\mathcal B$ equals: :$\mathbf M_{f, \mathcal B} = \mathbf M_{\mathcal A, \mathcal B}^\intercal \mathbf M_{f, \mathcal A} \mathbf M_{\mathcal A, \mathcal B}$


Symmetric Bilinear Form is Reflexive

Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$. Let $b$ be a bilinear form on $V$. Let $b$ be symmetric. Then $b$ is reflexive.


Alternating Bilinear Form is Reflexive

Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$. Let $b$ be a bilinear form on $V$. Let $b$ be alternating. Then $b$ is reflexive.


Bilinear Form is Reflexive iff Symmetric or Alternating

Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$. Let $b$ be a bilinear form on $V$. Then the following are equivalent: :$(1): \quad$ $b$ is reflexive :$(2): \quad$ $b$ is symmetric or alternating


Reflexive Bilinear Form is Symmetric or Alternating

Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$. Let $f$ be a bilinear form on $V$. Let $f$ be reflexive. Then $f$ is symmetric or alternating.


Apéry's Theorem

Apéry's constant: :$\map \zeta 3 = \displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$ is irrational.


Reciprocals whose Decimal Expansion contain Equal Numbers of Digits from 0 to 9

The following positive integers $p$ have reciprocals whose decimal expansions: :$(1): \quad$ have the maximum period, that is: $p - 1$ :$(2): \quad$ have an equal number, $\dfrac {p - 1} {10}$, of each of the digits from $0$ to $9$: ::$61, 131,\ldots$


Reciprocal of 131

The decimal expansion of the reciprocal of $131$ has the maximum period, that is: $130$: :$\dfrac 1 {131} = 0 \cdotp \dot 00763 \, 35877 \, 86259 \, 54198 \, 47328 \, 24427 \, 48091 \, 60305 \, 34351 \, 14503 \, 81679 \, 38931 \, 29770 \, 99236 \, 64122 \, 13740 \, 45801 \, 52671 \, 75572 \, 51908 \, 39694 \, 65648 \, 85496 \, 18320 \, 61068 \, 7022 \dot 9$ {{OEIS|A007450}} It also contains an equal number ($13$) of each of the digits from $0$ to $9$.


Sequences of 4 Consecutive Integers with Rising Sigma

The following ordered quadruples of consecutive integers have sigma values which are strictly increasing: :$61, 62, 63, 64$ :$73, 74, 75, 76$


Symmetric Bilinear Form can be Diagonalized

Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$ of finite dimension $n>0$. Let $f$ be a symmetric bilinear form on $V$. Then there exists an ordered basis for which the relative matrix of $f$ is diagonal.


Dimension of Radical of Bilinear Form

Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$ of finite dimension $n > 0$. Let $f$ be a bilinear form on $V$. Let $\operatorname{rad} \left({V}\right)$ be the radical of $V$. Let $\operatorname{rk} \left({f}\right)$ be the rank of $f$. Then: : $\dim \left({\operatorname{rad} \left({V}\right)}\right) = n - \operatorname{rk} \left({f}\right)$ where $\dim$ denotes dimension.


Dimension of Orthogonal Complement With Respect to Bilinear Form

Let $\mathbb K$ be a field. Let $V$ be a vector space over $\mathbb K$ of finite dimension. Let $f$ be a nondegenerate bilinear form on $V$. Let $U\subset V$ be a subspace. Let $U^\perp$ be its orthogonal complement. Then: :$\map \dim U + \map \dim U^\perp = \map \dim V$


Anisotropic Vector Gives Composition of Bilinear Space

Let $\mathbb K$ be a field. Let $\left({V, f}\right)$ be a bilinear space over $\mathbb K$. Let $v \in V$ be anisotropic. Let $\left\langle{v}\right\rangle$ be its span. Let $v^\perp$ be its orthogonal complement. Then $\left({V, f}\right)$ is the internal orthogonal sum of $\left\langle{v}\right\rangle$ and $v^\perp$: :$V = \left\langle{v}\right\rangle \oplus v^\perp$


Basis of Free Module is No Greater than Generator

Let $R$ be a commutative ring with unity. Let $M$ be a free $R$-module with basis $B$. Let $S$ be a generating set for $M$. Then: :$\size B \le \size S$. That is, there exists an injection from $B$ to $S$.


Extended Rolle's Theorem

Let $f: D \to \R$ be differentiable on a closed interval $I \subseteq \R$. Let $x_0 < x_1 < \dots < x_n \in I$. Let $f \left({x_i}\right) = 0$ for $i = 0, \ldots, n$. Then for all $i = 0, \ldots, n-1$: : $\exists \xi_i \in \left({x_i \,.\,.\, x_{i+1}}\right): f' \left({\xi_i}\right) = 0$


Pell's Equation/Examples/61

:$x^2 - 61 y^2 = 1$ has the smallest positive integral solution: :$x = 1 \, 766 \, 319 \, 049$ :$y = 226 \, 153 \, 980$


No Infinitely Descending Membership Chains/Corollary

There cannot exist a sequence $\left\langle{x_n}\right\rangle$ whose domain is $\N_{\gt 0}$ such that: :$\forall n \in \N_{\gt 0}: x_{n+1} \in x_n$


Sequence of Inconsummate Numbers

The sequence of inconsummate numbers begins: :$62, 63, 65, 75, 84, 95, 161, 173, 195, 216, 261, 266, 272, 276, \ldots$ {{OEIS|A003635}}


Kaprekar's Process for 2-Digit Numbers

Kaprekar's process, when applied to a non-repdigit $2$-digit positive integer leads to the cycle: :$09 \to 81 \to 63 \to 27 \to 45 \to 09$ Note that it is important to retain the leading zero on the $9$, or the process trivially terminates in $0$.


Numbers with 6 or more Prime Factors

The sequence of positive integers with $6$ or more prime factors (not necessarily distinct) begins: :$64, 96, 128, 144, 160, 192, 216, 224, 240, 256, \ldots$ {{OEIS|A046305}}


Cube as Sum of Sequence of Centered Hexagonal Numbers

:$C_n = \displaystyle \sum_{i \mathop = 1}^n H_i$ where: :$C_n$ denotes the $n$th cube number :$H_i$ denotes the $i$th centered hexagonal number.


Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared

Let $p$ be a prime number. Then there exists at least one positive integer $n$ greater than $1$ such that: :$n^{p - 1} \equiv 1 \pmod {p^2}$


Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes

The smallest positive integer which is both the sum of $2$ square numbers in two distinct ways and also the sum of $2$ cube numbers is $65$: {{begin-eqn}} {{eqn | l = 65       | m = 16 + 49       | mo= =       | r = 4^2 + 7^2       | c =  }} {{eqn | m = 1 + 64       | mo= =       | r = 1^2 + 8^2       | c =  }} {{eqn | o =        | mo= =       | r = 1^3 + 4^3       | c =  }} {{end-eqn}}


Magic Constant of Order 5 Magic Square

The magic constant of the order $5$ magic square is $65$.


Triple of Triangular Numbers whose Pairwise Sums are Triangular

The following triplet of triangular numbers has the property that the sum of each pair of them, and their total, are all triangular numbers: :$66, 105, 105$


Largest Even Integer not expressible as Sum of 2 k Odd Composite Integers

Let $k \in \Z_{>0}$ be a (strictly) positive integer. The largest even integer which cannot be expressed as the sum of $2 k$ odd positive composite integers is $18 k + 20$.


Number whose Square and Cube use all Digits Once

The only integer whose square and cube use each of the digits from $0$ to $9$ exactly once each is $69$.


Cube of 71 is Odd Integers in Sequence

The cube of $71$, when expressed in decimal notation, is the odd integers from $3$ to $11$ written in sequence: ::$71^3 = 357 \, 911$


Prime Numbers which Divide Sum of All Lesser Primes

The following sequence of prime numbers has the property that each is a divisor of the sum of all primes smaller than them: :$2, 5, 71, 369 \, 119, 415 \, 074 \, 643$ {{OEIS|A007506}} As of time of writing (April $2020$), no others are known.


4 Positive Integers in Arithmetic Sequence which have Same Euler Phi Value

The following sets of $4$ positive integers which form an arithmetic sequence are the smallest which all have the same Euler $\phi$ value: :$72, 78, 84, 90$ :$216, 222, 228, 234$ :$76 \, 326, 76 \, 332, 76 \, 338, 76 \, 344$


Positive Integers which are Euler Phi Value for 17 Integers

There are $17$ positive integers which have an Euler $\phi$ value of the following: :$72, 96, 120, \ldots$


Euler Phi Function of Non-Square Semiprime

Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$. Let $\map \phi n$ denote the Euler $\phi$ function. Then: :$\map \phi n = \paren {p - 1} \paren {q - 1}$


Euler Phi Function of Square-Free Integer

Let $n$ be an integer such that $n \ge 2$. Let $n$ be square-free. Let $\map \phi n$ be the Euler $\phi$ function of $n$. That is, let $\map \phi n$ be the count of strictly positive integers less than or equal to $n$ which are prime to $n$. Then: :$\map \phi n = \displaystyle \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$ where $p \divides n$ denotes the primes which divide $n$.


Euler Phi Function of 2 times Odd Prime

Let $n \in \Z_{>0}$ be a semiprime of the form $2 p$, where $p$ is an odd prime. Let $\map \phi n$ denote the Euler $\phi$ function. Then: :$\map \phi n = p - 1$


Set is Not Element of Itself

There cannot exist a set which is an element of itself. That is: :$\neg \exists a: a \in a$


Product of Number of Edges, Edges per Face and Faces of Tetrahedron

The product of the number of edges, edges per face and faces of a tetrahedron is $72$.


Product of Number of Edges, Edges per Face and Faces of Regular Octahedron

The product of the number of edges, edges per face and faces of a regular octahedron is $288$.


Ordinals are Well-Ordered/Corollary

Let $A$ be a set of ordinals. Let $\Epsilon {\restriction_A}$ denote the epsilon restriction on $A$. Then $A$ is strictly well-ordered by $\Epsilon {\restriction_A}$.


Product of Number of Edges, Edges per Face and Faces of Regular Icosahedron

The product of the number of edges, edges per face and faces of a regular icosahedron is $1800$.


Smallest 5th Power equal to Sum of 5 other 5th Powers

The smallest positive integer whose fifth power can be expressed as the sum of $5$ other fifth powers is $72$: :$72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5$


Smallest Consecutive Even Nontotients

The smallest pair of consecutive even nontotients is $74$ and $76$.


Positive Integers Expressible by Sum of Integers whose Reciprocals Sum to 1

Every positive integer over $77$ can be expressed as the sum of positive integers whose reciprocals add up to $1$. The full sequence of numbers that cannot be expressed as such is: :$2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 33, 34, 35, 36, 39, 40, 41, 42, 44, 46, 47, 48, 49, 51, 56, 58, 63, 68, 70, 72, 77$ {{OEIS|A051882}}


Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers

:$79 = 15 \times 1^4 + 4 \times 2^4$


Reciprocal of Square of 1 Less than Number Base

Let $b \in \Z$ be an integer such that $b > 2$. Let $n = \paren {b - 1}^2$. The reciprocal of $n$, expressed in base $b$, recurs with period $b - 1$: :$\dfrac 1 n = \sqbrk {0.012 \ldots cd012 \ldots}_b$ where: :$c = b - 3$ :$d = b - 1$


Square of 1 Less than Number Base

Let $b \in \Z$ be an integer such that $b > 2$. Let $n = b - 1$. The square of $n$ is expressed in base $b$ as: :$n^2 = \left[{c1}\right]_b$ where $c = b - 2$.


Positive Integers whose Square Root equals Sum of Digits

The following positive integers have a square root that equals the sum of their digits: :$0, 1, 81$ and there are no more.


Closed Form for Heptagonal Numbers

The closed-form expression for the $n$th heptagonal number is: :$H_n = \dfrac {n \paren {5 n - 3} } 2$


Square Numbers whose Sigma is Square

The sequence of square numbers whose $\sigma$ value is square starts as follows: {{begin-eqn}} {{eqn | l = \map \sigma {1^2}       | r = 1^2       | c =  }} {{eqn | l = \map \sigma {9^2}       | r = 11^2       | c =  }} {{eqn | l = \map \sigma {20^2}       | r = 31^2       | c =  }} {{eqn | l = \map \sigma {180^2}       | r = 341^2       | c =  }} {{eqn | l = \map \sigma {1306^2}       | r = 1729^2       | c =  }} {{end-eqn}} {{OEIS|A008847}}


Integers for which Sigma of Phi equals Sigma

The following positive integers have the property that the $\sigma$ value of their Euler $\phi$ value equals their $\sigma$ value: :$\map \sigma {\map \phi n} = \map \sigma n$ :$1, 87, 362, 1257, 1798, 5002, 9374, \ldots$ {{OEIS|A033631}}


Consecutive Triplets not Sum of Pentagonal Numbers

The following triplets of consecutive positive integers are such that none is the sum of $3$ pentagonal numbers: :$\tuple {19, 20, 21}$ :$\tuple {88, 89, 90}$ :$\tuple {99, 100, 101}$ :$\tuple {111, 112, 113}$


Prime Gaps of 8

The following pairs of consecutive prime numbers are those whose difference is $8$: :$\left({89, 97}\right), \left({359, 367}\right), \left({389, 397}\right), \left({401, 409}\right), \ldots$ {{OEIS|A031926}}


Smallest Cunningham Chain of the First Kind of Length 6

The smallest Cunningham chain of the first kind of length $6$ is: :$\tuple {89, 179, 359, 719, 1439, 2879}$


Sequence of Sum of Squares of Digits

For a positive integer $n$, let $\map f n$ be the integer created by adding the squares of digits of $n$. Let $m \in \Z_{>0}$ be expressed in decimal notation. Let $\sequence {S_m}_{n \mathop \in \Z_{>0} }$ be the sequence defined as follows: :$n_k = \begin{cases} m & : n = 1 \\ \map f {n_{k - 1} } & : n > 1 \end{cases}$ Then eventually either $\sequence {S_m}$ sticks at $1$, or goes into the cycle: :$\ldots, 4, 16, 37, 58, 89, 145, 42, 20, 4, \ldots$


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

Let $m \in \Z_{>0}$ be a positive integer expressed in decimal notation. Let $\map r m$ be the reverse-and-add process on $m$. Let $r$ be applied iteratively to $m$. The $2$-digit integers $m$ which need the largest number of iterations before reaching a palindromic number are $89$ and $98$, both needing $24$ iterations.


Cauchy Sequence is Bounded/Real Numbers

Every Cauchy sequence in $\R$ is bounded.


Fermat Pseudoprime/Base 3/Examples/91

The smallest Fermat pseudoprime to base $3$ is $91$: :$3^{91} \equiv 3 \pmod {91}$ despite the fact that $91$ is not prime: :$91 = 7 \times 13$


Even Integers not Sum of 2 Twin Primes

The following even integers cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes: :$2, 4, 94, 96, 98, 400, 402, 404, \ldots$ {{OEIS|A007534}} {{expand|add a page to the effect that it is conjectured that this list is complete.}}


Central Field is Field of Functional

Let $\mathbf y$ be an $N$-dimensional vector. Let $J$ be a functional, such that: :$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ Let the following be a central field: :$\map {\mathbf y'} x = \map {\boldsymbol \psi} {x, \mathbf y}$ Then this central field is a field of functional $J$.


Continued Fraction Expansion of Fourth Power of Pi

The continued fraction expansion of the $4$th power of $\pi$ begins: :$\pi^4 \approx 97 + \cfrac 1 {2 + \cfrac 1 {2 + \cfrac 1 {3 + \cfrac 1 {1 + \cfrac 1 {16 \, 539} } } } }$ {{OEIS|A058286}}


Decimal Expansion of Fourth Power of Pi

The decimal expansion of the $4$th power of $\pi$ begins: :$97 \cdotp 40909 \, 10340 \, 0 \ldots$ {{OEIS|A092425}}


Even Integers not Expressible as Sum of 3, 5 or 7 with Prime

The even integers that cannot be expressed as the sum of $2$ prime numbers where one of those primes is $3$, $5$ or $7$ begins: :$98, 122, 124, 126, 128, 148, 150, \ldots$ {{OEIS|A283555}}


Repdigit Number consisting of Instances of 9 is Kaprekar

A repdigit number that consists entirely of the digit $9$ is a Kaprekar number.


Numbers equal to Sum of Primes not Greater than its Prime Counting Function Value

Let $\pi \left({n}\right): \Z_{\ge 0} \to \Z_{\ge 0}$ denote the prime-counting function: :$\pi \left({n}\right) =$ the count of the primes less than $n$ Consider the equation: :$n = \displaystyle \sum_{p \mathop \le \pi \left({n}\right)} p$ where $p \le \pi \left({n}\right)$ denotes the primes not greater than $\pi \left({n}\right)$. Then $n$ is one of: :$5, 17, 41, 77, 100$ {{OEIS|A091864}}


Smallest Seventh Power which is Sum of 8 other Seventh Powers

The smallest seventh power that can be expressed as the sum of $8$ other seventh powers is $102^7$: :$102^7 = 12^7 + 35^7 + 53^7 + 58^7 + 64^7 + 83^7 + 85^7 + 90^7$


Consecutive Integers with Same Euler Phi Value

Let $\phi: \Z_{>0} \to \Z_{>0}$ be the Euler $\phi$ function, defined on the strictly positive integers. The equation: :$\map \phi n = \map \phi {n + 1}$ is satisfied by integers in the sequence: :$1, 3, 15, 104, 164, 194, 255, 495, 584, 975, 2204, \ldots$ {{OEIS|A001274}}


Sequences of 3 Consecutive Integers with Rising Phi

The following ordered triples of consecutive integers have $\phi$ values which are strictly increasing: :$105, 106, 107$ :$165, 166, 167$ :$315, 316, 317$


Integers whose Sigma equals Half Phi times Tau

The following positive integers $n$ have the property where: :$\sigma \left({n}\right) = \dfrac {\phi \left({n}\right) \times \tau \left({n}\right)} 2$ where: :$\sigma \left({n}\right)$ denotes the $\sigma$ function: the sum of the divisors of $n$ :$\phi \left({n}\right)$ denotes the Euler $\phi$ function: the count of positive integers smaller than of $n$ which are coprime to $n$ :$\tau \left({n}\right)$ denotes the $\tau$ function: the count of the divisors of $n$: These positive integers are: :$35, 105, \ldots$


Reciprocals of Odd Numbers adding to 1

$105$ is the smallest positive integer $n$ such that $1$ can be expressed as the sum of reciprocals of distinct odd integers such that none are less than $\dfrac 1 n$: :$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {33} + \dfrac 1 {35} + \dfrac 1 {45} + \dfrac 1 {55} + \dfrac 1 {77} + \dfrac 1 {105}$ {{OEIS|A238795}} There are $5$ ways of expressing $1$ as the sum of reciprocals of only $9$ distinct odd integers, but then the least term is less than $\dfrac 1 {105}$. The $9$-term solutions are as follows: :$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {35} + \dfrac 1 {45} + \dfrac 1 {231}$ {{OEIS|A201644}} :$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {21} + \dfrac 1 {231} + \dfrac 1 {315}$ {{OEIS|A201648}} :$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {33} + \dfrac 1 {45} + \dfrac 1 {385}$ {{OEIS|A201649}} :$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {21} + \dfrac 1 {165} + \dfrac 1 {693}$ {{OEIS|A201647}} :$1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {21} + \dfrac 1 {135} + \dfrac 1 {10 \, 395}$ {{OEIS|A201646}}


Smallest Polyomino with Hole

The smallest polyomino with a hole is the heptomino in the form of a $3 \times 3$  square with the center $1 \times 1$ square and a corner $1 \times 1$ square missing: :200px


Integers whose Sigma Value is Cube

The following positive integers are those whose $\sigma$ value is a cube: :$1, 7, 102, 110, 142, 159, 187, 381, 690, 714, 770, 994, 1034, \ldots$ {{OEIS|A020477}}


Magic Constant of Smallest Prime Magic Square

The magic constant of the smallest prime magic square is $111$.


Sequence of Square Lucky Numbers

The sequence of lucky numbers which are also square begins: :$1, 9, 25, 49, 169, 289, 361, 529, \ldots$ {{OEIS|A031162}}


Conditions for Extremal Embedding in Field of Functional

Let $J$ be a functional such that: :$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ Let $\gamma$ be an extremal of $J$, defined by $\mathbf y = \map {\mathbf y} x$ for $x \in \closedint a b$. Suppose: :$\forall x \in \closedint a b: \det \paren {F_{\mathbf y' \mathbf y'} } \ne 0$ Suppose no points conjugate to $\paren {a, \map {\mathbf y} a}$ lie on $\gamma$. Then $\gamma$ can be embedded in a field.


Sequence of Smallest Consecutive Composite Numbers longer than 100

The $1$st prime gap greater than $100$ is between $370 \, 261$ and $370 \, 373$, of length $112$. That is, the sequence of the smallest consecutive composite positive integers longer than $100$ is that of $111$ such, from  $370 \, 262$ to $370 \, 372$.


Difference between Two Squares equal to Repunit

The sequence of differences of two squares that each make a repunit begins: {{begin-eqn}} {{eqn | l = 1^2 - 0^2       | r = 1       | c =  }} {{eqn | l = 6^2 - 5^2       | r = 11       | c =  }} {{eqn | l = 20^2 - 17^2       | r = 111       | c =  }} {{eqn | l = 56^2 - 45^2       | r = 1111       | c =  }} {{eqn | l = 56^2 - 55^2       | r = 111       | c =  }} {{eqn | l = 156^2 - 115^2       | r = 11 \, 111       | c =  }} {{eqn | l = 340^2 - 67^2       | r = 111 \, 111       | c =  }} {{eqn | l = 344^2 - 65^2       | r = 111 \, 111       | c =  }} {{eqn | l = 356^2 - 125^2       | r = 111 \, 111       | c =  }} {{end-eqn}}


Smallest Equilateral Triangle with Internal Point at Integer Distances from Vertices

The smallest equilateral triangle with sides of integer length which contains a point which is an integer distance from each vertex has a side length $112$: :500px There exists a point inside it which is $57$, $65$ and $73$ away from the three vertices.


Smallest 3-Digit Permutable Prime

The smallest $3$-digit permutable prime is $113$.


Infinite Group has Infinite Number of Subgroups

Let $\struct {G, \circ}$ be an infinite group. Then $\struct {G, \circ}$ has an infinite number of distinct subgroups.


Group is Finite iff Finite Number of Subgroups

Let $\struct {G, \circ}$ be a group. Then $G$ is finite {{iff}} $\struct {G, \circ}$ has a  finite number of subgroups.


3-Digit Permutable Primes

The $3$-digit permutable primes are: :$311, 199, 337$ and their anagrams, and no other.


Digits of Permutable Prime

Let $p$ be a permutable prime with more than $1$ digit. Then $p$ contains only digits from the set: :$\left\{ {1, 3, 7, 9}\right\}$


Divisibility by 5

An integer $N$ expressed in decimal notation is divisible by $5$ {{iff}} the units digit of $N$ is divisible by $5$. That is: :$N = \sqbrk {a_n \ldots a_2 a_1 a_0}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $5$  {{iff}}: :$a_0$ is divisible by $5$.


Prime Repdigit Number is Repunit

Let $b \in \Z_{>0}$ be an integer greater than $1$. Let $n \in \Z$ expressed in base $b$ be a repdigit number with more than $1$ digit. Let $n$ be prime. Then $n$ is a repunit (in base $b$).


Prime Gaps of 14

The following pairs of consecutive prime numbers are those whose difference is $14$: :$\tuple {113, 127}, \tuple {293, 307}, \tuple {317, 331}, \ldots$ {{OEIS|A031932}}


Prime Gaps of 18

After the prime gap of $14$ between the pairs of consecutive prime numbers: :$\left({113, 127}\right), \left({293, 307}\right), \left({317, 331}\right), \ldots$ the next prime gap which is greater than $14$ is between the pair of consecutive prime numbers: :$\left({523, 541}\right)$ for a prime gap of $18$.


Number of Different Ways to Colour the Faces of Cube with 3 Colours

The number of different ways to colour the faces of a cube with $3$ given colours, one colour per face, is $114$.


Number of Different Ways to Colour the Faces of Cube with 4 Colours

The number of different ways to colour the faces of a cube with $4$ given colours, one colour per face, is $2652$.


Number of Different Ways to Colour the Faces of Cube with 5 Colours

The number of different ways to colour the faces of a cube with $5$ given colours, one colour per face, is $29 \, 660$.


Smallest Number which is Sum of 4 Triples with Equal Products

The smallest positive integer which is the sum of $4$  distinct ordered triples, each of which has the same product, is $118$: {{begin-eqn}} {{eqn | l = 118       | r = 14 + 50 + 54 }} {{eqn | r = 15 + 40 + 63       | c =  }} {{eqn | r = 18 + 30 + 70       | c =  }} {{eqn | r = 21 + 25 + 72       | c =  }} {{end-eqn}}


Interval containing Prime Number of forms 4n - 1, 4n + 1, 6n - 1, 6n + 1

Let $n \in \Z$ be an integer such that $n \ge 118$. Then between $n$ and $\dfrac {4 n} 3$ there exists at least one prime number of each of the forms: :$4 m - 1, 4 m + 1, 6 m - 1, 6 m + 1$


Sum of Cubes of 5 Consecutive Integers which is Square

The following sequences of $5$ consecutive positive integers have cubes that sum to squares: :$1, 2, 3, 4, 5$ :$96, 96, 98, 99, 100$ :$118, 119, 120, 121, 122$ No other such sequence of $5$ consecutive positive integers has the same property.


Smallest Number to appear 6 Times in Pascal's Triangle

The smallest positive integer greater than $1$ to appear $6$ times in Pascal's Triangle is $120$.


Smallest n such that 6 n + 1 and 6 n - 1 are both Composite

The smallest positive integer $n$ such that $6 n + 1$ and $6 n - 1$ are both composite is $20$.


Smallest Number with 16 Divisors

The smallest positive integer with $16$ divisors is $120$.


Fermat Set is Diophantine Quadruple

The Fermat set $F = \left\{{1, 3, 8, 120}\right\}$ is a Diophantine quadruple: :$\forall a, b \in F: a \ne b: a b + 1 = n^2$ for some $n \in \Z$.


Fermat Set cannot be Extended to Diophantine Quintuple

The Fermat set $F = \left\{{1, 3, 8, 120}\right\}$ cannot be extended to a Diophantine quintuple.


Ratio of 360 to Aliquot Sum

$360$ has the property that its ratio to its aliquot sum is $4 : 9$.


Are All Triperfect Numbers Even?/Progress/Minimum Size

It has been established that an odd triperfect number, if one were to exist, would be greater than $10^{70}$. If it does not have $3$ as a prime factor, then it is greater than $10^{108}$.


Are All Triperfect Numbers Even?/Progress/Prime Factors

An odd triperfect number has: :at least $11$ distinct prime factors :at least $32$ distinct prime factors if $3$ is not one of them.


Are All Triperfect Numbers Even?/Progress/Form

An odd triperfect number is square.


Squares which are 4 Less than Cubes

The only two square numbers which are $4$ less than a cube are: :$2^2 + 4 = 2^3$ :$11^2 + 4 = 5^3$


Integer Greater than 121 is Sum of Distinct Primes of form 4 n + 1

Let $n$ be an integer greater than $121$. Then $n$ can be expressed as the sum of distinct prime numbers of the form $4 n + 1$.


Palindromes in Base 10 and Base 3

The following $n \in \Z$ are palindromic in both decimal and ternary: :$0, 1, 2, 4, 8, 121, 151, 212, 242, 484, 656, 757, \ldots$ {{OEIS|A007633}}


Square Number which is Sum of Consecutive Powers

The only square number which is the sum of consecutive powers of a positive integer is $121$: :$121 = 3^0 + 3^1 + 3^2 + 3^3 + 3^4$


Numbers whose Difference equals Difference between Cube and Seventh Power

The following $2$ pairs of integers are the only ones known which exhibit this pattern: :$\left\vert {5^3 - 2^7}\right\vert = 5 - 2$ :$\left\vert {13^3 - 3^7}\right\vert = 13 - 3$


Triangles with Integer Area and Integer Sides in Arithmetical Sequence

The triangles with the following sides in arithmetic sequence have integer areas: :$3, 4, 5$ :$13, 14, 15$ :$15, 28, 41$ :$15, 26, 37$ Their areas are: :$6, 84, 126, 156$


Equivalence of Definitions of Weierstrass E-Function

Let $\mathbf y, \mathbf z, \mathbf w$  be $n$-dimensional vectors. Let $\mathbf y$ be such that $\map{\mathbf y} a=A$ and $\map{\mathbf y} b=B$. Let $J$ be a functional such that: :$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ {{TFAE|def = Weierstrass E-Function}}


Conditions for Strong Minimum of Functional

Let $\mathbf y$ be an $n$-dimensional vector such that $\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$ Let $J$ be a functional such that: :$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ Let $\gamma$ be an extremal curve of $J$. Let the following be the field of the functional $J$: :$\mathbf y' = \map {\boldsymbol \psi} {x, \mathbf y}$ Let $R$ be an open region containing $\gamma$ and have the field $\boldsymbol \psi$ defined as $\forall \paren {x, \mathbf y} \in R$. Let $\mathbf w$ be a finite vector. Suppose that: :$\forall \paren {x, \mathbf y} \in R: \map E {x, \mathbf y, \boldsymbol \psi, \mathbf w}\ge 0$ where $E$ is Weierstrass E-Function. Then $J$ has a strong minimum for $\gamma$. {{explain|What is a strong minimum?}}


Factor of Mersenne Number Mp is of form 2kp + 1

:$q = 2 k p + 1$ for some integer $k$.


Factor of Mersenne Number Mp equivalent to 1 mod p

:$q \equiv 1 \pmod p$


Factor of Mersenne Number equivalent to +-1 mod 8

:$q \equiv \pm 1 \pmod 8$


Numbers with 7 or more Prime Factors

The sequence of positive integers with $7$ or more prime factors (not necessarily distinct) begins: :$128, 192, 256, 288, 320, 384, 432, 448, 480, 512, \ldots$ {{OEIS|A046307}}


Numbers not Sum of Distinct Squares

The positive integers which are not the sum of $1$ or more distinct squares are: :$2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128$ {{OEIS|A001422}}


132 is Sum of all 2-Digit Numbers formed from its Digits

$132$ is the smallest sum of all the $2$-digit (positive) integers formed from its own digits.


Harmonic Mean of Divisors in terms of Tau and Sigma

Let $n \in \Z_{>0}$ be a positive integer. The harmonic mean of the divisors of $n$ is given by: :$H \left({n}\right) = \dfrac {n \, \tau \left({n}\right)} {\sigma \left({n}\right)}$ where: :$\tau \left({n}\right)$ denotes the $\tau$ (tau) function: the number of divisors of $n$ :$\sigma \left({n}\right)$ denotes the $\sigma$ (sigma) function: the sum of the divisors of $n$.


Perfect Number is Ore Number

Let $n \in \Z_{>0}$ be a perfect number. Then $n$ is an Ore number.


Square Fibonacci Number

After $1$, there exists exactly one Fibonacci number which is also square: :$F_{12} = 144 = 12^2$ which is also coincidentally the square of its index.


Carmichael's Theorem

Let $n \in \Z$ such that $n > 12$. Then the $n$th Fibonacci number $F_n$ has at least one prime factor which does not divide any smaller Fibonacci number. The exceptions for $n \le 12$ are: :$F_1 = 1, F_2 = 1$: neither have any prime factors :$F_6 = 8$ whose only prime factor is $2$ which is $F_3$ :$F_{12} = 144$ whose only prime factors are $2$ (which is $F_3$) and $3$ (which is $F_4$).


Weierstrass's Necessary Condition

Let $\mathbf y: \R \to \R^n$ be an $n$-dimensional vector-valued function such that $\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$. Let $J$ be a functional such that: :$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ Let $\mathbf w$ be an $n$-dimensional vector such that $\mathbf w \in \R^n$. Let $\gamma$ be a strong minimum of $J$. {{explain|exactly what a ''strong'' minimum is, by means of a link to a definition page}} Then along $\gamma$ and for every $\mathbf w$: :$\map E {x, \mathbf y, \mathbf y', \mathbf w} \ge 0$ {{explain|exactly what ''along'' $\gamma$ means, by means of a link to a definition page}} where $E$ stands for the Weierstrass E-Function.


Squares Ending in Repeated Digits

A square number $n^2$ can end in a repeated digit {{iff}} either: :$(1): \quad n^2$ is a multiple of $100$, in which case $n$ is a multiple of $10$ :$(2): \quad n^2$ ends in $44$ and $n$ ends in $12, 38, 62$ or $88$.


Magic Constant of Smallest Prime Magic Square with Consecutive Primes

The magic constant of the smallest prime magic square whose elements are consecutive odd primes is $4 \, 440 \, 084 \, 513$.


Magic Constant of Smallest Prime Magic Square with Consecutive Primes from 3

The magic constant of the smallest prime magic square whose elements are consecutive odd primes from $3$ upwards is $4514$.


Sum of 2 Squares in 2 Distinct Ways/Examples/145

$145$ can be expressed as the sum of two square numbers in two distinct ways: {{begin-eqn}} {{eqn | l = 145       | r = 12^2 + 1^2 }} {{eqn | r = 9^2 + 8^2 }} {{end-eqn}}


Representation of 1 as Sum of n Unit Fractions

Let $U \left({n}\right)$ denote the number of different ways of representing $1$ as the sum of $n$ unit fractions. Then for various $n$, $U \left({n}\right)$ is given by the following table: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="right" style = "padding: 2px 10px" | $U \left({n}\right)$  |- | align="right" style = "padding: 2px 10px" | $1$ | align="right" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $2$ | align="right" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $3$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $14$ |- | align="right" style = "padding: 2px 10px" | $5$ | align="right" style = "padding: 2px 10px" | $147$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="right" style = "padding: 2px 10px" | $3462$ |} {{OEIS|A002966}}


Sum of 3 Unit Fractions that equals 1

There are $3$ ways to represent $1$ as the sum of exactly $3$ unit fractions.


Necessary Condition for Integral Functional to have Extremum for given function/Dependent on n Variables

Let $\mathbf x$ be an $n$-dimensional vector. Let $\map u {\mathbf x}$ be a real function. Let $R$ be a fixed region. Let $J$ be a functional such that :$\ds J \sqbrk u = \idotsint_R \map F {\mathbf x, u, u_{\mathbf x} } \rd x_1 \cdots \rd x_n$ Then a necessary condition for $J \sqbrk u$ to have an extremum (strong or weak) for a given mapping $\map u {\mathbf x}$ is that $\map u {\mathbf x}$ satisfies Euler's equation: :$F_u - \dfrac {\partial} {\partial \mathbf x} F_{u_{\mathbf x} } = 0$


Repeated Sum of Cubes of Digits of Multiple of 3

Let $k \in \Z_{>0}$ be a positive integer. Let $f: \Z_{>0} \to \Z_{>0}$ be the mapping defined as: :$\forall m \in \Z_{>0}: \map f m = $ the sum of the cubes of the digits of $n$. Let $n_0 \in \Z_{>0}$ be a (strictly) positive integer which is a multiple of $3$. Consider the sequence: :$s_n = \begin{cases} n_0 & : n = 0 \\ \map f {s_{n - 1} } & : n > 0 \end{cases}$ Then: :$\exists r \in \N_{>0}: s_r = 153$ That is, by performing $f$ repeatedly on a multiple of $3$ eventually results in the pluperfect digital invariant $153$.


Numbers for which Euler Phi Function of 2n + 1 is less than that of 2n

The sequence of positive integers for which: :$\map \phi {2 n + 1} < \map \phi {2 n}$ begins: :$157, 262, 367, 412, \ldots$ {{OEIS|A001837}}


Odd Numbers Not Expressible as Sum of 5 Distinct Non-Zero Coprime Squares

The largest odd positive integer that cannot be expressed as the sum of exactly $5$ non-zero square numbers all of which are coprime is $245$. {{expand|The full list of those numbers is to be investigated.}}


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

:$159 = 14 \times 1^4 + 4 \times 2^4 + 3^4$


Numbers not Expressible as Sum of Distinct Pentagonal Numbers

The positive integer which cannot be expressed as the sum of distinct pentagonal numbers are: :$2, 3, 4, 7, 8, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30,$ :$31, 32, 33, 37, 38, 42, 43, 44, 45, 46, 49, 50, 54, 55, 59, 60, 61, 65,$ :$66, 67, 72, 77, 80, 81, 84, 89, 94, 95, 96, 100, 101, 102, 107, 112, 116,$ :$124, 136, 137, 141, 142, 147, 159$


Sum of Distinct Primes of form 6n-1

$161$ is the largest integer that cannot be expressed as the sum of distinct primes of the form $6 n - 1$.


Equivalence of Definitions of Polynomial Ring in One Variable

Let $R$ be a commutative ring with unity. The following definitions of polynomial ring are equivalent in the following sense: : For every two constructions, there exists an $R$-isomorphism which sends indeterminates to indeterminates. {{explain|this statement has to be made more precise}} === Definition 1: As a Ring of Sequences === {{:Definition:Polynomial Ring/Sequences}} === Definition 2: As a Monoid Ring on the Natural Numbers === {{:Definition:Polynomial Ring/Monoid Ring on Natural Numbers}}


Smallest Triplet of Integers whose Product with Tau Value are Equal

Let $\map \tau n$ denote the $\tau$ function: the number of divisors of $n$. The smallest set of $3$ integers $T$ such that $m \, \map \tau m$ is equal for each $m \in T$ is: :$\set {168, 192, 224}$


General Variation of Integral Functional/Dependent on n Variables

Let $\mathbf x$ be an $n$-dimensional vector. Let $u = \map u {\mathbf x}$ be a real-valued function. Let $J$ be a functional such that: :$\ds J \sqbrk u = \int_R \map F {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } \rd x_1 \dotsm \rd x_n$ Let $\mathbf x^*, u^*$ be defined by the following transformations $\paren \star$: {{begin-eqn}} {{eqn | l = \mathbf x^*       | r = \map {\boldsymbol \Phi} {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; \epsilon} = \mathbf x + \epsilon \map {\boldsymbol \phi} {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } + \map \OO {\epsilon^2} }} {{eqn | l = u^*       | r = \map \Psi {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; \epsilon} = \mathbf x + \epsilon \map \psi {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x } } + \map \OO {\epsilon^2} }} {{end-eqn}} where: :$\boldsymbol \Phi$, $\Psi$ are differentiable {{WRT|Differentiation}} $\epsilon$ and: {{begin-eqn}} {{eqn | l = \map {\boldsymbol \Phi} {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; 0}       | r = \mathbf x }} {{eqn | l = \map \Psi {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x}; 0}       | r = u }} {{eqn | l = \map {\boldsymbol \phi} {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} }       | r = \valueat {\dfrac {\partial \boldsymbol \Phi} {\partial \epsilon} } {\epsilon \mathop = 0} }} {{eqn | l = \map \psi {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} }       | r = \valueat {\dfrac {\partial \Psi} {\partial \epsilon} } {\epsilon \mathop = 0} }} {{end-eqn}} Then the variation of the functional $J$ due to the original mapping being transformed by $\paren \star$ reads: {{explain|not clear what "aforementioned transformation" refers to. Best to identify it with a label and refer to that label.}} :$\ds \delta J = \epsilon \int_R \paren {F_u - \dfrac {\partial F_{u_{\mathbf x} } } {\partial \mathbf x} } \overline \psi \rd x_1 \dotsm \rd x_n + \epsilon \int_R \map {\dfrac {\partial} {\partial \mathbf x} } {F_{u_x} \overline {\boldsymbol \psi} + F \boldsymbol \phi} \rd x_1 \dotsm \rd x_n$ where: :$\overline \psi = \psi - u_{\mathbf x} \boldsymbol \phi$


Sequence of Record Peaks in Values of Sigma

Let $d: \Z_{>0} \to \Z_{>0}$ be the mapping defined as: :$d \left({n}\right) = \sigma \left({n}\right) - \sigma \left({n'}\right)$ where: :$n$ denotes a highly abundant number :$n'$ denotes the previous highly abundant number. The following $n \in \Z_{>0}$ have the property that they have a higher value of $d \left({n}\right)$ than any smaller $n$: :$\forall m \in \Z_{>0}: m < n \implies \sigma \left({m}\right) < \sigma \left({n}\right)$ That is, they are the peak $\sigma$ values which exceed the previous peak by a higher number than any previous peak. That is, they are highly abundant number which have $\sigma$ values  whose difference with that of the $\sigma$ (sigma) value of the previous highly abundant numbers is greater than that with the previous record difference. :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="right" style = "padding: 2px 10px" | $n'$  ! align="right" style = "padding: 2px 10px" | $\sigma \left({n}\right)$  ! align="right" style = "padding: 2px 10px" | $\sigma \left({n'}\right)$  ! style = "padding: 2px 10px" | $d \left({n}\right)$  |- | align="right" style = "padding: 2px 10px" | $2$ | align="right" style = "padding: 2px 10px" | $1$ | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $1$ | align="right" style = "padding: 2px 10px" | $2$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $7$ | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $3$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $12$ | align="right" style = "padding: 2px 10px" | $7$ | align="right" style = "padding: 2px 10px" | $5$ |- | align="right" style = "padding: 2px 10px" | $12$ | align="right" style = "padding: 2px 10px" | $10$ | align="right" style = "padding: 2px 10px" | $28$ | align="right" style = "padding: 2px 10px" | $18$ | align="right" style = "padding: 2px 10px" | $10$ |- | align="right" style = "padding: 2px 10px" | $24$ | align="right" style = "padding: 2px 10px" | $20$ | align="right" style = "padding: 2px 10px" | $60$ | align="right" style = "padding: 2px 10px" | $42$ | align="right" style = "padding: 2px 10px" | $18$ |- | align="right" style = "padding: 2px 10px" | $36$ | align="right" style = "padding: 2px 10px" | $30$ | align="right" style = "padding: 2px 10px" | $91$ | align="right" style = "padding: 2px 10px" | $72$ | align="right" style = "padding: 2px 10px" | $19$ |- | align="right" style = "padding: 2px 10px" | $48$ | align="right" style = "padding: 2px 10px" | $42$ | align="right" style = "padding: 2px 10px" | $124$ | align="right" style = "padding: 2px 10px" | $96$ | align="right" style = "padding: 2px 10px" | $28$ |- | align="right" style = "padding: 2px 10px" | $60$ | align="right" style = "padding: 2px 10px" | $48$ | align="right" style = "padding: 2px 10px" | $168$ | align="right" style = "padding: 2px 10px" | $124$ | align="right" style = "padding: 2px 10px" | $44$ |- | align="right" style = "padding: 2px 10px" | $120$ | align="right" style = "padding: 2px 10px" | $108$ | align="right" style = "padding: 2px 10px" | $360$ | align="right" style = "padding: 2px 10px" | $280$ | align="right" style = "padding: 2px 10px" | $80$ |}


Order Isomorphism forms Galois Connection

Let $L_1 = \left({S_1, \preceq_1}\right)$, $L_2 = \left({S_2, \preceq_2}\right)$ be ordered sets. Let $f:S_1 \to S_2$ be an order isomorphism between $L_1$ and $L_2$. Then $\left({f, f^{-1} }\right)$ is a Galois connection.


Numbers of Primes with at most n Digits

Let $p: \Z_{>0} \to \Z_{>0}$ be the mapping defined as: :$\forall n \in \Z_{>0}: p \left({n}\right) = $ the number of prime numbers with no more than $n$ digits Then the value of $p$ for the first few numbers is given below: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="right" style = "padding: 2px 10px" | $p \left({n}\right)$  |- | align="right" style = "padding: 2px 10px" | $1$ | align="right" style = "padding: 2px 10px" | $4$ |- | align="right" style = "padding: 2px 10px" | $2$ | align="right" style = "padding: 2px 10px" | $25$ |- | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $168$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $1229$ |- | align="right" style = "padding: 2px 10px" | $5$ | align="right" style = "padding: 2px 10px" | $9592$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="right" style = "padding: 2px 10px" | $78 \, 498$ |- | align="right" style = "padding: 2px 10px" | $7$ | align="right" style = "padding: 2px 10px" | $664 \, 579$ |- | align="right" style = "padding: 2px 10px" | $8$ | align="right" style = "padding: 2px 10px" | $5 \, 761 \, 455$ |- | align="right" style = "padding: 2px 10px" | $9$ | align="right" style = "padding: 2px 10px" | $50 \, 847 \, 534$ |- | align="right" style = "padding: 2px 10px" | $10$ | align="right" style = "padding: 2px 10px" | $455 \, 052 \, 511$ |} {{OEIS|A006880}}


Sequence of Square Centered Hexagonal Numbers

The sequence of centered hexagonal numbers which are also square begins: :$1, 169, 32 \, 761, 6 \, 355 \, 441, \ldots$ {{OEIS|A006051}}


169 as Sum of up to 155 Squares

$169$ can be expressed as the sum of $n$ non-zero squares for all $n$ from $1$ to $155$.


Lattice of Power Set is Arithmetic

Let $X$ be a set. Let $P = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be a lattice of power set. Then $P$ is arithmetic.


Nth Derivative of Natural Logarithm

The $n$th derivative of $\map \ln x$ for $n \ge 1$ is: :$\dfrac {\d^n} {\d x^n} \ln x = \dfrac {\paren {n - 1}! \paren {-1}^{n - 1} } {x^n}$


Sum of Factorials of Digits of 169

Let the factorials of the digits of $169$ be added. Let the same process be done on the result. Repeat. After $3$ iterations, the result will be $169$.


Squares which are Difference between Two Cubes

$169$ is the smallest square number which is the difference between two cubes: :$169 = 8^3 - 7^3$ {{expand|Add the rest of the sequence, having found out what they are.}}


Cube of 180 is Sum of Sequence of Consecutive Cubes

:$180^3 = \displaystyle \sum_{k \mathop = 6}^{69} k^3$ That is: :$180^3 = 6^3 + 7^3 + \cdots + 67^3 + 68^3 + 69^3$


Numbers Expressible as Sum of Five Distinct Squares

The largest positive integer which cannot be expressed as the sum of no more than $5$ distinct squares is $188$. Both $188$ and $124$ require as many as $6$ distinct squares to represent them: {{begin-eqn}} {{eqn | l = 124       | r = 1 + 4 + 9 + 25 + 36 + 49       | c =  }} {{eqn | r = 1^2 + 2^2 + 3^2 + 5^2 + 6^2 + 7^2       | c =  }} {{eqn | l = 188       | r = 1 + 4 + 9 + 25 + 49 + 100       | c =  }} {{eqn | r = 1^2 + 2^2 + 3^2 + 5^2 + 7^2 + 10^2       | c =  }} {{end-eqn}}


Closed Form for Heptagonal Pyramidal Numbers

The closed-form expression for the $n$th heptagonal pyramidal number is: :$Q_n = \dfrac {n \paren {n + 1} \paren {5 n - 2} } 6$


Heptagonal Pyramidal Numbers which are Square

The sequence of heptagonal pyramidal numbers which also have the property of being square begins: :$0, 1, 196, \ldots$ {{expand|Further research needed}}


Conditions for Limit Function to be Limit Minimizing Function of Functional

Let $y$ be a real function. Let $J \sqbrk y$ be a functional. Let $\sequence {y_n}$ be a minimizing sequence of $J$. Let: :$\ds \lim_{n \mathop \to \infty} y_n = \hat y$ Suppose $J$ is lower semicontinuous at $y = \hat y$. Then: :$\ds J \sqbrk {\hat y} = \lim_{n \mathop \to \infty} J \sqbrk {y_n}$


Smallest 10 Primes in Arithmetic Sequence

The smallest $10$ primes in arithmetic sequence are: :$199 + 210 n$ for $n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9$. These are also the smallest $8$ and $9$ primes in arithmetic sequence.


Sum of Cubes of 3 Consecutive Integers which is Square

The following sequence of $3$ consecutive positive integers have cubes that sum to a square: :$23, 24, 25$


Integer Greater than 205 is Sum of Distinct Primes of form 6 n + 1

Let $n$ be an integer greater than $205$. Then $n$ can be expressed as the sum of distinct prime numbers of the form $6 n + 1$.


Numbers such that Tau divides Phi divides Sigma

The sequence of integers $n$ with the property that: :$\map \tau n \divides \map \phi n \divides \map \sigma n$ where: :$\divides$ denotes divisibility :$\tau$ denotes the $\tau$ (tau) function: the count of divisors of $n$ :$\phi$ denotes the Euler $\phi$ (phi) function: the count of smaller integers coprime to $n$ :$\sigma$ denotes the $\sigma$ (sigma) function: the sum of divisors of $n$ begins: :$1, 3, 15, 30, 35, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, \ldots$ {{OEIS|A020493}}


Number of Representations as Sum of Two Primes

The number of ways an integer $n$ can be represented as the sum of two primes is no greater than the number of primes in the interval $\closedint {\dfrac n 2} {n - 2}$.


Integers whose Number of Representations as Sum of Two Primes is Maximum

$210$ is the largest integer which can be represented as the sum of two primes in the maximum number of ways. The full list of such numbers is as follows: :$1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 24, 30, 36, 42, 48, 60, 90, 210$ {{OEIS|A141340}} The list contains: :$n \le 8$ :$n \le 18$ where $2 \divides n$ :$n \le 48$ where $2 \times 3 \divides n$ :$n \le 90$ where $2 \times 3 \times 5 \divides n$ :$210 = 2 \times 3 \times 5 \times 7$ {{WIP|Add this sequence to the number pages}}


25 as Sum of 4 to 11 Squares

$25$ can be expressed as the sum of $n$ non-zero squares for all $n$ from $4$ to $11$.


Cubes which are Sum of Three Cubes

The following cube numbers can be expressed as the sum of $3$ positive cube numbers: :$6^3, 9^3, 12^3, 18^3, 19^3, 20^3, 24^3, 25^3, \ldots$ {{OEIS|A066890}} with cube roots: :$6, 9, 12, 18, 19, 20, 24, 25, \ldots$ {{OEIS|A023042}} {{WIP|Add these sequences to the separate pages for the numbers themselves}}


Ritz Method implies Not Worse Approximation with Increased Number of Functions

Consider the Ritz method. Let $\eta_n = \boldsymbol \alpha \boldsymbol \phi$. Let $J \sqbrk {\eta_n} = \mu_n$. Then: :$\mu_n \ge \mu_{n + 1}$


Dissection of Cube into 3 Cubes using 8 Pieces

A cube can be dissected into $3$ smaller cubes by cutting it into $8$ pieces and reassembling them.


Smallest Multiplicative Magic Square is of Order 3

The order of the smallest multiplicative magic square is $3$, for example: {{:Multiplicative Magic Square/Examples/Order 3/Smallest}}


230 Fedorov Groups where Chiral Pairs are Distinct

There are $219$ Fedorov groups, if chiral copies are considered distinct.


Equivalence of Definitions of Amicable Pair

Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers. {{TFAE|def = Amicable Pair}}


Thabit's Rule

Let $n$ be a positive integer such that: {{begin-eqn}} {{eqn | l = a       | r = 3 \times 2^n - 1       | c =  }} {{eqn | l = b       | r = 3 \times 2^{n - 1} - 1       | c =  }} {{eqn | l = c       | r = 9 \times 2^{2 n - 1} - 1       | c =  }} {{end-eqn}} are all prime. Then: :$\tuple {2^n a b, 2^n c}$ forms an amicable pair.


Smaller of Thabit Pair is Tetrahedral

Let $\tuple {m_1, m_2}$ be a Thabit pair such that $m_1 < m_2$. Then $m_1$ is a tetrahedral number.


Amicable Pair with Smallest Common Prime Factor 5

A amicable pair whose smallest common prime factor is greater than $3$ has the elements: :$m_1 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 101 \times 8693 \times 19 \, 479 \, 382 \, 229$ and: :$m_2 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 365 \, 147 \times 47 \, 307 \, 071 \, 129$ This was at one point the smallest known counterexample to the observation that: :most amicable pairs consist of even integers :most of the rest, whose elements are odd, have both elements divisible by $3$.


Equivalence of Definitions of Amicable Triplet

Let $m_1, m_2, m_3 \in \Z_{>0}$ be (strictly) positive integers. {{TFAE|def = Amicable Triplet}}


Largest Number not Expressible as Sum of Less than 37 Positive Fifth Powers

The largest positive integer which cannot be expressed as the sum of less than $37$ positive fifth powers is $223$: :$223 = 31 \times 1^5 + 6 \times 2^5$


Smallest Triple of Consecutive Sums of Squares

The smallest triple of consecutive positive integers each of which is the sum of two squares is: :$\tuple {232, 233, 234}$


No Quadruple of Consecutive Sums of Squares Exists

It is not possible for a quadruple of consecutive positive integers each of which is the sum of two squares.


Directed iff Filtered in Dual Ordered Set

Let $\struct {S, \preceq_1}$ be an ordered set. Let $\struct {S, \preceq_2}$ be a dual ordered set of $\struct {S, \preceq_1}$ Let $X \subseteq S$. Then: :$X$ is directed in $\struct {S, \preceq_1}$ {{iff}}: :$X$ is filtered in $\struct {S, \preceq_2}$.


Sum of Two Squares not Congruent to 3 modulo 4

Let $n \in \Z$ such that $n = a^2 + b^2$ where $a, b \in \Z$. Then $n$ is not congruent modulo $4$ to $3$.


Numbers not Expressible as Sum of Less than 9 Positive Cubes

The following are the only positive integers cannot be expressed as the sum of less than $9$ positive cubes: {{begin-eqn}} {{eqn | l = 23       | r = 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3       | c =  }} {{eqn | l = 239       | r = 4^3 + 4^3 + 3^3 + 3^3 + 3^3 + 3^3 + 1^3 + 1^3 + 1^3       | c =  }} {{end-eqn}}


Largest Prime Factor of n squared plus 1

Let $n \in \Z$ be greater than $239$. Then the largest prime factor of $n^2 + 1$ is at least $17$.


Solution of Ljunggren Equation

The only solutions of the Ljunggren equation: :$x^2 + 1 = 2 y^4$ are: :$x = 1,  y = 1$ :$x = 239,  y = 13$ {{OEIS|A229384}}


Solutions of Diophantine Equation x^4 + y^4 = z^2 + 1 for x = 239

Consider the indeterminate Diophantine equation: :$x^4 + y^4 = z^2 + 1$ When $x = 239$ and $x > y$, there are $3$ solutions: {{begin-eqn}} {{eqn | l = 239^4 + 104^4       | r = 58 \, 136^2 + 1       | c =  }} {{eqn | l = 239^4 + 143^4       | r = 60 \, 671^2 + 1       | c =  }} {{eqn | l = 239^4 + 208^4       | r = 71 \, 656^2 + 1       | c =  }} {{end-eqn}}


Number with over 240 Divisors is greater than 1,000,000

Let $n \in \Z$ be an integer with over $240$ divisors. Then: :$n > 1 \, 000 \, 000$


Sequence of 4 Consecutive Integers with Equal Number of Divisors

The following sequence of integers are sets of $4$ consecutive integers which all have the same number of divisors: :$\map \tau m = \map \tau {m + 1} = \map \tau {m + 2} = \map \tau {m + 3}$ where $\map \tau n$ denotes the $\tau$ function. :$242, 243, 244, 245, 3655, 3656, 3657, 3658, 4503, 4504, 4505, 4506, \ldots$ {{OEIS|A039665}}


Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes/Sequence

The  sequence of positive integers which are both the sum of $2$ square numbers in two distinct ways and also the sum of $2$ cube numbers begins: :$65, 250, \ldots$


Sum of 3 Cubes in 2 Distinct Ways

The sequence of positive integers which can be expressed as the sum of $3$ cubes numbers in two or more different ways begins: {{begin-eqn}} {{eqn | l = 251       | r = 5^3 + 5^3 + 1^3       | rr= = 6^3 + 3^3 + 2^3 }} {{end-eqn}} {{OEIS|???}}


78,557 is Sierpiński

$78 \, 557$ is a Sierpiński number of the second kind.


Existence of Infinite Number of Numbers that are Riesel, Carmichael and Sierpiński

There exists an infinite number of integers which are simultaneously Sierpiński, Riesel and Carmichael.


Equivalence of Definitions of Balanced Prime

The following definitions of a balanced prime are equivalent:


Divisor of Fermat Number

Let $F_n$ be a Fermat number. Let $m$ be divisor of $F_n$.


Prime Decomposition of 6th Fermat Number

The prime decomposition of the $6$th Fermat number is given by: {{begin-eqn}} {{eqn | l = 2^{\paren {2^6} } + 1       | r = 18 \, 446 \, 744 \, 073 \, 709 \, 551 \, 617       | c = Sequence of Fermat Numbers }} {{eqn | r = 274 \, 177 \times 67 \, 280 \, 421 \, 310 \, 721       | c =  }} {{eqn | r = \paren {3^2 \times 7 \times 17 \times 2^8 + 1} \times \paren {5 \times 47 \times 373 \times 2 \, 998 \, 279 \times 2^8 + 1}       | c =  }} {{end-eqn}}


Prime Decomposition of 7th Fermat Number

The prime decomposition of the $7$th Fermat number is given by: {{begin-eqn}} {{eqn | l = 2^{\paren {2^7} } + 1       | r = 340 \, 282 \, 366 \, 920 \, 938 \, 463 \, 463 \, 374 \, 607 \, 431 \, 768 \, 211 \, 457       | c = Sequence of Fermat Numbers }} {{eqn | r = 59 \, 649 \, 589 \, 127 \, 497 \, 217 \times 5 \, 704 \, 689 \, 200 \, 685 \, 129 \, 054 \, 721       | c =  }} {{eqn | r = \paren {116 \, 503 \, 103 \, 764 \, 643 \times 2^9 + 1} \paren {3^5 \times 5 \times 12497 \times 733803 839347 \times 2^9 + 1}       | c =  }} {{end-eqn}}


Filtered iff Directed in Dual Ordered Set

Let $\left({S, \preceq_1}\right)$ be an ordered set. Let $\left({S, \preceq_2}\right)$ be a dual ordered set of $\left({S, \preceq_1}\right)$ Let $X \subseteq S$. Then $X$ is filtered in $\left({S, \preceq_1}\right)$ {{iff}} $X$ is directed in $\left({S, \preceq_2}\right)$


Filters equal Ideals in Dual Ordered Set

Let $L_1 = \left({S, \preceq_1}\right)$ be an ordered set. Let $L_2 = \left({S, \preceq_2}\right)$ be a dual ordered set of $L_1$. Then $\mathit{Filt}\left({L_1}\right) = \mathit{Ids}\left({L_2}\right)$ where :$\mathit{Filt}\left({L_1}\right)$ denotes the set of all filters of $L_1$, :$\mathit{Ids}\left({L_2}\right)$ denotes the set of all ideals of $L_2$.


Pépin's Test

Let $F_n = 2^{2^n} + 1$ be a Fermat number. Then $F_n$ is prime {{iff}}: :$3^{\paren {F_n - 1} / 2} \equiv -1 \pmod {F_n}$


Triangular Fermat Number

The only one Fermat number which is triangular is $3$.


Fermat Number is not Square

There exist no Fermat numbers which are square.


Fermat Number is not Cube

There exist no Fermat numbers which are cubes.


Ideals equal Filters in Dual Ordered Set

Let $L_1 = \left({S, \preceq_1}\right)$ be an ordered set. Let $L_2 = \left({S, \preceq_2}\right)$ be a dual ordered set of $L_1$. Then $\mathit{Ids}\left({L_1}\right) = \mathit{Filt}\left({L_2}\right)$ where :$\mathit{Filt}\left({L_2}\right)$ denotes the set of all filters of $L_2$, :$\mathit{Ids}\left({L_1}\right)$ denotes the set of all ideals of $L_1$.


Power Set is Filter in Lattice of Power Set

Let $X$ be a set. Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be a inclusion lattice of power set of $X$. Then $\mathcal P\left({X}\right)$ is a filter on $L$.


Triplet in Arithmetic Sequence with equal Sigma

The smallest triple of integers in arithmetic sequence which have the same $\sigma$ (sigma) value is: :$\map \sigma {267} = \map \sigma {295} = \map \sigma {323} = 360$


Singleton of Set is Filter in Lattice of Power Set

Let $X$ be a set. Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be an inclusion lattice of power set of $X$. Then $\left\{ {X}\right\}$ is a filter on $L$.


Existence of Arbitrarily Long Aliquot Sequences

It is possible to construct an aliquot sequence of arbitrary length which is monotonically increasing.


Consecutive Triple of Repeated Digit-Products

The triplet of integers $281, 282, 283$ have the property that if their digits are multiplied, and the process repeated on the result until only $1$ digit remains, that final digit is the same for all three, that is, $6$. There does not exist an set of four consecutive integers which also all end up at the same single digit.


Consecutive Powerful Numbers

The following pairs are of consecutive positive integers both of which are powerful: :$\left({8, 9}\right), \left({288, 289}\right), \left({675, 676}\right), \left({9800, 9801}\right), \left({332 \, 928, 332 \, 929}\right), \ldots$ {{OEIS|A060355}}


Smallest Multiple of 9 with all Digits Even

$288$ is the smallest integer multiple of $9$ all of whose digits are even.


Kaprekar's Symmetry

Let $n$ be a Kaprekar number with $D$ digits. Then $10^D - n$ is also a Kaprekar number.


Cyclic Permutation of Kaprekar Number

Let $n$ be a Kaprekar number of $k$ digits. Let $m$ be an integer formed from a cyclic permutation of the digits of $n$. Let $m$ be squared and the result split into $2$ parts, where the $2$nd part is of $k$ digits. Let these two parts be added, in the way of operating on a Kaprekar number. If the result is more than $k$ digits long, split that into $2$ parts, where the $2$nd part is of $k$ digits, and add the parts. The result will be another cyclic permutation of the digits of $n$.


Filters of Lattice of Power Set form Bounded Above Ordered Set

Let $X$ be a set. Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be an inclusion lattice of power set of $X$. Let $F = \left({\mathit{Filt}\left({L}\right), \subseteq}\right)$ be an inclusion ordered set, where $\mathit{Filt}\left({L}\right)$ denotes the set of all filters on $L$. Then $F$ is bounded above and $\top_F = \mathcal P\left({X}\right)$ where $\top_F$ denotes the greatest element of $F$.


Palindromic Primes in Base 10 and Base 2

The following $n \in \Z$ are prime numbers which are palindromic in both decimal and binary: :$3, 5, 7, 313, 7 \, 284 \, 717 \, 174 \, 827, 390 \, 714 \, 505 \, 091 \, 666 \, 190 \, 505 \, 417 \, 093, \ldots$ {{OEIS|A046472}} It is not known whether there are any more.


Ideals form Complete Lattice

Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice. Let $\mathcal I = \struct {\mathit{Ids}\left({L}\right), \subseteq}$ be an inclusion ordered set, where $\mathit{Ids}\left({L}\right)$ denotes the set of all ideals in $L$. Then $\mathcal I$ is complete lattice.


Filters form Complete Lattice

Let $L = \left({S, \wedge, \preceq}\right)$ be a bounded above meet semilattice. Let $F = \left({\mathit{Filt}\left({L}\right), \subseteq}\right)$ be an inclusion ordered set, where $\mathit{Filt}\left({L}\right)$ denotes set of all filters on $L$. Then $F$ is a complete lattice.


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

:$319 = 15 \times 1^4 + 3 \times 2^4 + 4^4$ or: :$319 = 12 \times 1^4 + 4 \times 2^4 + 3 \times 3^4$


Positive Integers not Expressible as Sum of Fewer than 19 Fourth Powers

The following positive integers are the only ones which cannot be expressed as the sum of fewer than $19$ fourth powers: === $79$ as Sum of $19$ Fourth Powers === {{:Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers}} === $159$ as Sum of $19$ Fourth Powers === {{:159 is not Expressible as Sum of Fewer than 19 Fourth Powers}} === $319$ as Sum of $19$ Fourth Powers === {{:319 is not Expressible as Sum of Fewer than 19 Fourth Powers}} === $399$ as Sum of $19$ Fourth Powers === {{:399 is not Expressible as Sum of Fewer than 19 Fourth Powers}}


Existence of n such that M - 2^n or M + 2^n has no Prime factors less than 331

Let $M \in \Z$ be an integer. Then there exists a positive integer $n \in \Z_{\ge 0}$ such that either $M - 2^n$ or $M + 2^n$ has no prime factors less than $331$.


Completely Irreducible implies Infimum differs from Element

Let $\struct {S, \preceq}$ be an ordered set. Let $p \in S$ such that :$p$ is completely irreducible. Then $\map \inf {p^\succeq \setminus \set p} \ne p$ where $p^\succeq$ donotes the upper closure of $p$.


Not Every Number is the Sum or Difference of Two Prime Powers

Not every positive integer can be expressed in the form $p^m \pm q^n$ where $p, q$ are prime and $m, n$ are positive integers.


Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element

Let $L = \struct {S, \preceq}$ be an ordered set. Let $p \in S$. Then $p$ is completely irreducible {{iff}} :$\exists q \in S: p \prec q \land \paren {\forall s \in S: p \prec s \implies q \preceq s} \land p^\succeq = \set p \cup q^\succeq$ where $p^\succeq$ denotes the upper closure of $p$.


Poulet Number/Examples/341

The smallest Poulet number is $341$: :$2^{341} \equiv 2 \pmod {341}$ despite the fact that $341$ is not prime: :$341 = 11 \times 31$


Fourth Powers which are Sum of 4 Fourth Powers

The following positive integers are such that their fourth powers can be expressed as the sum of the fourth powers of $4$ other positive integers with no common factors: :$353, 651, 2487, 2501, 2829, \ldots$ {{OEIS|A039664}}


Sums of both 2 and 3 Consecutive Squares

The following are the smallest positive integers that are the sum of both $2$ and $3$ consecutive non-zero square numbers: :$365, 35 \, 645, 3 \, 492 \, 725, 342 \, 251 \, 285, 33 \, 537 \, 133 \, 085, 3 \, 286 \, 296 \, 790 \, 925, \ldots$ {{OEIS|A007667}}


Sturm-Liouville Problem/Unit Weight Function

Let $P, Q: \R \to \R$ be real mappings such that $P$ is smooth and positive, while $Q$ is continuous: :$\map P x \in C^\infty$  :$\map P x > 0$ :$\map Q x \in C^0$ Let the Sturm-Liouville equation, with $\map w x = 1$, be of the form: :$-\paren {P y'}' + Q y = \lambda y$ where $\lambda \in \R$. Let it satisfy the following boundary conditions: :$\map y a = \map y b = 0$ Then all solutions of the Sturm-Liouville equation, together with their eigenvalues, form infinite sequences $\sequence {y^{\paren n} }$ and $\sequence {\lambda^{\paren n} }$. Furthermore, each $\lambda^{\paren n}$ corresponds to an eigenfunction $y^{\paren n}$, unique up to a constant factor.


Discrepancy between Julian Year and Tropical Year

The Julian year and the tropical year differ such that the Julian calendar becomes $1$ day further out approximately every $128$ years.


Completely Irreducible and Subset Admits Infimum Equals Element implies Element Belongs to Subset

Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $x \in S$ such that :$x$ is completely irreducible. Let $X \subseteq S$ such that :$X$ admits an infimum and $x = \inf X$ Then $x \in X$


Order Generating Includes Completely Irreducible Elements

Let $\left({S, \preceq}\right)$ be an ordered set. Let $X \subseteq S$ be an order generating subset of $S$. Let $X \in S$ be a completely irreducible element of $S$. Then $x \in X$.


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

:$399 = 14 \times 1^4 + 3 \times 2^4 + 3^4 + 4^4$ or: :$399 = 11 \times 1^4 + 4 \times 2^4 + 4 \times 3^4$


Maximal implies Difference equals Intersection

Let $\left({S, \preceq}\right)$ be an ordered set. Let $x, y \in S$ such that :$x$ is maximal in $S \setminus y^\succeq$ Then $x^\succeq \setminus \left\{ {x}\right\} = x^\succeq \cap y^\succeq$


Square of n such that 2n-1 is Composite is not Sum of Square and Prime

Let $n^2$ be a square such that $2 n - 1$ is composite. Then $n^2$ cannot be expressed as the sum of a square and a prime.


40 times Heptagonal Numbers plus 9 gives Squares of Numbers ending in 7

Consider the heptagonal numbers: :$\displaystyle H_n = \sum_{k \mathop = 1}^n \left({5 k - 4}\right)$ Let $S_n$ be the sequence defined as: :$\forall n \in \Z_{>1}: S_n = 40 \times H_n + 9$ Then $S_n$ consists of the squares of all the positive integers which end in a $7$: :$49, 289, 729, 1369, 2209, 3249, 4489, 5929, 7569, \ldots$ that is: :$7^2, 17^2, 27^2, 37^2, 47^2, 57^2, 67^2, 77^2, 87^2, 97^2, \ldots$ {{OEIS|A017354}}


Maximal implies Completely Irreducible

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Let $p \in S$ such that :$\exists k \in S: p$ is maximal in $S \setminus k^\succeq$ Then $p$ is completely irreducible.


Infimum of Intersection of Upper Closures equals Join Operands

Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice. Let $x, y \in S$. Then $\inf\left({x^\succeq \cap y^\succeq}\right) = x \vee y$


Sturm-Liouville Problem/Unit Weight Function/Lemma

Let $\map \alpha x: \R \to \R$ such that $\map \alpha x \in C^2 \closedint a b$. Suppose: :$\displaystyle \forall \map h x \in C^2 \closedint a b: \map h a = \map h b = \map {h'} a = \map {h'} b = 0: \int_a^b \map \alpha x \, \map {h''} x \rd x = 0$ Then: :$\forall x \in \closedint a b: \map \alpha x = c_0 + c_1 x$ where $ c_0, c_1 \in \R $ are constants.


Intersection of Upper Closures is Upper Closure of Join Operands

Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice. Let $x, y \in S$. Then $\left({x \vee y}\right)^\succeq = x^\succeq \cap y^\succeq$


Strictly Precede and Step Condition and not Precede implies Joins are equal

Let $\struct {S, \vee, \preceq}$ be a join semilattice. Let $p, q, u \in S$ be such that: :$p \prec q$ and $\paren {\forall s \in S: p \prec s \implies q \preceq s}$ and $u \npreceq p$ Then: :$p \vee u = q \vee u$


Not Preceding implies Exists Completely Irreducible Element in Algebraic Lattice

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below algebraic lattice. Let $x, y \in S$ such that :$y \npreceq x$ Then :$\exists p \in S: p$ is completely irreducible $\land ~x \preceq p \land y \npreceq p$


Set of All Completely Irreducible Elements is Smallest Order Generating

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below algebraic lattice. Then $\mathit{Irr}\left({L}\right)$ is order generating and :$\forall X \subseteq S: X$ is order generating $\implies \mathit{Irr}\left({L}\right) \subseteq X$ where $\mathit{Irr}\left({L}\right)$ denotes the set of all completely irreducible elements of $L$.


Preimage of Lower Set under Increasing Mapping is Lower

Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be preordered sets. Let $f: S \to T$ be an increasing mapping. Let $X \subseteq T$ be a lower subset of $T$. Then $f^{-1} \sqbrk X$ is lower where $f^{-1} \sqbrk X$ denotes the preimage of $X$ under $f$.


Preimage of Upper Set under Increasing Mapping is Upper

Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be preordered sets. Let $f: S \to T$ be an increasing mapping. Let $X \subseteq T$ be a upper subset of $T$. Then $f^{-1} \sqbrk X$ is upper where $f^{-1} \sqbrk X$ denotes the preimage of $X$ under $f$.


Directed Suprema Preserving Mapping at Element is Supremum

Let $\left({S, \vee, \wedge, \preceq}\right)$ and $\left({T, \vee_2, \wedge_2, \precsim}\right)$ be bounded below continuous lattices. Let $f: S \to T$ be a mapping such that :$f$ preserves directed suprema. Let $x \in S$. Then $f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$


Preceding implies Way Below Closure is Subset of Way Below Closure

Let $\struct {S, \preceq}$ be an ordered set. Let $x, y \in S$ such that :$x \preceq y$ Then $x^\ll \subseteq y^\ll$ where $x^\ll$ denotes the way below closure of $x$.


Mapping at Element is Supremum implies Mapping is Increasing

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Let $\left({T, \vee_2, \wedge_2, \precsim}\right)$ be a complete lattice. Let $f: S \to T$ be a mapping such that :$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$ Then $f$ is an increasing mapping.


Semiperfect Number is not Deficient

Let $n \in \Z_{>0}$ be a semiperfect number. Then $n$ is not deficient.


Perfect Number is Primitive Semiperfect

Let $n \in \Z_{>0}$ be a perfect number. Then $n$ is also a primitive semiperfect number.


Mapping at Element is Supremum implies Way Below iff There Exists Element that Way Below and Way Below

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $\left({T, \vee_2, \wedge_2, \precsim}\right)$ be a continuous complete lattice. Let $f: S \to T$ be a mapping such that :$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$ Let $x \in S, y \in T$. Then :$y \ll f\left({x}\right) \iff \exists w \in S: w \ll x \land y \ll f\left({w}\right)$


Subset and Image Admit Suprema and Mapping is Increasing implies Supremum of Image Precedes Mapping at Supremum

Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets. Let $f: S \to T$ be a increasing mapping. Let $D \subseteq S$ such that :$D$ admits a supremum in $S$ and $f\left[{D}\right]$ admits a supremum in $T$. Then $\sup \left({f\left[{D}\right]}\right) \precsim f\left({\sup D}\right)$


Characteristic Subgroup of Normal Subgroup is Normal

Let $G$ be a group. Let $N\leq G$ be normal. Let $H\leq N$ be characteristic. Then $H$ is normal in $G$.


Center is Characteristic Subgroup

Let $G$ be a group. Then its center $\map Z G$ is characteristic in $G$.


Semidirect Product of Groups is Group

Let $H$ and $N$ be groups. Let $\operatorname{Aut}(N)$ denote the automorphism group of $N$. Let $\phi : H\to \operatorname{Aut}(N)$ be a group homomorphism, that is, let $H$ act on $N$. Let $N\rtimes_\phi H$ be the semidirect product of $N$ and $H$ with respect to $\phi$, that is: :$N\rtimes_\phi H = (N\times H, \circ)$ where :$(n_1, h_1) \circ (n_2, h_2) = (n_1\cdot \phi(h_1)(n_2), h_1\cdot h_2)$ Then $N\rtimes_\phi H$ is a group.


Inverse of Element in Semidirect Product

Let $N$ and $H$ be groups. Let $H$ act by automorphisms on $N$ via $\phi$. Let $N \rtimes_\phi H$ be the corresponding (outer) semidirect product. Let $\tuple {n, h} \in N \rtimes_\phi H$. Then: {{begin-eqn}} {{eqn | l = \tuple {n, h}^{-1}       | r = \tuple {\map {\phi_{h^{-1} } } {n^{-1} }, h^{-1} }       | c =  }} {{eqn | r = \tuple {\paren {\map {\phi_{h^{-1} } } n}^{-1}, h^{-1} }       | c =  }} {{eqn | r = \tuple {\map { {\phi_h}^{-1} } {n^{-1} }, h^{-1} }       | c =  }} {{eqn | r = \tuple {\paren {\map { {\phi_h}^{-1} } n}^{-1}, h^{-1} }       | c =  }} {{end-eqn}}


Semidirect Product with Trivial Action is Direct Product

Let $H$ and $N$ be groups. Let $\Aut N$ denote the automorphism group of $N$. Let $\phi: H \to \Aut N$ be defined as: :$\forall h \in H: \map \phi h = I_N$ for all $h \in H$ where $I_N$ denotes the identity mapping on $N$. Let $N \rtimes_\phi H$ be the corresponding semidirect product. Then $N \rtimes_\phi H$ is the direct product of $N$ and $H$.


Semidirect Product is Abelian iff Components are Abelian and Action is Trivial

Let $N$ and $H$ be groups. Let $H$ act by automorphisms on $N$ via $\phi$. Let $N\rtimes_\phi H$ be the corresponding semidirect product. Then the following are equivalent: :$(1): \quad$ $N\rtimes_\phi H$ is abelian :$(2): \quad$ $N$ and $H$ are abelian and $H$ acts trivially


Integers whose Squares end in 444

The sequence of positive integers whose square ends in $444$ begins: :$38, 462, 538, 962, 1038, 1462, 1538, 1962, 2038, 2462, 2538, 2962, 3038, 3462, \ldots$ {{OEIS|A039685}}


Subset and Image Admit Infima and Mapping is Increasing implies Infimum of Image Succeeds Mapping at Infimum

Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be ordered sets. Let $f: S \to T$ be a increasing mapping. Let $D \subseteq S$ such that :$D$ admits a infimum in $S$ and $f \sqbrk D$ admits a infimum in $T$. Then $\map f {\inf D} \precsim \map \inf {f \sqbrk D}$


Equivalence of Definitions of Finite Galois Extension

Let $L/K$ be a finite field extension. {{TFAE|def = Finite Galois Extension}}


Finite Field Extension has Finite Galois Group

Let $E / F$ be a finite field extension. Then its automorphism group is finite.


Primitive Element Theorem

Let $E/F$ be a separable field extension of finite degree. Then $E/F$ is simple: there exists $\alpha\in E$ such that $E=F(\alpha)$.


Largest Number not Expressible as Sum of Less than 32 Positive Fifth Powers

The largest positive integer which cannot be expressed as the sum of less than $32$ positive fifth powers is $466$: :$466 = 18 \times 1^5 + 14 \times 2^5$


Smallest Square which is Sum of 3 Fourth Powers

The smallest positive integer whose square is the sum of $3$ fourth powers is $481$: :$481^2 = 12^4 + 15^4 + 20^4$


Solutions to p^2 Divides 10^p - 10

The known prime numbers $p$ which satisfy the equation: :$p^2 \divides \paren {10^p - 10}$ where $\divides$ denotes divisibility, are: :$3, 487, 56 \, 598 \, 313$ {{OEIS|A045616}}


Kaprekar's Process on 3 Digit Number ends in 495

Let $n$ be a $3$-digit integer whose digits are not all the same. Kaprekar's process, when applied to $n$, results in $495$ after no more than $6$ iterations.


Nilpotent Elements of Commutative Ring form Ideal

Let $\struct {R, +, \circ}$ be a commutative ring whose zero is $0_R$ and whose unity is $1_R$. The subset of nilpotent elements of $R$ form an ideal of $R$.


Algebraic Closure of Field is Unique

Let $F$ be a field. Let $K$ and $L$ be algebraic closures of $F$. Then $K$ and $L$ are $F$-isomorphic.


Multiplicative Group of Galois Field is Cyclic

Let $\GF$ be a Galois field of order $q$. Then its multiplicative group is cyclic of order $q-1$: :$\GF^\times \cong C_{q - 1}$


Automorphism Group of C Over R

The field extension $\C / \R$ of complex numbers $\C$ over real numbers $\R$ has automorphism group $\operatorname{Aut}$: :$\operatorname{Aut} \paren {\C / \R} = \set {\operatorname{id}, \sigma}$ where: :$\operatorname{id}$ denotes the identity mapping :$\sigma$ denotes complex conjugation


Image under Inclusion Mapping

Let $X$ be a set. Let $S \subseteq X$, $Z \subseteq S$. Then $i_S\left[{Z}\right] = Z$ where :$i_S$ denotes the inclusion mapping of $S$, :$i_S\left[{Z}\right]$ denotes the image of $Z$ under $i_S$.


Limit Inferior of Inclusion Moore-Smith Sequence is Supremum of Directed Subset

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be an up-complete lattice. Let $D \subseteq S$ be a directed subset of $S$. Let $\left({D, \preceq'}\right)$ be a directed ordered subset of $L$. Let $i_D: D \to S$, the inclusion mapping, be a Moore-Smith sequence in $S$. Then $\liminf i_D = \sup D$


Correspondence Between Group Actions and Permutation Representations

Let $G$ be a group. Let $X$ be a set. There is a one-to-one correspondence between group actions of $G$ on $X$ and permutation representations of $G$ in $X$, as follows: Let $\phi : G \times X \to X$ be a group action. Let $\rho : G \to \struct {\map \Gamma X, \circ}$ be a permutation representation. The following are equivalent: :$(1): \quad \rho$ is the permutation representation associated to $\phi$ :$(2): \quad \phi$ is the group action associated to $\rho$


Trivial Field Extension is Galois

Let $F$ be a field. The trivial field extension $F/F$ is Galois.


Automorphism Group Acts Faithfully on Generating Set

Let $E/F$ be a field extension. Let $\operatorname{Aut}(E/F)$ be its automorphism group. Let $S\subset E$ be a generating set of the extension. Let $S$ be stable under the group action of $\operatorname{Aut}(E/F)$. Then the induced group action on $S$ is faithful.


Upper Closure in Ordered Subset is Intersection of Subset and Upper Closure

Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $\left({T, \precsim}\right)$ be an ordered subset of $L$. Let $t \in T$. Then $t^\succsim = T \cap t^\succeq$


Approximation to Power of 7 by Power of 10

:$7^{510} \approx 1 \cdotp 00000 \, 09377 \, 76536 \ldots \times 10^{431}$ This is the closest known approximation of a power of $7$ by a power of $10$.


Limit Inferior of Repetition Moore-Smith Sequence

Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $N = \left({\N, \le}\right)$ be a directed ordered set. Let $a, b \in S$. Let $f = \left({c_i}\right)_{i \in \N} = \left({a, b, a, b, \dots}\right):\N \to S$ be a Moore-Smith sequence. Then $\liminf \left({c_i}\right)_{i \in \N} = a \wedge b$


Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping is Increasing

Let $\left({S, \vee_1, \wedge_1, \preceq_1}\right)$ and $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be lattices. Let $f: S \to T$ be a mapping such that :for all directed set $\left({D, \precsim}\right)$ and Moore-Smith sequence $N:D \to S$ in $S$: $f\left({\liminf N}\right) \preceq_2 \liminf\left({f \circ N}\right)$ Then $f$ is an increasing mapping.


Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Supremum of Image is Mapping at Supremum of Directed Subset

Let $\left({S, \vee_1, \wedge_1, \preceq_1}\right)$ and $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be up-complete lattices. Let $f: S \to T$ be a mapping such that :for all directed set $\left({D, \precsim}\right)$ and Moore-Smith sequence $N:D \to S$ in $S$: $f\left({\liminf N}\right) \preceq_2 \liminf\left({f \circ N}\right)$ Let $D$ be a directed subset of $S$. Then $\sup \left({f\left[{D}\right]}\right) = f\left({\sup D}\right)$ where $f\left[{D}\right]$ denotes the image of $D$ under $f$.


Prime Values of Double Factorial plus 1

Let $n!!$ denote the double factorial function. The sequence of positive integers $n$ such that $n!! + 1$ is prime begins: :$0, 1, 2, 518, 33 \, 416, 37 \, 310, 52 \, 608, 123 \, 998, 220 \, 502, \ldots$ {{OEIS|A080778}}


Limit Inferior of Restriction Moore-Smith Sequence is Supremum of Image of Directed Subset

Let $L = \left({S, \vee_1, \wedge_1, \preceq_1}\right)$ and $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be up-complete lattices. Let $f:S \to T$ be an increasing mapping. Let $D \subseteq S$ be a directed subset of $S$. Let $\left({D, \preceq'}\right)$ be a directed ordered subset of $L$. Let $f \restriction D: D \to T$, the restriction of mapping, be a Moore-Smith sequence in $T$. Then $\liminf \left({f \restriction D}\right) = \sup \left({f\left[{D}\right]}\right)$


Infimum of Image of Upper Closure of Element under Increasing Mapping

Let $\left({S, \preceq}\right)$ and $\left({T, \precsim}\right)$ be ordered set. Let $f:S \to T$ be an increasing mapping. Let $x \in S$. Then $\inf \left({f\left[{x^\succeq}\right]}\right) = f\left({x}\right)$


Composition of Mapping and Inclusion is Restriction of Mapping

Let $S, T$ be sets. Let $f:S \to T$ be a mapping. Let $A \subseteq S$. Then $f \circ i_A = f \restriction A$ where :$i_A$ denotes the inclusion mapping of $A$, :$f \restriction A$ denotes the restriction of $f$ to $A$.


Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping Preserves Directed Suprema

Let $\struct {S, \vee_1, \wedge_1, \preceq_1}$ and $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be complete lattices. Let $f: S \to T$ be a mapping such that :for all directed sets $\struct {D, \precsim}$ and Moore-Smith sequences $N:D \to S$ in $S$: $\map f {\liminf N} \preceq_2 \map \liminf {f \circ N}$ Then $f$ preserves directed suprema.


Thurston's Geometrization Conjecture

When a topological manifold of dimension $3$ has been split into its connected sum and the Jaco-Shalen-Johannson torus decomposition, the remaining components each admit exactly one of the following geometries: :$(1): \quad$ Euclidean geometry :$(2): \quad$ Hyperbolic geometry :$(3): \quad$ Spherical geometry :$(4): \quad$ The geometry of $\mathbb S^2 \times \R$ :$(5): \quad$ The geometry of $\mathbb H^2 \times \R$ :$(6): \quad$ The geometry of the universal cover $S L_2 \R^~$ of the Lie group $S L_2 \R$ :$(7): \quad$ Nil geometry :$(8): \quad$ Sol geometry where: :$\mathbb S^2$ is the 2-sphere :$\mathbb H^2$ is the hyperbolic plane.


Poincaré Conjecture/Historical Note

The Poincaré Conjecture was first posed in $1904$ by {{AuthorRef|Jules Henri Poincaré}}. It was finally resolved by the work of {{AuthorRef|Grigori Perelman}}, who solved Thurston's Geometrization Conjecture in $2003$.


Definition:Stabilizer/Linguistic Note

The British English spelling for '''stabilizer''' is '''stabiliser'''.


Mapping Preserves Directed Suprema implies Mapping is Continuous

Let $\left({S, \preceq_1, \tau_1}\right)$ and $\left({T, \preceq_2, \tau_2}\right)$ be up-complete ordered sets with Scott topologies. Let $f: S \to T$ be a directed suprema preserving mapping. Then $f$ is continuous.


Secant of Complex Number

Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\sec \paren {a + b i} = \dfrac {\cos a \cosh b + i \sin a \sinh b} {\cos^2 a \cosh^2 b + \sin^2 a \sinh^2 b}$ where: :$\sec$ denotes the complex secant function. :$\sin$ denotes the real sine function :$\cos$ denotes the real cosine function :$\sinh$ denotes the hyperbolic sine function :$\cosh$ denotes the hyperbolic cosine function


Tangent of Complex Number/Formulation 1

:$\tan \paren {a + b i} = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}$


Tangent of Complex Number/Formulation 2

:$\tan \paren {a + b i} = \dfrac {\tan a + i \tanh b} {1 - i \tan a \tanh b}$


Tangent of Complex Number/Formulation 3

:$\tan \paren {a + b i} = \dfrac {\tan a - \tan a \tanh ^2 b} {1 + \tan ^2 a \tanh ^2 b} + \dfrac {\tanh b + \tan ^2 a \tanh b} {1 + \tan ^2 a \tanh ^2 b} i$


Cosecant of Complex Number

Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\csc \paren {a + b i} = \dfrac {\sin a \cosh b - i \cos a \sinh b} {\sin^2 a \cosh^2 b + \cos^2 a \sinh^2 b}$ where: :$\csc$ denotes the complex cosecant function. :$\sin$ denotes the real sine function :$\cos$ denotes the real cosine function :$\sinh$ denotes the hyperbolic sine function :$\cosh$ denotes the hyperbolic cosine function


Cotangent of Complex Number

Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then:


Cotangent of Complex Number/Formulation 1

:$\cot \paren {a + b i} = \dfrac {\cos a \cosh b - i \sin a \sinh b} {\sin a \cosh b + i \cos a \sinh b}$


Cotangent of Complex Number/Formulation 2

:$\map \cot {a + b i} = \dfrac {i \cot a \coth b - 1} {\cot a - i \coth b}$


Continuous iff Mapping at Limit Inferior Precedes Limit Inferior of Composition of Mapping and Sequence

Let $\left({S, \preceq_1, \tau_1}\right)$ and $\left({T, \preceq_2, \tau_2}\right)$ be complete topological lattices with Scott topologies. Let $f: S \to T$ be a mapping. Then $f$ is continuous {{iff}} :for all directed set $\left({D, \precsim}\right)$ and Moore-Smith sequence $N:D \to S$ in $S$: $f\left({\liminf N}\right) \preceq_2 \liminf\left({f \circ N}\right)$


Mapping is Increasing implies Mapping at Infimum for Sequence Precedes Infimum for Composition of Mapping and Sequence

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 an increasing mapping. Let $\left({D, \precsim}\right)$ be a directed set. Let $N: D \to S$ be a Moore-Smith sequence in $S$. Let $j \in D$. Then $f\left({\inf\left({N\left[{\precsim \left({j}\right)}\right]}\right)}\right) \preceq_2 \inf \left({\left({f \circ N}\right)\left[{\precsim \left({j}\right)}\right]}\right)$


Set of Infima for Sequence is Directed

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $\left({A, \precsim}\right)$ be a non-empty directed set. Let $Z: A \to S$ be a Moore-Smith sequence. Let $D = \left\{ {\inf \left({Z\left[{\precsim \left({j}\right)}\right]}\right): j \in A}\right\}$ be a subset of $S$. Then $D$ is directed.


Preceding implies Image is Subset of Image

Let $\left({S, \precsim}\right)$ be a preordered set. Let $x, y \in S$ such that :$x \precsim y$ Then $\precsim\left({y}\right) \subseteq \mathord\precsim\left({x}\right)$ where $\precsim\left({y}\right)$ denotes the image of $y$ under $\precsim$.


Universal Property of Group Ring

Let $R$ be a commutative ring with unity. Let $G$ be a group. Let $R \sqbrk G$ be the corresponding group ring. Let $S$ be a commutative ring with unity. Let $\phi: R \to S$ be a ring homomorphism. Let $\beta : G \to R^\times$ be a group homomorphism, where $R^\times$ is the multiplicative group of $R$. Then there exists a unique ring homomorphism from $R \sqbrk G$ to $S$ which extends $\phi$ and $\beta$.


Continuous iff Directed Suprema Preserving

Let $\left({S, \preceq_1, \tau_1}\right)$ and $\left({T, \preceq_2, \tau_2}\right)$ be complete topological lattices with Scott topologies. Let $f: S \to T$ be a mapping. Then $f$ is continuous {{iff}} $f$ preserves directed suprema.


Universal Property of Quotient of Topological Group

Let $G$ and $H$ be topological groups. Let $N$ be a normal subgroup of $G$. Let $\pi : G \to G/N$ be the quotient mapping. Let $f : G \to H$ be a continuous group homomorphism whose kernel contains $N$. Then there exists a unique continuous group homomorphism $\overline f : G/N \to H$ such that $f = \overline f \circ \pi$.


Universal Property of Quotient Set

Let $X$ and $Y$ be sets. Let $\sim$ be an equivalence relation on $X$. Let $\pi : X \to X/\sim$ be the quotient mapping to the quotient set. Let $f : X \to Y$ be $\sim$-invariant. Then there exists a unique mapping $\overline f : X/\!\sim \to Y$ such that $f = \overline f \circ \pi$. :$\xymatrix{ X \ar[d]_\pi \ar[r]^f & Y\\ X/\sim \ar@{.>}[ru]_{\overline f} }$


Continuous iff Way Below iff There Exists Element that Way Below and Way Below

Let $\struct {S, \preceq_1, \tau_1}$ and $\struct {T, \preceq_2, \tau_2}$ be complete continuous topological lattices with Scott topologies. Let $f: S \to T$ be a mapping. Then $f$ is continuous {{iff}} :$\forall x \in S, y \in T: y \ll \map f x \iff \exists w \in S: w \ll x \land y \ll \map f w$ {{explain|link to definition of $\ll$ and $\gg$}}


Finite Symmetric Group is Ambivalent

Let $n$ be a natural number. Let $S_n$ be a symmetric group of order $n$. Then $S_n$ is ambivalent.


Alternating Groups that are Ambivalent

Let $n$ be a natural number. Then the $n$th alternating group $A_n$ is ambivalent {{iff}} $n\in \{1, 2, 5, 6, 10, 14\}$. {{OEIS|A115200}}


Transitivity of Big-O Estimates/General

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


Transitivity of Big-O Estimates/Sequences

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


Conjugate of Cycle

Let $n \ge 1$ be a natural number. Let $S_n$ be the symmetric group on $n$ letters. Let $\pi, \sigma \in S_n$. Let $\sigma$ be a cycle of length $k$. Then the conjugate $\pi \sigma \pi^{-1}$ is a cycle of length $k$.