hcju/proofwikips huggingface dataset:
Order of Reciprocal of Entire Function
Let $f: \C \to \C$ be an entire function of order $\rho$. Let $f$ have no zeroes. Then $1/f$ has order $\rho$.
Jacobi's Necessary Condition/Dependent on N Functions
Let $J$ be a functional, such that: :$J \sqbrk {\mathbf y} = \displaystyle \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ where $\mathbf y = \paren {\sequence {y_i}_{1 \le i \le N} }$ is an N-dimensional real vector. Let $\map {\mathbf y} x$ correspond to the minimum of $J$. Let the $N\times N$ matrix $\mathbf P = F_{y_i' y_j'}$ be positive definite along $\map {\mathbf y} x$. Then the open interval $\openint a b$ contains no points conjugate to $a$.
Probability of no 2 People out of 53 Sharing the Same Birthday
Let there be $53$ people in a room. The probability that no $2$ of them have the same birthday is approximately $\dfrac 1 {53}$.
Primes not Sum of or Difference between Powers of 2 and 3
The sequence of prime numbers which cannot be expressed as either sum of or the difference between a power of $2$ and a power of $3$ begins: :$53, 71, 103, 107, 109, 149, 151, \ldots$ {{OEIS|A007644}}
Triangular Fibonacci Numbers
The only Fibonacci numbers which are also triangular are: :$0, 1, 3, 21, 55$ {{OEIS|A039595}}
Repdigit Triangular Numbers
The only repdigit numbers which are also triangular are: :$55, 66, 666$ {{OEIS|A045914}}
Increasing and Ordering on Mappings implies Mapping is Composition
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 :$g$ and $d$ are increasing mappings and :$d \circ g \preceq I_S$ and $I_T \precsim g \circ d$ where $\preceq, \precsim$ denotes the orderings on mappings. Then $d \circ \left({g \circ d}\right)$ and $g = \left({g \circ d}\right) \circ g$
Closed Form for Square Pyramidal Numbers
The closed-form expression for the $n$th square pyramidal number is: :$S_n = \dfrac {n \paren {n + 1} \paren {2 n + 1} } 6$
Closed Form for Pentagonal Pyramidal Numbers
The closed-form expression for the $n$th pentagonal pyramidal number is: :$Q_n = \dfrac {n^2 \paren {n + 1} } 2$
Surjection from Class to Proper Class
Let $A$ be a class. Let $\mathrm P$ be a proper class. Let $f: A \to \mathrm P$ be a surjection. Then $A$ is proper.
Image of Set under Mapping is Set
Let $A$ be a class. Let $\mathrm U$ denote the universal class. Let $f: A \to \mathrm U$ be a class mapping. Let $S$ be a subset of $A$. {{explain|Note that further work is needed on the Subset page to clarify that a "subset" of a "proper class" is indeed a "set". Some words are already on that page, but this needs to be formalised and separated out into its own transcluded "class theoretic" page.}} Then the image $f \sqbrk S$ is also a set. If $A$ is a set, then this result is known as the Axiom of Replacement in Zermelo-Fraenkel set theory.
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 : :$\forall i, k \in \N: 1 \le i, k \le N: \valueat {\dfrac {\partial p_i \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } } {\partial y_k} } {x \mathop = a} = \valueat {\dfrac {\partial p_k \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } } {\partial y_i} } {x \mathop = a}$
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
:$79 = 15 \times 1^4 + 4 \times 2^4$