hcju/proofwikips huggingface dataset:
Discrete Group Acts Continuously iff Acts by Homeomorphisms
Let $G$ be a discrete group acting on a topological space $X$. Then the following are equivalent: :$G$ acts continuously :$G$ acts by homeomorphisms
Euler Phi Function of 3
:$\map \phi 3 = 2$
Number of Distinct Parenthesizations on Word
Let $w_n$ denote an arbitrary word of $n$ elements. The number of distinct parenthesizations of $w_n$ is the Catalan number $C_{n - 1}$: :$C_{n - 1} = \dfrac 1 n \dbinom {2 \paren {n - 1} } {n - 1}$
Number of Paths on Graph along X-axis using Diagonal Steps
The number of different paths that can be taken on a Cartesian plane from the origin to $\tuple {2 n + 2, 0}$, using only diagonal steps, and never touching the $x$-axis except at the beginning and the end of the path, is the Catalan number $C_n$: :400px
Non-Integral Value of Göbel's Sequence
Consider Göbel's sequence defined recursively as: :$x_n = \begin{cases} 1 & : n = 0 \\ \ds \paren {1 + \sum_{k \mathop = 0}^{n - 1} {x_k}^2} / n & : n > 0 \end{cases}$ The smallest $n$ such that $x_n$ is not an integer is $43$.
Non-Integral Value of 3-Göbel Sequence
Consider the $3$-Göbel sequence defined recursively as: :$x_n = \begin {cases} 1 & : n = 0 \\ \displaystyle \paren {1 + \sum_{k \mathop = 0}^{n - 1} {x_k}^3} / n & : n > 0 \end {cases}$ The smallest $n$ such that $x_n$ is not an integer is $88$.
Best Approximation from Below to 1 as Sum of Minimal Number of Unit Fractions
:$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 7 + \dfrac 1 {43} = 1 -  \dfrac 1 {1806}$ This is the best approximation from below to $1$ as the minimal sum of unit fractions.
Cuboid with Integer Edges and Face Diagonals
The smallest cuboid whose edges and the diagonals of whose faces are all integers has edge lengths $44$, $117$ and $240$. Its space diagonal, however, is not an integer. {{expand|Add the definition of Definition:Euler Brick, and rewrite and rename as appropriate.}}
Largest n such that 1 to n can be Partitioned for no Element to be Sum of 2 Elements in Same Set
$44$ is the largest integer $n$ such that the set of integers from $1$ to $n$ can be partitioned into $4$ subsets such that no integer in any of these subsets is the sum of $2$ other integers in the same subset: :$\set {1, 3, 5, 15, 17, 19, 26, 28, 40, 42, 44}$ :$\set {2, 7, 8, 18, 21, 24, 27, 33, 37, 38, 43}$ :$\set {4, 6, 13, 20, 22, 23, 25, 30, 32, 39, 41}$ :$\set {9, 10, 11, 12, 14, 16, 29, 31, 34, 35, 36}$
Conjugate Transpose is Involution
Let $\mathbf A$ be a complex-valued matrix. Let $\mathbf A^*$ denote the Hermitian conjugate of $\mathbf A$. Then the operation of Hermitian conjugate is an involution: :$\paren {\mathbf A^*}^* = \mathbf A$
Number as Sum of Distinct Primes greater than 11
Every number greater than $45$ can be expressed as the sum of distinct primes greater than $11$.
Subclass of Set is Set
Let $A$ be a set. Let $\map \phi x$ be a condition in which $x$ is taken to be a set. Then there exists a set that consists of all of the elements of $A$ that satisfies this condition. In ZF, this result is known as the Axiom of Specification.
Hexagonal Number as 4 times Triangular Number plus n
Let $H_n$ be the $n$th hexagonal number. Then: :$H_n = 4 T_{n - 1} + n$ where $T_{n - 1}$ is the $n - 1$th triangular number.
Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Dependent on N Functions
Let $K$ be a (real) functional, such that: :$\ds K \sqbrk {\mathbf h} = \int_a^b \paren {\mathbf h' \mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$ where: :$\mathbf h$ is an $N$-dimensional vector :$\mathbf Q$ is a $N \times N$ matrix :$\mathbf P$ is a $N\times N$ symmetric positive definite matrix. Let $\closedint a b$ be such that it does not contain a point conjugate to $a$. Then: :$\forall \mathbf h: \map {\mathbf h} a = \map {\mathbf h} b = 0: K \sqbrk {\mathbf h} > 0$  the above holds. {{explain|"the above" -- reference the specific statement, there is plenty of "above" and it may not be immediately clear exactly what}}
Class is Proper iff Bijection from Class to Proper Class
Let $A$ be a class. Let $\mathrm P$ be a proper class. Then $A$ is proper if and only if there exists a bijection from $A$ to $\mathrm P$.
Infinite Product of Analytic Functions
Let $D \subset \C$ be an open connected set. Let $\left\langle{f_n}\right\rangle$ be a sequence of analytic functions $f_n: D \to \C$ that are not identically zero. Let $\displaystyle \sum_{n \mathop = 1}^\infty \left({f_n - 1}\right)$ converge locally uniformly absolutely on $D$. Then: :$(1): \quad f = \displaystyle \prod_{n \mathop = 1}^\infty f_n$ converges locally uniformly absolutely on $D$ :$(2): \quad f$ is analytic :$(3): \quad$ For each $z \in D$, $f_n \left({z}\right) = 0$ for finitely many $n \in \N$ :$(4): \quad$ For each $z \in D$, $\operatorname {mult}_z \left({f}\right) = \displaystyle \sum_{n \mathop = 1}^\infty \operatorname{mult}_z \left({f_n}\right)$ where $\operatorname{mult}$ denotes multiplicity.
Uniformly Continuous Function Preserves Uniform Convergence
Let $X$ be a set. Let $M$ and $N$ be metric spaces. Let $(g_n)$ be a sequence of mappings $g_n:X\to M$. Let $g_n$ converge uniformly to $g:X\to M$. Let $f:M\to N$ be uniformly continuous. Then $(f\circ g_n)$ converges uniformly to $f\circ g$.
Complex Exponential is Uniformly Continuous on Half-Planes
Let $a\in\R$. Then $\exp$ is uniformly continuous on the half-plane $\set {z \in \C : \map \Re z \le a}$.
Local Uniform Convergence Implies Compact Convergence
Let $X$ be a topological space. Let $M$ be a metric space. Let $(f_n)$ be a sequence of mappings $f_n : X\to M$. Let $f_n$ converge locally uniformly to $f:X\to M$. Then $f_n$ converges compactly to $f$.
Bounds for Complex Logarithm
Let $\ln$ denote the complex logarithm. Let $z \in \C$ with $\cmod z \le \dfrac 1 2$. Then: :$\dfrac 1 2 \cmod z \le \cmod {\map \ln {1 + z} } \le \dfrac 3 2 \cmod z$
Reciprocal of Absolutely Convergent Product is Absolutely Convergent
Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\sequence {1 + a_n}$ be a sequence of nonzero elements of $\mathbb K$. Let the infinite product $\ds \prod_{n \mathop = 1}^\infty \left({1 + a_n}\right)$ converge absolutely to $a \in \mathbb K \setminus \set 0$. Then $\ds \prod_{n \mathop = 1}^\infty \frac 1 {1 + a_n}$  converges absolutely to $1 / a$.
Equivalence of Definitions of Absolute Convergence of Product
Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\sequence{a_n}$ be a sequence in $\mathbb K$. {{TFAE|def = Absolute Convergence of Product}} === Definition 1 === {{:Definition:Absolute Convergence of Product/General Definition/Definition 1}} === Definition 2 === {{:Definition:Absolute Convergence of Product/General Definition/Definition 2}}
Factors in Convergent Product Converge to One
Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ be convergent. Then $a_n \to 1$.
Logarithm of Convergent Product of Real Numbers
The following are equivalent: :The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converges to $a \in \R_{\ne 0}$. :The series $\displaystyle \sum_{n \mathop = 1}^\infty \ln a_n$ converges to $\ln a$.
Factors in Absolutely Convergent Product Converge to One
Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be absolutely convergent. Then: :$a_n \to 0$
Logarithm of Divergent Product of Real Numbers
Let $\left\langle{a_n}\right\rangle$ be a sequence of strictly positive real numbers.
Logarithm of Infinite Product of Real Numbers
Let $(a_n)$ be a sequence of strictly positive real numbers.
Absolutely Convergent Product Does not Diverge to Zero
Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be absolutely convergent. Then it is not divergent to $0$.
Absolute Value of Absolutely Convergent Product is Absolutely Convergent
Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converge absolutely to $a\in\mathbb K$. Then $\displaystyle \prod_{n \mathop = 1}^\infty \norm{a_n}$ converges absolutely to $\norm{a}$.
Logarithm of Divergent Product of Real Numbers/Infinity
: :$(1): \quad$ The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $+\infty$. :$(2): \quad$ The series $\displaystyle \sum_{n \mathop = 1}^\infty \log a_n$ diverges to $+\infty$.
Product of Convergent Products is Convergent
Let $\struct {\mathbb K, \norm{\,\cdot\,} }$ be a valued field. Let $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converge to $a$. Let $\displaystyle \prod_{n \mathop = 1}^\infty b_n$ converge to $b$. Then $\displaystyle \prod_{n \mathop = 1}^\infty a_nb_n$ converges to $ab$.
Product of Convergent and Divergent Product is Divergent
Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field. Let $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ be convergent. Let $\displaystyle \prod_{n \mathop = 1}^\infty b_n$ be divergent. Then $\displaystyle \prod_{n \mathop = 1}^\infty a_n b_n$ is divergent.
Trimorphic Number is not necessarily Automorphic
Let $n \in \Z_{>0}$ be a trimorphic number. Then it is not necessarily the case that $n$ is also an automorphic number.
Reciprocal of 49 shows Powers of 2 in Decimal Expansion
The decimal expansion of the reciprocal of $49$ contains the powers of $2$: :$\dfrac 1 {49} = 0 \cdotp \dot 02040 \, 81632 \, 65306 \, 12244 \, 89795 \, 91836 \, 73469 \, 38775 \, 5 \dot 1$ {{OEIS|A007450}}
Numbers the Multiple of whose Reciprocal are Cyclic Permutations
Let $m \in \Z_{>0}$. Consider the reciprocal of $m$. Let $n \in \Z$ such that $1 \le n < m$. Then: :The digits in the decimal expansion of the rational number $\dfrac n m$ form a cyclic permutation of the digits in the decimal expansion of $\dfrac 1 m$ : :$(1): \quad m$ is the integer power of a prime number $p$ :$(2): \quad$ The period of recurrence of the decimal expansion of $1 / p$ is $p - 1$ :$(3): \quad p$ is not a divisor of $n$.
Uniform Product of Continuous Functions is Continuous
Let $X$ be a metric space. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\left\langle{f_n}\right\rangle$ be a sequence of bounded continuous mappings $f_n: X \to \mathbb K$. Let the product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge uniformly to $f$. Then $f$ is continuous.
Logarithm of Infinite Product of Complex Functions
Let $X$ be a weakly locally compact topological space. Let $\sequence {f_n}$ be a sequence of everywhere nonzero continuous mappings $f_n: X \to \C$. Then the following are equivalent: :$(1): \quad$ The product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converges locally uniformly to $f$. :$(2): \quad$ The series $\displaystyle \sum_{n \mathop = 1}^\infty \ln f_n$ converges locally uniformly to $\ln f + 2k\pi i$ for some mapping $k:K\to\Z$.
Logarithmic Derivative of Infinite Product of Analytic Functions
Let $D\subset\C$ be open. Let $(f_n)$ be a sequence of analytic functions $f_n:D\to\C$. Let none of the $f_n$ be identically zero on any open subset of $D$. Let the product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$. Then $\displaystyle \frac{f'}f = \sum_{n \mathop = 1}^\infty \frac{f_n'}{f_n}$ and the series converges locally uniformly in $D\setminus\{z\in D : f(z) = 0\}$.
Galois Connection implies Upper Adjoint is Surjection iff Lower Adjoint is Injection
Let $L = \struct {S, \preceq}$ and $R = \struct {T, \precsim}$ be ordered sets. Let $g: S \to T$ and $d: T \to S$ be mappings such that $\struct {g, d}$ is a Galois connection. Then $g$ is a surjection  $d$ is an injection.
Derivative of Uniform Limit of Analytic Functions
Let $U$ be an open subset of $\C$. Let $\left\langle{f_n}\right\rangle_{n \mathop \in \N}$ be a sequence of analytic functions $f_n : U \to \C$. Let $\left\langle{f_n}\right\rangle$ converge locally uniformly to $f$ on $U$. Then the sequence $\left\langle{f_n'}\right\rangle_{n \mathop \in \N}$ converges locally uniformly to $f'$.
Derivative of Infinite Product of Analytic Functions
Let $D \subset \C$ be open. Let $\left\langle{f_n}\right\rangle$ be a sequence of analytic functions $f_n: D \to \C$. Let the product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$. Then: :$\displaystyle f' = \sum_{n \mathop = 1}^\infty f_n'\cdot \prod_{\substack{k \mathop = 1 \\ k\mathop \ne n} }^\infty f_k$ and the series converges locally uniformly in $D$.
Uniformly Convergent Sequence Multiplied with Function
Let $X$ be a set. Let $V$ be a normed vector space over $\mathbb K$. Let $\sequence {f_n}$ be a sequence of mappings $f_n: X \to V$. Let $\sequence {f_n}$ be uniformly convergent. Let $g: X \to \mathbb K$ be bounded. Then $\sequence {f_n g}$ is uniformly convergent.
Logarithmic Derivative of Product of Analytic Functions
Let $D \subset \C$ be open. Let $f, g: D \to \C$ be analytic. Let $z \in D$ with $f \left({z}\right) \ne 0 \ne g \left({z}\right)$. Then: :$\dfrac{\left({f g}\right)' \left({z}\right)} {\left({f g}\right) \left({z}\right)} = \dfrac{f' \left({z}\right)} {f \left({z}\right)} + \dfrac {g' \left({z}\right)} {g \left({z}\right)}$ {{explain|Link to a clarifying definition of what $\left({f g}\right)$ is -- presumably it's the pointwise product, but in the context there's nothing to say it can't be $\left({f \circ g}\right)$, that is, $f$ of $g$ of $z$.}}
Uniformly Convergent Sequence on Dense Subset
Let $X$ be a metric space. Let $Y \subset X$ be dense. Let $V$ be a Banach space. Let $\sequence {f_n}$ be a sequence of continuous mappings $f_n : X\to V$. Let $\sequence {f_n}$ be uniformly convergent on $Y$. Then $\sequence {f_n}$ is uniformly convergent on $X$.
Tail of Uniformly Convergent Product Converges Uniformly to One
Let $X$ be a compact topological space. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\sequence {f_n}$ be a sequence of continuous mappings $f_n: X \to \mathbb K$. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge uniformly on $X$. Then for all $N \in \N$, $\displaystyle \prod_{n \mathop = N}^\infty f_n$ converges uniformly and the sequence $\displaystyle \prod_{n \mathop = N}^\infty f_n$ converges uniformly to $1$.
Factors in Uniformly Convergent Product Converge Uniformly to One
Let $X$ be a set. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\left\langle{f_n}\right\rangle$ be a sequence of bounded mappings $f_n: X \to \mathbb K$. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge uniformly on $X$. Then $f_n$ converges uniformly to $1$.
Quintuplets of Consecutive Integers which are not Sigma Values
The elements of the following $5$-tuples of consecutive integers have the property that they are not values of the $\sigma$ function $\map \sigma n$ for any $n$: :$\tuple {49, 50, 51, 52, 53}$ :$\tuple {115, 116, 117, 118, 119}$ :$\tuple {145, 146, 147, 148, 149}$
Positive Integers which are not Sigma Values
The following positive integers are not the values of the $\sigma$ function $\sigma \left({n}\right)$ for any $n$: :$2, 5, 9, 10, 11, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 37, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 58, 59, 61, 64, 65, \ldots$ {{OEIS|A007369}}
Uniformly Absolutely Convergent Product is Uniformly Convergent
Let $X$ be a set. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\mathbb K$ be complete. Let $\left\langle{f_n}\right\rangle$ be a sequence of bounded mappings $f_n: X \to \mathbb K$. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge uniformly absolutely on $X$. Then it converges uniformly.
Uniformly Convergent Product Satisfies Uniform Cauchy Criterion
Let $X$ be a compact topological space. Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field. Let $\sequence {f_n}$ be a sequence of continuous mappings $f_n: X \to \mathbb K$. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge uniformly on $X$. Then $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ satisfies the uniform Cauchy condition for products.
Absolute Value of Uniformly Convergent Product
Let $X$ be a compact topological space. Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\left\langle{f_n}\right\rangle$ be a sequence of continuous mappings $f_n: X \to \mathbb K$. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge uniformly to $f$. Then $\displaystyle \prod_{n \mathop = 1}^\infty \norm{f_n}$ converges uniformly to $ \norm{f}$.
Squares whose Digits can be Separated into 2 other Squares
The decimal representation of the following square numbers can be split into two parts which are each themselves square: {{begin-eqn}} {{eqn | l = 7^2       | r = 49       | c = $4 = 2^2$,       | cc= $9 = 3^2$ }} {{eqn | l = 13^2       | r = 169       | c = $16 = 4^2$,       | cc= $9 = 3^2$ }} {{eqn | l = 19^2       | r = 361       | c = $36 = 6^2$,       | cc= $1 = 1^2$ }} {{eqn | l = 35^2       | r = 1225       | c = $1 = 1^2$,       | cc= $225 = 15^2$ }} {{eqn | l = 38^2       | r = 1444       | c = $144 = 12^2$,       | cc= $4 = 2^2$ }} {{eqn | l = 41^2       | r = 1681       | c = $16 = 4^2$,       | cc= $81 = 9^2$ }} {{eqn | l = 57^2       | r = 3249       | c = $324 = 18^2$,       | cc= $9 = 3^2$ }} {{eqn | l = 65^2       | r = 4225       | c = $4 = 2^2$,       | cc= $225 = 15^2$ }} {{eqn | l = 70^2       | r = 4900       | c = $4 = 2^2$,       | cc= $900 = 30^2$ }} {{eqn | l = 125^2       | r = 15 \, 625       | c = $1 = 1^2$,       | cc= $5625 = 75^2$ }} {{eqn | l = 130^2       | r = 16 \, 900       | c = $16 = 4^2$,       | cc= $900 = 30^2$ }} {{eqn | l = 190^2       | r = 36 \, 100       | c = $36 = 6^2$,       | cc= $100 = 10^2$ }} {{eqn | l = 205^2       | r = 42 \, 025       | c = $4 = 2^2$,       | cc= $2025 = 45^2$ }} {{eqn | l = 223^2       | r = 49 \, 729       | c = $49 = 7^2$,       | cc= $729 = 27^2$ }} {{end-eqn}} {{OEIS|A048375}}
Zeroes of Infinite Product of Analytic Functions
Let $D \subset \C$ be an open connected set. Let $\left\langle{f_n}\right\rangle$ be a sequence of analytic functions $f_n: D \to \C$. Let $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$. Let $z_0\in D$. Then: :$(1): \quad$ $f$ is identically zero  some $f_n$ is identically zero  :$(2): \quad$ $f_n \left({z_0}\right) = 0$ for finitely many $n \in \N$ :$(3): \quad$ If $f$ is not identically zero, $\operatorname {mult}_{z_0} \left({f}\right) = \displaystyle \sum_{n \mathop = 1}^\infty \operatorname{mult}_{z_0} \left({f_n}\right)$ where $\operatorname{mult}$ denotes multiplicity.
Infinite Product of Analytic Functions is Analytic
Let $D \subset \C$ be an open set. Let $\left\langle{f_n}\right\rangle$ be a sequence of analytic functions $f_n: D \to \C$. Let $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$. Then $f$ is analytic.
Sum of 2 Squares in 2 Distinct Ways
Let $m, n \in \Z_{>0}$ be distinct positive integers that can be expressed as the sum of two distinct square numbers. Then $m n$ can be expressed as the sum of two square numbers in at least two distinct ways.
Bounds for Finite Product of Real Numbers
Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Then: :$\displaystyle \sum_{k \mathop = 1}^n a_k \le \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \map \exp {\sum_{k \mathop = 1}^n a_k}$
Bounds for Complex Exponential
Let $\exp$ denote the complex exponential. Let $z \in \C$ with $\cmod z \le \dfrac 1 2$. Then :$\dfrac 1 2 \cmod z \le \cmod {\exp z - 1} \le \dfrac 3 2 \cmod z$
Sum of 2 Squares in 2 Distinct Ways/Sequence
The sequence of positive integers which can be expressed as the sum of two square numbers in two or more different ways begins: {{begin-eqn}} {{eqn | l = 50       | r = 7^2 + 1^2       | rr= = 5^2 + 5^2 }} {{eqn | l = 65       | r = 8^2 + 1^2       | rr= = 7^2 + 4^2 }} {{eqn | l = 85       | r = 9^2 + 2^2       | rr= = 7^2 + 6^2 }} {{eqn | l = 125       | r = 11^2 + 2^2       | rr= = 10^2 + 5^2 }} {{eqn | l = 130       | r = 11^2 + 3^2       | rr= = 9^2 + 7^2 }} {{eqn | l = 145       | r = 12^2 + 1^2       | rr= = 9^2 + 8^2 }} {{eqn | l = 170       | r = 13^2 + 1^2       | rr= = 11^2 + 7^2 }} {{end-eqn}}
Sum of 2 Squares in 2 Distinct Ways/Examples/50
$50$ is the smallest positive integer which can be expressed as the sum of two square numbers in two distinct ways: {{begin-eqn}} {{eqn | l = 50       | r = 5^2 + 5^2 }} {{eqn | r = 7^2 + 1^2 }} {{end-eqn}}
Lower Adjoint at Element is Minimum of Preimage of Singleton of Element implies Composition is Identity
Let $L = \struct {S, \preceq}, R = \struct {T, \precsim}$ be ordered sets. Let $g: S \to T, d: T \to S$ be mappings such that :$\forall t \in T: \map d t = \min \set {g^{-1} \sqbrk {\set t} }$ Then $g \circ d = I_T$ where $I_T$ denotes the identity mapping of $T$.
Bounds for Weierstrass Elementary Factors
Let $E_p: \C \to \C$ denote the $p$th Weierstrass elementary factor: :$\map {E_p} z = \begin{cases} 1 - z & : p = 0 \\ \paren {1 - z} \map \exp {z + \dfrac {z^2} 2 + \cdots + \dfrac {z^p} p} & : \text{otherwise}\end{cases}$ Let $z \in \C$.
Order of Product of Entire Functions
Let $f, g: \C \to \C$ be entire functions of order $\alpha$ and $\beta$. Then $f g$ has order at most $\map \max {\alpha, \beta}$.
Numbers Partitioned into up to 4 Squares in 5 Ways
The following positive integers can be expressed as the sum of no more than $4$ squares in $5$ distinct ways: :$50, 52, 54, 58, \ldots$
Order of Product of Entire Function with Polynomial
Let $f:\C\to\C$ be an entire function of order $\omega$. Let $P:\C\to\C$ be a nonzero polynomial. Then $f\cdot P$ has order $\omega$.
Order of Shifted Entire Function
Let $f: \C \to \C$ be an entire function of order $\alpha$. Let $a \in \C$. Then $f \left({z + a}\right)$ has order $\alpha$.
Zerofree Entire Function of Finite Order is Exponential of Polynomial
Let $f:\C\to\C$ be an entire function of finite order. Let $f$ have no zeroes. Then $f=\exp P$ for some polynomial $P$.
Borel-Carathéodory Lemma
Let $D \subset \C$ be an open set with $0 \in D$. Let $R > 0$ be such that $\map B {0, R} \subset D$. {{explain|Domain of $R$, definition of $B$}} Let $f: D \to \C$ be analytic with $\map f 0 = 0$. Let $\map \Re {\map f z} \le M$ for $\cmod z \le R$. Let $0 < r < R$. Then for $\cmod z \le r$: :$(1): \quad \cmod {\map f z} \le \dfrac {2 M r} {R - r}$ :$(2): \quad \cmod {\map {f^{\paren k} } z} \le \dfrac {2 M R k!} {\paren {R - r}^{k + 1} }$ for all $k \ge 1$
Order of Sum of Entire Functions
Let $f, g: \C \to \C$ be entire functions of order $\alpha$ and $\beta$. Then $f + g$ has order at most $\map \max {\alpha, \beta}$, with equality if $\alpha \ne \beta$.
Exponent of Convergence is Less Than Order
Let $f: \C \to \C$ be an entire function. Let $\omega$ be its order. Let $\tau$ be its exponent of convergence. Then $\tau \le \omega$.
Class is Proper iff Bijection from Class to Proper Class/Corollary
$A$ is proper if and only if there exists a bijection from $\mathrm P$ to $A$.
Injection from Proper Class to Class
Let $A$ be a class. Let $\mathrm P$ be a proper class. Let $f: \mathrm P \to A$ be an injection. Then $A$ is proper.
Polynomial has Order Zero
Let $P:\C\to\C$ be a polynomial function. Then $P$ has order $0$.
Order is Maximum of Exponent of Convergence and Degree
Let $f: \C \to \C$ be an entire function. Let $\omega$ be its order. Let $\tau$ be its exponent of convergence. Let $h$ be the degree of the polynomial in its canonical factorization. Then: :$\omega = \map \max {\tau, h}$
Recurring Parts of Multiples of One Fifty-Third
The multiples of $\dfrac 1 {53}$ from $\dfrac 1 {53}$ to $\dfrac {52} {53}$ can be divided into $4$ sets of equal size: :one where the digits of the recurring part consists of a cyclic permutation of $01886 \, 79245 \, 283$ :one where the digits of the recurring part consists of a cyclic permutation of $03773 \, 58490 \, 566$ :one where the digits of the recurring part consists of a cyclic permutation of $07547 \, 16981 \, 132$ :one where the digits of the recurring part consists of a cyclic permutation of $09433 \, 96226 \, 415$.