hcju/proofwikips huggingface dataset:
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 $k$.
Center of Group is Kernel of Conjugacy Action
Let $G$ be a group. Let $Z$ be the kernel of the conjugacy action. Then $Z$ is the center of $G$.
Little-O Implies Big-O/General Result
Let $X$ be a topological space. Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\,\cdot\,}$ Let $f, g: X \to V$ be mappings. Let $x_0 \in X$. Let $f = \map o g$ as $x \to x_0$, where $o$ denotes little-O notation. Then $f = \map \OO g$ as $x \to x_0$, where $\OO$ denotes big-$\OO$ notation.
Little-O Implies Big-O/Sequences
Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers. Let $a_n = \map o {b_n}$ where $o$ denotes little-O notation. Then $a_n = \map \OO {b_n}$ where $\OO$ denotes big-$\OO$ notation.
Substitution in Big-O Estimate/Sequences
Let $(a_n)$ and $(b_n)$ be sequences of real or complex numbers. Let $a_n = O(b_n)$ where $O$ denotes big-O notation. Let $(n_k)$ be a diverging sequence of natural numbers. Then $a_{n_k} = O(b_{n_k})$.
First Sequence of Three Consecutive Strictly Decreasing Euler Phi Values
The first sequence of $3$ consecutive positive integers whose Euler $\phi$ values are strictly decreasing is: :$\map \phi {523} > \map \phi {524} > \map \phi {525}$
Bases of Finitely Generated Free Module have Equal Cardinality
Let $R$ be a commutative ring with unity. Let $M$ be a free $R$-module. Let $M$ be finitely generated. Let $B$ and $C$ be bases of $M$. Then $B$ and $C$ are finite and have the same cardinality.
Bases of Free Module have Equal Cardinality
Let $R$ be a commutative ring with unity. Let $M$ be a free $R$-module. Let $B$ and $C$ be bases of $M$. Then $B$ and $C$ are equinumerous.
Sequences of Three Consecutive Strictly Increasing Euler Phi Values
The following sequences of $3$ consecutive positive integers have the property that their Euler $\phi$ values are strictly increasing: :$\tuple {105, 106, 107}, \tuple {165, 166, 167}, \tuple {315, 316, 317}, \tuple {525, 526, 527}, \dots$ {{expand|Related sequences: A161962 (superset), A161963}}
Basis of Vector Space Injects into Generator
Let $K$ be a division ring. Let $V$ be a vector space over $K$. Let $B$ be a basis of $V$. Let $G$ be a generator of $V$. Then there exists an injection from $B$ to $G$.
Increasing Union of Ideals is Ideal/Sequence
Let $R$ be a ring. Let $S_0 \subseteq S_1 \subseteq S_2 \subseteq \dotsb \subseteq S_i \subseteq \dotsb$ be ideals of $R$. Then the increasing union $S$: :$\displaystyle S = \bigcup_{i \mathop \in \N} S_i$ is an ideal of $R$.
Increasing Union of Ideals is Ideal/Chain
Let $R$ be a ring. Let $\left({P, \subseteq}\right)$ be the ordered set consisting of all ideals of $R$, ordered by inclusion. Let $\left\{{I_\alpha}\right\}_{\alpha \in A}$ be a non-empty chain of ideals in $P$. Let $\displaystyle I = \bigcup_{\alpha \in A} I_\alpha$ be their union. Then $I$ is an ideal of $R$.
Union of Chain of Proper Ideals is Proper Ideal
Let $R$ be a ring with unity. Let $\struct {P, \subseteq}$ be the ordered set consisting of all ideals of $R$, ordered by inclusion. Let $\sequence {I_\alpha}_{\alpha \mathop \in A}$ be a non-empty chain of proper ideals in $P$. Let $\displaystyle I = \bigcup_{\alpha \mathop \in A} I_\alpha$ be their union. Then $I$ is a proper ideal of $R$.
Dimension of Free Vector Space on Set
Let $k$ be a division ring. Let $X$ be a set. Let $k^{\paren X}$ be the free vector space on $X$. The vector space $k^{\paren X}$ has dimension the cardinality of $X$.
239 is not Expressible as Sum of Fewer than 19 Fourth Powers
:$239 = 13 \times 1^4 + 4 \times 2^4 + 2 \times 3^4$
479 is not Expressible as Sum of Fewer than 19 Fourth Powers
:$479 = 13 \times 1^4 + 3 \times 2^4 + 2 \times 3^4 + 4^4$ or: :$479 = 10 \times 1^4 + 4 \times 2^4 + 5 \times 3^4$
559 is not Expressible as Sum of Fewer than 19 Fourth Powers
:$559 = 15 \times 1^4 + 2 \times 2^4 + 2 \times 4^4$ or: :$559 = 9 \times 1^4 + 4 \times 2^4 + 6 \times 3^4$
Canonical Basis of Free Module on Set is Basis
Let $R$ be a ring with unity. Let $I$ be a set. Let $R^{(I)}$ be the free $R$-module on $I$. Let $B$ be its canonical basis. Then $B$ is a basis of $R^{(I)}$.
Korselt's Theorem
Let $n \ge 2$ be an integer. Then $n$ is a '''Carmichael number''' : : $(1): \quad n$ is odd and the following conditions hold for every prime factor $p$ of $n$: : $(2): \quad p^2 \nmid n$ : $(3): \quad \paren {p - 1} \divides \paren {n - 1}$ where: :$\divides$ denotes divisibility :$\nmid$ denotes non-divisibility.
Intersection of Submodules is Submodule
Let $R$ be a ring. Let $M$ be an $R$-module. Let $S$ be a set of submodules of $M$. Then the intersection $\bigcap S$ is a submodule of $M$.
There are Infinitely Many Carmichael Numbers
There are infinitely many Carmichael numbers.
Product of Big-O Estimates/Sequences
Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be sequences of real or complex numbers. Let: :$a_n = \map \OO {b_n}$ :$c_n = \map \OO {d_n}$ where $\OO$ denotes big-O notation. Then: :$a_n c_n = \map \OO {b_n d_n}$
Equivalence of Definitions of Order of Entire Function
Let $f: \C \to \C$ be an entire function. Let $\ln$ denote the natural logarithm. {{TFAE| def = Order of Entire Function}}
Little-O Times Big-O is Little-O/Sequences
Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be sequences of real or complex numbers. Let: :$a_n = \map \OO {b_n}$ :$c_n = \map o {d_n}$ where: :$\OO$ denotes big-O notation :$o$ denotes little-o notation. Then: :$a_n c_n = \map o {b_n d_n}$
Field Norm of Complex Number Equals Field Norm
Let $z = a + i b$ be a complex number, where $a, b \in \R$. Then the field norm of $z$ is the field norm with respect to the field extension $\C / \R$.
Big-O Notation for Sequences Coincides with General Definition
Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ be sequences of real or complex numbers. Let $\N$ be given the discrete topology.  :$(1): \quad$ $a_n = O(b_n)$, where $O$ denotes big-O notation for sequences :$(2): \quad$ $a_n = O(b_n)$, where $O$ stands for the general definition of big-O notation
Bounded iff Big-O of 1/Sequences
Let $\sequence {a_n}$ be a sequence of real or complex numbers.  :$(1): \quad a_n$ is bounded :$(2): \quad a_n = \map \OO 1$, where $\OO$ denotes big-$\OO$ notation
Sum of Little-O Estimates/Sequences
Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be sequences of real or complex numbers. Let: :$a_n = \map o {b_n}$ :$c_n = \map o {d_n}$ where $o$ denotes little-o notation. Then: :$a_n + c_n = \map o {\size {b_n} + \size {d_n} }$
Equivalence of Definitions of Asymptotically Equal Sequences
Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences in $\R$. {{TFAE|def = Asymptotically Equal Sequences}}
Largest Number not Expressible as Sum of Multiples of 23 and 28
The largest integer $n$ that cannot be expressed in the form: :$n = 23 x + 28 y$ for $x, y \in \Z_{>0}$ is $593$.
Open implies There Exists Way Below Element
Let $L = \left({S, \preceq, \tau}\right)$ be a continuous topological lattice with Scott topology. Let $p \in S, A \subseteq S$ such that :$A$ is open and $p \in A$. Then :$\exists q \in A: q \ll p$
Smallest Fourth Power which is Sum of 5 Fourth Powers
$625$ is the smallest fourth power which is the sum of $5$ fourth powers: :$625 = 5^4 = 2^4 + 2^4 + 3^4 + 4^4 + 4^4$
Interior is Union of Way Above Closures
Let $\left({S, \preceq, \tau}\right)$ be a complete continuous topological lattice with Scott topology. Let $X \subseteq S$. Then $X^\circ = \bigcup \left\{ {x^\gg: x \in S \land x^\gg \subseteq X}\right\}$ where :$X^\circ$ denotes the interior of $X$, :$x^\gg$ denotes the way above closure of $x$.
Way Above Closures Form Basis
Let $L = \left({S, \preceq, \tau}\right)$ be a complete continuous topological lattice with Scott topology. Then $\left\{ {x^\gg: x \in S}\right\}$ is basis of $L$.
Record Gaps between Twin Primes
The gaps between the following pairs of twin primes are larger than those for all smaller pairs: {{begin-eqn}} {{eqn | l = \tuple {3, 5}       | o = \to       | r = \tuple {5, 7}       | c = a gap of $0$ }} {{eqn | l = \tuple {5, 7}       | o = \to       | r = \tuple {11, 13}       | c = a gap of $4$ }} {{eqn | l = \tuple {17, 19}       | o = \to       | r = \tuple {29, 31}       | c = a gap of $10$ }} {{eqn | l = \tuple {41, 43}       | o = \to       | r = \tuple {59, 61}       | c = a gap of $16$ }} {{eqn | l = \tuple {69, 73}       | o = \to       | r = \tuple {101, 103}       | c = a gap of $28$ }} {{eqn | l = \tuple {311, 313}       | o = \to       | r = \tuple {347, 349}       | c = a gap of $34$ }} {{eqn | l = \tuple {347, 349}       | o = \to       | r = \tuple {419, 421}       | c = a gap of $70$ }} {{eqn | l = \tuple {659, 661}       | o = \to       | r = \tuple {809, 811}       | c = a gap of $148$ }} {{end-eqn}} {{OEIS|A036061}} {{finish|Add this list to the Number Sequence}}
Sophie Germain Prime cannot be 6n+1
Let $p$ be a Sophie Germain prime. Then $p$ cannot be of the form $6 n + 1$, where $n$ is a positive integer.
Way Above Closure is Open
Let $L = \left({S, \preceq, \tau}\right)$ be a complete continuous topological lattice with Scott topology. Let $x \in S$. Then $x^\gg$ is open where $x^\gg$ denotes the way above closure of $x$.
Way Above Closure is Upper
Let $\left({S, \preceq}\right)$ be an ordered set. Let $x \in S$. Then $x^\gg$ is upper where $x^\gg$ denotes the way above closure of $x$.
Way Above Closures that Way Below Form Local Basis
Let $L = \left({S, \preceq, \tau}\right)$ be a complete continuous topological lattice with Scott topology. Let $p \in S$. Then $\left\{ {q^\gg: q \in S \land q \ll p}\right\}$ is local basis at $p$.
Characterization of Analytic Basis by Local Bases
Let $T = \struct {S, \tau}$ be a topological space. Let $P$ be a set of subsets of $S$ such that :$P \subseteq \tau$ and :for all $p \in S$: there exists local basis $B$ at $p: B \subseteq P$ Then $P$ is basis of $T$.
Approximation to Golden Ratio using 666
The Golden Ratio is approximated closely by the following formula: :$\phi \approx -2 \sin \left({666}\right) = 1.61803 \, 39887 \, 5 \ldots$
Euler Phi Function of 666 equals Product of Digits
The number $666$ has the following interesting property: :$\map \phi {666} = 6 \times 6 \times 6$ where $\phi$ denotes the Euler $\phi$ function.
Sum of Sequence of Squares of Primes
Let $S = \sequence {s_n}$ be the integer sequence defined as: :$\displaystyle s_n = \sum_{i \mathop = 1}^n {p_i}^2$ where $P_i$ denotes the $i$th prime number. Then $S$ begins: :$4, 13, 38, 87, 208, 377, 666, 1027, 1556, 2397, 3358, 4727, 6408, 8257, 10466, \ldots$ {{OEIS|A024450}}
Pair of Consecutive Powerful Numbers whose First is Odd
The only known pair of consecutive integers which are both powerful numbers such that the first of the pair is odd is: :$\tuple {675, 676}$
Interior is Union of Elements of Basis
Let $T = \left({S, \tau}\right)$ be a topological space. Let $B$ be a basis of $T$. Let $V$ be a subset of $S$. Then $V^\circ = \bigcup \left\{ {G \in B: G \subseteq V}\right\}$ where $V^\circ$ denotes the interior of $V$.
Palindromic Squares with Non-Palindromic Roots
The sequence of palindromic squares with non-palindromic square roots begins: :$676, 69 \, 696, 94 \, 249, 698 \, 896, 5 \, 221 \, 225, 6 \, 948 \, 496, 522 \, 808 \, 225, \ldots$ This sequence is not explicitly given in . The sequence of those corresponding non-palindromic square roots begins: :$26, 264, 307, 836, 2285, 2636, 22 \, 865, 24 \, 846, 30 \, 693, \ldots$ {{OEIS|A251673}}
Tetrahedral Numbers which are Sum of 2 Tetrahedral Numbers
The sequence of tetrahedral numbers which are the sum of two other tetrahedral numbers begins: :$680, \ldots$ {{expand|To be expanded when we have some more.}}
Open Set is Union of Elements of Basis
Let $T = \left({S, \tau}\right)$ be a topological space. Let $B$ be a basis of $T$. Let $V$ be an open subset of $S$. Then $V = \bigcup \left\{ {G \in B: G \subseteq V}\right\}$
Equivalence of Definitions of Compatible Atlases
Let $M$ be a topological space. Let $\mathscr F, \mathscr G$ be $d$-dimensional atlases of class $C^k$ on $M$. {{TFAE|def = Compatible Atlases}}
Atlas is Contained in Unique Maximal Atlas
Let $M$ be a topological space. Let $A$ be a $d$-dimensional atlas of class $C^k$. Then $A$ is contained in a unique maximal atlas of class $C^k$.
Locally Euclidean iff has C0-Atlas
Let $M$ be a topological space.  :$(1):\quad$ $M$ is locally euclidean. :$(2):\quad$ There exists a $C^0$-atlas on $M$.
Differentiable Structure Contains Unique Maximal Atlas
Let $M$ be a topological space. Let $k$ and $d$ be natural numbers. Let $S$ be a $d$-dimensional differentiable structure of class $C^k$ on $M$. Then $S$ contains a unique maximal $C^k$-atlas.
Consecutive Integers whose Product is Primorial
The following primorials can be expressed as the product of consecutive integers: :$2, 6, 30, 210, 510 \, 510$ {{OEIS|A161620}} No others are known. The corresponding indices of those primorials are: :$2, 3, 5, 7, 17$ {{OEIS|A215658}} The corresponding values of $n$ such that $p\# = \paren {n - 1} n$ are: :$2, 3, 6, 15, 715$ {{OEIS|A215659}}
Integers whose Ratio between Sigma and Phi is Square
The sequence of integers whose $\sigma$ value divided by its Euler $\phi$ value is a square begins: :$1, 14, 30, 105, 248, 264, 418, 714, 1485, 3080, \ldots$ {{OEIS|A293391}} {{finish|This sequence needs to be added to the individual integer pages.}}
Implicit Function Theorem/Real Functions
Let $n$ and $k$ be natural numbers. Let $\Omega \subset \R^{n + k}$ be open. Let $f: \Omega \to \R^k$ be continuous. Let the partial derivatives of $f$ with respect to $\R^k$ be continuous. Let $\tuple {a, b} \in \Omega$, with $a\in \R^n$ and $b\in \R^k$. Let $\map f {a, b} = 0$. For $\tuple {x_0, y_0} \in \Omega$, let $D_2 \map f {x_0, y_0}$ denote the total derivative of the function $y \mapsto \map f {x_0, y}$ at $y_0$. Let the linear map $D_2 \map f {a, b}$ be invertible. Then there exist neighborhoods $U \subset \Omega$ of $a$ and $V \subset \R^k$ of $b$ such that there exists a unique function $g: U \to V$ such that $\map f {x, \map g x} = 0$ for all $x \in U$. Moreover, $g$ is continuous.
Uniform Contraction Mapping Theorem
Let $M$ and $N$ be metric spaces. Let $M$ be complete. Let $f : M \times N \to M$ be a continuous uniform contraction. Then for all $t \in N$ there exists a unique $g \left({t}\right) \in M$ such that $f(g \left({t}\right), t) = g \left({t}\right)$, and the mapping $g: N \to M$ is continuous.
Mapping at Element is Supremum of Compact Elements implies Mapping at Element is Supremum that Way Below
Let $\left({S, \vee_1, \wedge_1, \preceq_1}\right)$ and $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be complete lattices. Let $f: S \to T$ be a mapping such that :$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \preceq_1 x \land w}\right.$ is compact$\left.{}\right\}$ Then :$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
Inverse Function Theorem for Real Functions
Let $n\geq1$ and $k\geq1$ be natural numbers. Let $\Omega\subset \R^n$ be open. $f : \Omega \to \R^n$ be a vector-valued function of class $C^k$. Let $a\in\Omega$. Let the differential $Df(a)$ of $f$ at $a$ be invertible. Then there exist open sets $U\subset\Omega$ and $V\subset\R^n$ such that the restriction of $f$ to $U$ is a $C^k$-diffeomorphism $f:U\to V$.
720 is Product of Consecutive Numbers in Two Ways
:$720 = 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8$
Factorial which is Sum of Two Squares
The only factorial which can be expressed as the sum of two squares is: {{begin-eqn}} {{eqn | l = 6!       | r = 12^2 + 24^2       | c =  }} {{end-eqn}}
Multiplicity of 720 in 720 Factorial
The multiplicity of $720$ in $720!$ is $178$. That is: :$720^{178} \divides 720!$ but: :$720^{179} \nmid 720!$ where: :$720!$ denotes $720$ factorial :$\divides$ denotes divisibility :$\nmid$ denotes non-divisibility.
Continuous iff Mapping at Element is Supremum
Let $\left({S, \preceq_1, \tau_1}\right)$ and $\left({T, \preceq_2, \tau_2}\right)$ be complete continuous topological lattices with Scott topologies. Let $f: S \to T$ be a mapping. Then $f$ is continuous  :$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
Equivalence of Definitions of Generated Submodule
Let $R$ be a ring. Let $M$ be an $R$-module. Let $S\subset M$ be a subset. {{TFAE|def = Generated Submodule}}
Equivalence of Definitions of Basis of Vector Space
Let $K$ be a division ring. Let $\struct {G, +_G, \circ}_K$ be an vector space over $K$. {{TFAE| def = Basis of Vector Space}}
Equivalence of Definitions of Differentiable Real Function at Point
Let $\openint a b \subset \R$ be an open interval. Let $\xi$ be a point in $\openint a b$. {{TFAE|def = Differentiable Real Function at Point|view = differentiable real function at a point}}
Solutions to Approximate Fermat Equation x^3 = y^3 + z^3 Plus or Minus 1
The approximate Fermat equation: :$x^3 = y^3 + z^3 \pm 1$ has the solutions: {{begin-eqn}} {{eqn | l = 9^3       | r = 6^3 + 8^3 + 1 }} {{eqn | l = 103^3       | r = 64^3 + 94^3 - 1       | c =  }} {{end-eqn}}
Cubes which are Sum of Five Cubes
The following cube numbers can be expressed as the sum of $5$ positive cube numbers: :$9^3, \ldots$ {{expand|More terms needed. It seems that: </br> $4$ and all numbers $> 8$ can be so expressed </br> only $4, 8, 10, 11, 13$ require repeated cubes}}
Period of Reciprocal of 729 is 81
The decimal expansion of the reciprocal of $729$ has $\dfrac 1 9$ the maximum period, that is, $81$: :$\dfrac 1 {729} = 0 \cdotp \dot 00137 \, 17421 \, 12482 \, 85322 \, 35939 \, 64334 \, 70507 \, 54458 \, 16186 \, 55692 \, 72976 \, 68038 \, 40877 \, 91495 \, 19890 \, 26063 \, \dot 1$ The recurring part can be arranged in groups of $9$ digits each, revealing an interesting pattern: {{begin-eqn}} {{eqn | l = 001 \, 371 \, 742       | o =  }} {{eqn | l = 112 \, 482 \, 853       | o =  }} {{eqn | l = 223 \, 593 \, 964       | o =  }} {{eqn | l = 334 \, 705 \, 075       | o =  }} {{eqn | l = 445 \, 816 \, 186       | o =  }} {{eqn | l = 556 \, 927 \, 297       | o =  }} {{eqn | l = 668 \, 038 \, 408       | o =  }} {{eqn | l = 779 \, 149 \, 519       | o =  }} {{eqn | l = 890 \, 260 \, 631       | o =  }} {{end-eqn}} that is, each row (apart from the last) can be obtained from the previous one by adding $111 \, 111 \, 111$ to it. {{OEIS|A021733}}
Implicit Function Theorem for Differentiable Real Functions
Let $\Omega \subset \R^{n+k}$ be open. Let $f : \Omega \to \R^k$ be differentiable. Let the $i$th partial derivatives of $f$ be continuous in $\Omega$ for $n+1 \leq i \leq n+k$. Let $(a,b) \in \Omega$, with $a\in \R^n$ and $b\in \R^k$. Let $f(a,b) = 0$. For $(x_0,y_0)\in\Omega$, let $D_2 f(x_0,y_0)$ denote the differential of the function $y\mapsto f(x_0, y)$ at $y_0$. Let the linear map $D_2 f(a,b)$ be invertible. Then there exist neighborhoods $U\subset\Omega$ of $a$ and $V\subset\R^k$ of $b$ such that there exists a unique function $g : U \to V$ such that $f(x, g(x)) = 0$ for all $x\in U$. Moreover, $g$ is differentiable, and its differential satisfies: :$dg (x) = - \left( (D_2f)(x, g(x)) \right)^{-1} \circ (D_1 f)(x, g(x))$ for all $x\in U$.
Implicit Function Theorem for Smooth Real Functions
Let $\Omega \subset \R^{n+k}$ be open. Let $f : \Omega \to \R^k$ be smooth. Let $(a,b) \in \Omega$, with $a\in \R^n$ and $b\in \R^k$. Let $f(a,b) = 0$. For $(x_0,y_0)\in\Omega$, let $D_2 f(x_0,y_0)$ denote the differential of the function $y\mapsto f(x_0, y)$ at $y_0$. Let the linear map $D_2 f(a,b)$ be invertible. Then there exist neighborhoods $U\subset\Omega$ of $a$ and $V\subset\R^k$ of $b$ such that there exists a unique function $g : U \to V$ such that $f(x, g(x)) = 0$ for all $x\in U$. Moreover, $g$ is smooth, and its differential satisfies: :$dg (x) = - \left( (D_2f)(x, g(x)) \right)^{-1} \circ (D_1 f)(x, g(x))$ for all $x\in U$.
Sum of 4 Consecutive Binomial Coefficients forming Square
Consider the Diophantine equation: :$\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 = m^2$ where: :$\dbinom a b$ denotes a binomial coefficient :$n$ is an integer :$m$ is a non-negative integer. Then $n$ has one of the following values: :$-1, 0, 2, 7, 15, 74, 767$ {{OEIS|A047694}} The corresponding values of $m$ are: :$0, 1, 2, 8, 24, 260, 8672$ {{OEIS|A047695}}
Implicit Function Theorem for Lipschitz Contractions
Let $M$ and $N$ be metric spaces. Let $M$ be complete. Let $f : M \times N \to M$ be a Lipschitz continuous uniform contraction. Then for all $t\in N$ there exists a unique $g(t) \in M$ such that $f(g(t), t) = g(t)$, and the mapping $g : N \to M$ is Lipschitz continuous.
Local Normal Form for Immersions
Let $\Omega\subset\R^k$ be open. Let $f: \Omega \to \R^n$ be an immersion. Let $p \in \Omega$. Then: :$k \le n$ and there exists a local diffeomorphism $\phi$ around $\map f p$ such that: :$\phi \circ \map f x = \tuple {x, 0}$ for all $x$ in a neighborhood of $p$.
Local Normal Form for Submersions
Let $\Omega\subset\R^n$ be open. Let $f : \Omega \to \R^k$ be an submersion. Let $p\in\Omega$. Then $n\geq k$, and there exists a local diffeomorphism $\phi$ around $f(p)$ such that :$\phi\circ f (x, y) = x$ for all $(x, y)$ in a neighborhood of $p$.
Smallest Square Inscribed in Two Pythagorean Triangles
The smallest square with integer sides that can be inscribed within two different Pythagorean triangles so that one side of the square lies on the hypotenuse has side length $780$. The two Pythagorean triangles in question have side lengths $\tuple {1443, 1924, 2405}$ and $\tuple {1145, 2748, 2977}$.
Sequence of Numbers Divisible by Sequence of Primes
The integers in this sequence: :$788, 789, 790, 791, 792, 793$ are divisible by: :$2, 3, 5, 7, 11, 13$ respectively.
Numbers whose Square is Palindromic with Even Number of Digits
The sequence of positive integers whose square is a palindromic number with an even number of digits begins: :$836, 798 \, 644, 64 \, 030 \, 648, 83 \, 163 \, 115 \, 486, 6 \, 360 \, 832 \, 925 \, 898, \ldots$ {{OEIS|A016113}}
Sum of Sequence of Factorials
The sequence $S = \sequence {s_n}$ defined as: :$\displaystyle s_n = \sum_{k \mathop = 1}^n k!$ begins: :$1, 3, 9, 33, 153, 873, 5913, 46 \, 233, 409 \, 113, 4 \, 037 \, 913, \ldots$ {{OEIS|A007489}}
Number of Magic Squares of Order 4
The number of different magic squares of order $4$, up to rotation and reflection, is $880$.
Primitive Semiperfect Numbers which are not Primitive Abundant
The sequence of primitive semiperfect numbers which are not also primitive abundant starts: :$6, 28, 350, 490, 496, 770, 910, 1190, \ldots$ These are semiperfect numbers which are either: : perfect or: : whose only abundant aliquot parts are weird.
Sequence of Odd Abundant Numbers
The sequence of odd abundant numbers begins: :$945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, \ldots$ {{OEIS|A005231}}
Set of 5 Triplets whose Sums and Products are Equal
The following set of $5$ triplets of integers have the property that: :the sum of the integers in each triplet are equal and: :the product of the integers in each triplet are equal: :$\tuple {6, 480, 495}$, $\tuple {11, 160, 810}$, $\tuple {12, 144, 825}$, $\tuple {20, 81, 880}$, $\tuple {33, 48, 900}$ The sum is $981$, and the product is $1 \, 425 \, 600$. This is the only known such set of $5$ triplets of integers with this property.