hcju/proofwikips huggingface dataset:
Continued Fraction Expansion of Euler's Number/Convergents
The convergents of the continued fraction expansion to Euler's number $e$ are: :$2, 3, \dfrac 8 3, \dfrac {11} 4, \dfrac {19} 7, \dfrac {87} {32}, \dfrac {106} {39}, \dfrac {193} {71}, \dfrac {1264} {465}, \dfrac {1457} {536}, \dfrac {2721} {1001}, \ldots$ {{OEIS-Numerators|A007676}} {{OEIS-Denominators|A007677}} These best rational approximations are accurate to $0, 0, 1, 1, 2, 3, 3, 4, 5, 5, \ldots$ decimals. {{OEIS|A114539}} The fraction $\dfrac {878} {323}$ is exceptionally easy to remember: :$\dfrac {878} {323} = 2 \cdotp 71826 \, 625 \ldots$ although this does not occur in the above continued fraction expansion.
Number of Random Fractional Reals whose Total Exceeds 1
Let real numbers be selected at random following a continuous uniform distribution from the interval $\closedint 0 1$ until their total sum is greater than $1$. The expectation of the number of selections is Euler's number $e$.
Sine of X over X as Infinite Product
Let $z$ be a non-zero Complex Number. Then: :$\displaystyle \frac {\sin z} z = \cos \frac z 2 \cos \frac z 4 \cos \frac z 8 \cdots = \prod_{i \mathop = 1}^{\infty} \cos \frac z {2^i}$ where $\sin$ denotes the sine function and $\cos$ denotes the cosine function.
Maximum of Three Mutually Perpendicular Lines in Ordinary Space
In ordinary space, there can be no more than $3$ straight lines which are pairwise perpendicular. Thus, in a configuration of $4$ straight lines in space, at least one pair will not be perpendicular to each other.
Complex Logarithm of 1
:$\ln 1 = \set {2 k \pi i: k \in \Z}$
Sum Rule for Complex Derivatives
Let $\map f z, \map j z, \map k z$ be single-valued continuous complex functions in a domain $D \subseteq \C$, where $D$ is open. Let $f$, $j$, and $k$ be complex-differentiable at all points in $D$. Let $\map f z = \map j z + \map k z$. Then: : $\forall z \in D: \map {f'} z = \map {j'} z + \map {k'} z$
Product Rule for Complex Derivatives
Let $\map f z, \map j z, \map k z$ be single-valued continuous complex functions in a domain $D \subseteq \C$, where $D$ is open. Let $f$, $j$, and $k$ be complex-differentiable at all points in $D$. Let $\map f z = \map j z \, \map k z$. Then: :$\forall z \in D: \map {f'} z = \map j z \, \map {k'} z + \map {j'} z \, \map k z$
Derivative of Constant/Complex
Let $f_c \left({z}\right)$ be the constant function on an open domain $D \in \C$, where $c \in \C$. Then: :$\forall z \in D : f_c' \left({z}\right) = 0$
Derivative of Identity Function/Real
Let $I_\R: \R \to \R$ be the identity mapping on the real numbers $\R$. Then: :$\map {I_\R'} x = 1$
Derivative of Identity Function/Complex
Let $I_\C: \C \to \C$ be the identity function. Then: :$\map {I_\C'} z = 1$
Chain Rule for Real-Valued Functions/Corollary
Let $\Psi$ represent a differentiable function of $x$ and $y$. Let $y$ represent a differentiable function of $x$. Then: {{begin-eqn}} {{eqn | l = \frac {\d \Psi} {\d x}       | r = \frac {\partial \Psi} {\partial x} + \frac {\partial \Psi} {\partial y} \frac {\d y} {\d x} }} {{end-eqn}}
Trisecting the Angle/Hyperbola
Let $\alpha$ be an angle which is to be trisected. This can be achieved by means of a hyperbola.
Trisecting the Angle/Quadratrix of Hippias
Let $\alpha$ be an angle which is to be trisected. This can be achieved by means of a quadratrix of Hippias.
Trisecting the Angle/Cissoid of Diocles
Let $\alpha$ be an angle which is to be trisected. This can be achieved by means of a cissoid of Diocles.
Trisecting the Angle/Conchoid of Nicomedes
Let $\alpha$ be an angle which is to be trisected. This can be achieved by means of a conchoid of Nicomedes.
Laplace Transform of Periodic Function
Let $f$ be periodic, that is: :$\exists T \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + T}$ Then: :$\laptrans {\map f t} = \dfrac 1 {1 - e^{-s T} } \displaystyle \int_0^T e^{-s t} \map f t \rd t$ where $\laptrans {\map f t}$ denotes the Laplace transform.
Triangle is Medial Triangle of Larger Triangle
Let $\triangle ABC$ be a triangle. $\triangle ABC$ is the medial triangle of a larger triangle.
Altitudes of Triangle Meet at Point
Let $\triangle ABC$ be a triangle. The altitudes of $\triangle ABC$ all intersect at the same point.
Perpendicular Bisectors of Triangle Meet at Point
Let $\triangle ABC$ be a triangle. The perpendicular bisectors of $AB$, $BC$ and $AC$ all intersect at the same point.
Centroid of Triangle is Centroid of Medial
Let $\triangle ABC$ be a triangle. Let $\triangle DEF$ be the medial triangle of $\triangle ABC$. Let $G$ be the centroid of $\triangle ABC$. Then $G$ is also the centroid of $\triangle DEF$.
Diameters of Parallelogram Bisect each other
Let $\Box ABCD$ be a parallelogram with diameters $AC$ and $BD$. Let $AC$ and $BD$ intersect at $E$. Then $E$ is the midpoint of both $AC$ and $BD$.
Position of Centroid on Euler Line
Let $\triangle ABC$ be a triangle which is not equilateral. Let $O$ be the circumcenter of $\triangle ABC$. Let $G$ be the centroid of $\triangle ABC$. Let $H$ be the orthocenter of $\triangle ABC$. Then $G$ lies on the straight line connecting $O$ and $H$ such that: :$OG : GH = 1 : 2$ The line $OGH$ is the '''Euler line''' of $\triangle ABC$.
Perpendicular Bisector of Triangle is Altitude of Medial Triangle
Let $\triangle ABC$ be a triangle. Let $\triangle DEF$ be the medial triangle of $\triangle ABC$. Let a perpendicular bisector be constructed on $AC$ at $F$ to intersect $DE$ at $P$. Then $FP$ is an altitude of $\triangle DEF$.
Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral
Let $\triangle ABC$ be a triangle. Let $O$ be the circumcenter of $\triangle ABC$. Let $G$ be the centroid of $\triangle ABC$. Let $H$ be the orthocenter of $\triangle ABC$. Then $O$, $G$ and $H$ are the same points  $\triangle ABC$ is equilateral. If $\triangle ABC$ is not equilateral, then $O$, $G$ and $H$ are all distinct.
Altitude, Median and Perpendicular Bisector Coincide iff Triangle is Isosceles
Let $\triangle ABC$ be a triangle. Then: ::the altitude from $AB$ to $C$ ::the median from $AB$ to $C$ ::the perpendicular bisector of $AB$ :are all the same straight line : :$\triangle ABC$ is isosceles where $AB$ is the base.
Three Points Describe a Circle
Let $A$, $B$ and $C$ be points which are not collinear. Then there exists exactly one circle whose circumference passes through all $3$ points $A$, $B$ and $C$.
Three Regular Tessellations
There exist exactly $3$ regular tessellations of the plane.
Integer is Sum of Three Triangular Numbers
Let $n$ be a positive integer. Then $n$ is the sum of $3$ triangular numbers.
Fermat's Last Theorem/Cubic
The Diophantine equation $a^3 + b^3 = c^3$ has no solutions in strictly positive integers.
Smallest Magic Square is of Order 3
Apart from the trivial order $1$ magic square: {{:Magic Square/Examples/Order 1}} the smallest magic square is the order $3$ magic square: {{:Magic Square/Examples/Order 3}}
Euler Phi Function of n equal to Euler Phi Function of n+3
Let $\phi$ denote the Euler $\phi$ function. The only solutions to the equation: :$\phi \left({n}\right) = \phi \left({n + 3}\right)$ less than $1 \, 000 \, 000$ are: :$\phi \left({3}\right) = \phi \left({6}\right) = 2$ :$\phi \left({5}\right) = \phi \left({8}\right) = 4$
Friendship Theorem
Let there be a group of $6$ people. The traditional setting is that these $6$ people are at a party. Then (at least) one of the following $2$ statements is true: :$(1): \quad$ At least $3$ of these $6$ people have all met each other before :$(2): \quad$ At least $3$ of these $6$ people have never met each other before. That is, either there exists a set of $3$ mutual acquaintances, or there exists a set of $3$ mutual strangers.
Volume of Smallest Tetrahedron with Integer Edges and Integer Volume
The volume of the smallest tetrahedron with integer edges and integer volume is $3$. There are $2$ possible sets of edges: :$32, 33, 35, 40, 70, 76$ :$21, 32, 47, 56, 58, 76$
Barbier's Theorem
Let $K$ be a (closed) curve of constant diameter. {{explain|"constant diameter"}} Let the circumference of $K$ be $c$. Let the diameter of $K$ be $d$. Then: :$\dfrac c d = \pi$
Machin's Formula for Pi
:$\dfrac \pi 4 = 4 \arctan \dfrac 1 5 - \arctan \dfrac 1 {239} \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$
Ordinals under Addition form Ordered Semigroup
$\left({\operatorname{On}, +, \le}\right)$ forms an ordered semigroup, where: : $\operatorname{On}$ denotes the ordinal class, and : $+$ denotes ordinal addition.
Subset is Left Compatible with Ordinal Addition
Let $x, y, z$ be ordinals. Then: :$x \le y \implies \paren {z + x} \le \paren {z + y}$
Subset is Compatible with Ordinal Addition
Let $x, y, z$ be ordinals. Then: : $(1): x \le y \implies \left({z + x}\right) \le \left({z + y}\right)$ : $(2): x \le y \implies \left({x + z}\right) \le \left({y + z}\right)$
Ordinals under Addition form Semigroup
$\left({\operatorname{On}, +}\right)$ forms an semigroup, where: : $\operatorname{On}$ denotes the ordinal class, and : $+$ denotes ordinal addition.
Ordinals under Multiplication form Semigroup
$\struct {\On, \times}$ forms an semigroup, where: : $\On$ denotes the ordinal class, and : $\times$ denotes ordinal multiplication.
Subset is Left Compatible with Ordinal Multiplication
Let $x, y, z$ be ordinals. Then: :$x \le y \implies \left({z \cdot x}\right) \le \left({z \cdot y}\right)$
Subset is Compatible with Ordinal Multiplication
Let $x, y, z$ be ordinals. Then: : $(1): x \le y \implies \left({z \cdot x}\right) \le \left({z \cdot y}\right)$ : $(2): x \le y \implies \left({x \cdot z}\right) \le \left({y \cdot z}\right)$
Ordinals under Multiplication form Ordered Semigroup
$\struct {\On, \times, \le}$ forms an ordered semigroup, where: : $\On$ denotes the ordinal class, and : $\times$ denotes ordinal multiplication.
Ordinals under Addition form Monoid
$\struct {\On, +}$ forms an monoid, where: :$\On$ denotes the ordinal class :$+$ denotes ordinal addition.
Ordinals under Addition form Ordered Monoid
$\left({\operatorname{On}, +, \le}\right)$ forms an ordered monoid, where: : $\operatorname{On}$ denotes the ordinal class, and : $+$ denotes ordinal addition.
Ordinals under Multiplication form Monoid
$\left({\operatorname{On}, \cdot}\right)$ forms an monoid, where: : $\operatorname{On}$ denotes the ordinal class, and : $\cdot$ denotes ordinal multiplication.
Ordinals under Multiplication form Ordered Monoid
$\struct {\On, \cdot, \le}$ forms an ordered monoid, where: :$\On$ denotes the ordinal class, and :$\cdot$ denotes ordinal multiplication.
Buffon's Needle
Let a horizontal plane be divided into strips by a series of parallel lines a fixed distance apart, like floorboards. Let a needle whose length equals the distance between the parallel lines be dropped onto the plane randomly from a random height. Then the probability that the needle falls across one of the parallel lines is $\dfrac 2 \pi$.
Pi as Sum of Odd Reciprocals Alternating in Sign in Pairs
:$\dfrac {\pi \sqrt 2} 4 = 1 + \dfrac 1 3 - \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} - \dfrac 1 {13} - \dfrac 1 {15} \cdots$
Sum of Reciprocals of Squares of Odd Integers
{{begin-eqn}} {{eqn | l = \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^2}       | r = 1 + \dfrac 1 {3^2} + \dfrac 1 {5^2} + \dfrac 1 {7^2} + \dfrac 1 {9^2} + \cdots }} {{eqn | r = \dfrac {\pi^2} 8 }} {{end-eqn}}
Basel Problem as Infinite Product
:$\displaystyle \dfrac {\pi^2} 6 = \prod_{p \mathop \in \mathbb P} \dfrac {p^2} {p^2 - 1}$
Convergence of Taylor Series of Function Analytic on Disk
Let $F$ be a complex function. Let $x_0$ be a point in $\R$. Let $R$ be an extended real number greater than zero. Let $F$ be analytic at every point $z \in \C$ satisfying $\cmod {z - x_0} < R$. Let $f = F {\restriction_\R}$ be a real function. Then: :the Taylor series of $f$ about $x_0$ converges to $f$ at every point $x \in \R$ satisfying $\size {x - x_0} < R$
Taylor Series of Analytic Function has infinite Radius of Convergence
Let $F$ be a complex function. Let $F$ be analytic everywhere. Let $f = F {\restriction_{\R}}$ be a real function. Let $x_0$ be a point in $\R$. Then the Taylor series of $f$ about $x_0$ converges to $f$ at every point in $\R$.
Number of Binary Digits in Power of 10
Let $n$ be a positive integer. Expressed in binary notation, the number of digits in the $n$th power of $10$: :$10^n$ is equal to: :$\ceiling {n \log_2 10}$ where $\ceiling x$ denotes the ceiling of $x$.
Regular Octahedron is Dual of Cube
The regular octahedron is the dual of the cube.
Plane Figure with Bilateral Symmetry about Two Lines has 4 Congruent Parts
Let $F$ be a plane figure. Let $F$ have two different axes of bilateral symmetry. Then those two axes divide $F$ into $4$ congruent parts. :500px
Hyperbola can be Drawn through Four Non-Collinear Points
Let $A, B, C, D$ be points in the plane of which no $3$ are collinear. Then a hyperbola can be drawn so that it passes through all points $A, B, C, D$.
Classification of Finite Simple Groups
The finite simple groups can be classified as follows:
Divisibility of n-1 Factorial by Composite n
Let $n \in \Z$ be composite. Then: :$n \divides \paren {n - 1}! \iff n \ne 4$ where: :$\divides$ denotes divisibility :$n!$ denotes the factorial of $n$.
Divisibility by 4
An integer $N$ expressed in decimal notation is divisible by $4$  the $2$ least significant digits of $N$ form a $2$-digit integer divisible by $4$. 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 $4$  : :$10 a_1 + a_0$ is divisible by $4$.
Ratio of Number to Reversal which is Multiple
Take a (strictly) positive integer $n$, written in conventional decimal notation. Let $m$ be the reversal of $n$. Let $m = k n$ where $k$ is an integer. Then $k$ is either $4$ or $9$.
Smallest Pythagorean Triangle is 3-4-5
The smallest Pythagorean triangle has sides of length $3$, $4$ and $5$. :300px
Pythagorean Triangle with Sides in Arithmetic Sequence
The $3-4-5$ triangle is the only Pythagorean triangle such that: :the lengths of whose sides are in arithmetic sequence and: :the lengths of whose sides form a primitive Pythagorean triple.
Injectivity of Laplace Transform
Let $f$, $g$ be functions from $\left [{0 \,.\,.\, \to} \right ) \to \mathbb F$ of a real variable $t$, where $\mathbb F \in \left\{ {\R, \C}\right\}$. Further let $f$ and $g$ be continuous everywhere on their domains. Let $f$ and $g$ both admit Laplace transforms. Suppose that the Laplace transforms $\mathcal L \left\{{f}\right\}$ and $\mathcal L \left\{{g}\right\}$ satisfy: :$\forall t \ge 0: \mathcal L \left\{{f\left({t}\right)}\right\} = \mathcal L \left\{{g\left({t}\right)}\right\}$ Then $f = g$ everywhere on $\left[{0 \,.\,.\, \to}\right)$.
Pythagorean Triangle whose Area is Half Perimeter
The $3-4-5$ triangle is the only Pythagorean triangle whose area is half its perimeter.
Square Modulo 5/Corollary
When written in conventional base $10$ notation, no square number ends in one of $2, 3, 7, 8$.
Prime equals Plus or Minus One modulo 6
Let $p$ be a prime number greater than $3$. Then $p$ is either of the form: :$p = 6 n + 1$ or: :$p = 6 n - 1$ That is: :$p = \pm 1 \pmod 6$
Conic Section through Five Points
Let $A, B, C, D, E$ be distinct points in the plane such that no $3$ of them are collinear. Then it is possible to draw a conic section that passes through all $5$ points.
Lamé's Theorem
Let $a, b \in \Z_{>0}$ be (strictly) positive integers. Let $c$ and $d$ be the number of digits in $a$ and $b$ respectively when expressed in decimal notation. Let the Euclidean Algorithm be employed to find the GCD of $a$ and $b$. Then it will take no more than $5 \times \min \set {c, d}$ cycles around the Euclidean Algorithm to find $\gcd \set {a, b}$.
Volume of Unit Hypersphere
The volume of the unit sphere in $n$-dimensional space increases as $n$ goes up to $5$, but decreases thereafter.
Finite Subset Bounds Element of Finite Infima Set and Upper Closure
Let $L = \left({S, \wedge, \preceq}\right)$ be meet semilattice. Let $F$ be filter in $L$. Let $X$ be non empty finite subset of $S$. Let $x \in S$ such that : $x \in \left({\operatorname{fininfs}\left({F \cup X}\right)}\right)^\succeq$ where :$\operatorname{fininfs}$ denotes the finite infima set, :$X^\succeq$ denotes the upper closure of $X$. Then there exists $a \in S$: $a \in F \land x \succeq a \wedge \inf X$
Length of Reciprocal of Product of Powers of 2 and 5
Let $n \in \Z$ be an integer. Let $\dfrac 1 n$, when expressed as a decimal expansion, terminate after $m$ digits. Then $n$ is of the form $2^p 5^q$, where $m$ is the greater of $p$ and $q$.
Structure of Recurring Decimal
Let $\dfrac 1 m$, when expressed as a decimal expansion, recur with a period of $p$ digits. Let $\dfrac 1 n$, when expressed as a decimal expansion, terminate after $q$ digits. Then $\dfrac 1 {m n}$ has a nonperiodic part of $q$ digits, and a recurring part of $p$ digits.
Characteristics of Pentatope
A pentatope has $5$ cells, $10$ faces, $10$ edges and $5$ vertices.
Pentatope is Self-Dual
A pentatope is self-dual.
Riemann Hypothesis implies Odd Number is Sum of at most 5 Primes
Let the truth of the Riemann Hypothesis be assumed. Let $n$ be an odd integer. Then $n$ is the sum of at most $5$ primes.
Numbers of Zeroes that Factorial does not end with
Let $n \in \Z_{\ge 0}$ be a positive integer. Let $n!$ denote the factorial of $n$. Let $n!$ be expressed in decimal notation. Then $n!$ cannot end in the following numbers of zeroes: :$5, 11, 17, 23, 29, 30, 36, 42, \ldots$ {{OEIS|A000966}}
Euler-Binet Formula/Historical Note
The Euler-Binet Formula, derived by {{AuthorRef|Jacques Philippe Marie Binet|Binet}} in $1843$, was already known to {{AuthorRef|Leonhard Paul Euler|Euler}}, {{AuthorRef|Abraham de Moivre|de Moivre}} and {{AuthorRef|Daniel Bernoulli}} over a century earlier. However, it was {{AuthorRef|Jacques Philippe Marie Binet|Binet}} who derived the more general Binet Form of which this is an elementary application.
Cosine is of Exponential Order Zero
Let $\cos t$ be the cosine of $t$, where $t \in \R$. Then $\cos t$ is of exponential order $0$.
Sine is of Exponential Order Zero
Let $\sin t$ be the sine of $t$, where $t \in \R$. Then $\sin t$ is of exponential order $0$.
Scalar Multiple of Function of Exponential Order
Let $f: \R \to \F$ be a function, where $\F \in \set {\R, \C}$. Let $\lambda$ be a complex constant. Suppose $f$ is of exponential order $a$. Then $\lambda f$ is also of exponential order $a$.
Function of Exponential Order of Scalar Multiple
Let $f: \R \to \F$ be a function, where $\F \in \set {\R, \C}$. Let $\lambda$ be a real constant. Let $\map f t$ be of exponential order $a$. Then the function defined by $t \mapsto \map f {\lambda t}$ is of exponential order $a\lambda$.
Identity is of Exponential Order Epsilon
Let $I_\R: t \mapsto t$ be the identity mapping on $\R_{\ge 0}$. Then $I_\R$ is of exponential order $\epsilon$ for any  $\epsilon > 0$ arbitrarily small in magnitude.
Product of Functions of Exponential Order
Let $f, g: \R \to \F$ be functions, where $\F \in \set {\R, \C}$. Let $f$ be of exponential order $a$ and $g$ be of exponential order $b$. Then $f g: t \mapsto \map f t \, \map g t$ is of exponential order $a+b$.
Sum of Functions of Exponential Order
Let $f, g: \R \to \F$ be functions, where $\F \in \set {\R, \C}$. Suppose $f$ is of exponential order $a$ and $g$ is of exponential order $b$. Then $f + g: t \mapsto \map f t + \map g t$ is of exponential order $\max \set {a, b}$.
Linear Combination of Functions of Exponential Order
Let $f, g: \R \to \F$ be functions, where $\F \in \set {\R, \C}$. Let $\lambda, \mu$ be complex numbers. Suppose $f$ is of exponential order $a$ and $g$ is of exponential order $b$. Then $\map {\paren {\lambda f + \mu g} } t = \lambda \, \map f t + \mu \, \map g t$ is of exponential order $\max \set {a, b}$.
Constant Function is of Exponential Order Zero
Let $f_C: \R \to \mathbb F: t \mapsto C$ be a constant function, where $\mathbb F \in \set {\R, \C}$. Then $f_C$ is of exponential order $0$.
Polynomial is of Exponential Order Epsilon
Let $P: \R \to \mathbb F$ be a polynomial, where $\mathbb F \in \set {\R, \C}$. Then $P$ is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude.
Fibonacci Number by Power of 2
{{begin-eqn}} {{eqn | ll= \forall n \in \Z_{\ge 0}:       | l = 2^{n - 1} F_n       | r = \sum_k 5^k \dbinom n {2 k + 1}       | c =  }} {{eqn | r = \dbinom n 1 + 5 \dbinom n 3 + 5^2 \dbinom n 5 + \cdots       | c =  }} {{end-eqn}} where: :$F_n$ denotes the $n$th Fibonacci number :$\dbinom n {2 k + 1} \ $ denotes a binomial coefficient.
Approximation to Golden Rectangle using Fibonacci Squares
An approximation to a golden rectangle can be obtained by placing adjacent to one another squares with side lengths corresponding to consecutive Fibonacci numbers in the following manner: :800px It can also be noted, as from Sequence of Golden Rectangles, that an equiangular spiral can be approximated by constructing quarter circles as indicated.
Fibonacci Number of Index 2n as Sum of Squares of Fibonacci Numbers
Let $F_n$ denote the $n$th Fibonacci number. Then: :$F_{2 n} = {F_{n + 1} }^2 - {F_{n - 1} }^2$
Euler-Binet Formula/Also known as
The '''Euler-Binet Formula''' is also known as '''{{AuthorRef|Jacques Philippe Marie Binet|Binet}}'s formula'''.
If Ideal and Filter are Disjoint then There Exists Prime Ideal Including Ideal and Disjoint from Filter
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a distributive lattice. Let $I$ be an ideal in $L$. Let $F$ be a filter on $L$ such that :$I \cap F = \O$ Then there exists a prime ideal $P$ in $L$: $I \subseteq P$ and $P \cap F = \O$
No 4 Fibonacci Numbers can be in Arithmetic Sequence
Let $a, b, c, d$ be distinct Fibonacci numbers. Then, except for the trivial case: :$a = 0, b = 1, c = 2, d = 3$ it is not possible that $a, b, c, d$ are in arithmetic sequence.
Sum of Alternating Sign Reciprocals of Sequence of Pairs of Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared
{{begin-eqn}} {{eqn | l = \sum_{k \mathop \ge 2} \left({-1}\right)^k \dfrac 1 {F_k F_{k + 1} }       | r = \dfrac 1 {1 \times 2} - \dfrac 1 {2 \times 3} + \dfrac 1 {3 \times 5} - \dfrac 1 {5 \times 8} + \cdots       | c =  }} {{eqn | r = \phi^{-2}       | c =  }} {{end-eqn}} where: :$F_k$ denotes the $k$th Fibonacci number :$\phi$ denotes the golden mean.
Sum of Reciprocals of Sequence of Pairs of Even Index Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared
{{begin-eqn}} {{eqn | l = \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k} F_{2 k + 2} }       | r = \dfrac 1 {1 \times 3} + \dfrac 1 {3 \times 8} + \dfrac 1 {8 \times 21} + \dfrac 1 {21 \times 55} + \cdots       | c =  }} {{eqn | r = \phi^{-2}       | c =  }} {{end-eqn}} where: :$F_k$ denotes the $k$th Fibonacci number :$\phi$ denotes the golden mean.
Number of Fibonacci Numbers between n and 2n
Let $n \in \Z_{> 0}$ be a (strictly) positive integer. Then there exists either one or two Fibonacci numbers between $n$ and $2 n$ inclusive.
Square Root is of Exponential Order Epsilon
The positive square root function: :$t \mapsto \sqrt t$ is of exponential order $\epsilon$ for any  $\epsilon > 0$ arbitrarily small in magnitude.
Number of Fibonacci Numbers with Same Number of Decimal Digits
Let $n$ be an integer such that $n > 1$. When expressed in decimal notation, there are either $4$ or $5$ Fibonacci numbers with $n$ digits.
Prime Number divides Infinite Number of Fibonacci Numbers
Let $p$ be a prime number. Then there exist an infinite number of Fibonacci numbers which are divisible by $p$.
Prime Number divides Fibonacci Number
For $n \in \Z$, let $F_n$ denote the $n$th Fibonacci number. Let $p$ be a prime number. Then: :$p \equiv \pm 1 \pmod 5 \implies p \divides F_{p - 1}$ :$p \equiv \pm 2 \pmod 5 \implies p \divides F_{p + 1}$ where $\divides$ denotes divisibility. Thus in all cases, except where $p = 5$ itself: :$p \divides F_{p \pm 1}$
Existence of Fibonacci Number Divisible by Number
Let $m \in \Z$ be an integer. Then in the first $m^2$ Fibonacci numbers there exists at least one Fibonacci number which is divisible by $m$.
Fibonacci Prime has Prime Index except for 3
Let $F_n$ denote the $n$th Fibonacci number. Let $F_n$ be a prime number. Then, apart from $F_4 = 3$, $n$ is a prime number.