hcju/proofwikips huggingface dataset:
Function that Satisfies Axioms of Uncertainty
Let $n \in \N$ be a natural number. Let $p_1, p_2, \dotsc, p_n$ be real numbers such that: :$\forall i \in \set {1, 2, \dotsc, n}: p_i \ge 0$ :$\displaystyle \sum_{i \mathop = 1}^n p_i = 1$ Let $\map H {p_1, p_2, \ldots, p_n}$ be a real-valued function which satisfies the axioms of uncertainty. Then: :$\map H {p_1, p_2, \ldots, p_n} = \displaystyle -\lambda \sum_{i \mathop = 1}^n p_i \log_b p_i$ where: :$\lambda \in \R_{>0}$ :$b \in \R_{>1}$
Uncertainty Function satisfies Axioms of Uncertainty
Let $X$ be a random variable. Let $X$ take a finite number of values with probabilities $p_1, p_2, \dotsc, p_n$. Let $\map H X$ be the '''uncertainty function''' of $X$: :$\map H X = \displaystyle -\sum_k p_k \lg p_k$ where: :$\lg$ denotes logarithm base $2$ :the summation is over those $k$ where $p_k > 0$. Then the uncertainty function satisfies the Axioms of Uncertainty.
Number of k-Cycles in Symmetric Group
Let $n \in \N$ be a natural number. Let $S_n$ denote the symmetric group on $n$ letters. Let $k \in N$ such that $k \le n$. The number of elements $m$ of $S_n$ which are $k$-cycles is given by: :$m = \paren {k - 1}! \dbinom n k = \dfrac {n!} {k \paren {n - k}!}$
Number of k-Cycles on Set of k Elements
Let $k \in \N$ be a natural number. Let $S_k$ denote the symmetric group on $k$ letters. The number of elements of $S_k$ which are $k$-cycles is $\paren {k - 1}!$.
Zero Padded Basis Representation
Let $b \in \Z: b > 1$. Let $m \in \Z_{> 0}$. For every $n \in \Z_{\ge 0}$ such that $n < b^m$, there exists one and only one sequence $\sequence {r_j}_{0 \mathop \le j \mathop \le m - 1}$ such that: : $(1): \quad \displaystyle n = \sum_{j \mathop = 0}^{m - 1} r_j b^j$ : $(2): \quad \displaystyle \forall j \in \closedint 0 {m - 1}: r_j \in \N_b$
Sum of Elements of Inverse of Matrix with Column of Ones
Let $\mathbf B = \sqbrk b_n$ denote the inverse of a square matrix $\mathbf A$ of order $n$. Let $\mathbf A$ be such that it has a row or column of all ones. Then the sum of elements in $\mathbf B$ is one: :$\displaystyle \sum_{i \mathop  = 1}^n \sum_{j \mathop = 1}^n b_{ij}  = 1$
Coherent Sequence is Partial Sum of P-adic Expansion/Lemma
:$\forall n \in \N: \alpha_n = \displaystyle \sum_{i \mathop = 0}^n b_{i, i} p^i$
Limit to Infinity of Number of p-Groups of Order p^m
Let $p$ be a prime number. Let $m \in \N$ be a natural number Let $\map \nu {p^n}$ denote the $\nu$ function of $p^n$: the number of group types of order $p^m$. Then: :$\map \nu {p^m} = p^{A m^3}$ where: :$\displaystyle \lim_{m \mathop \to \infty} A = \dfrac 2 {27}$
Copeland-Erdős Constant is Normal
The Copeland-Erdős constant whose decimal expansion is formed by concatenating the prime numbers in ascending order: :$0 \cdotp 23571 \, 11317 \, 1923 \ldots$ is normal with respect to base $10$.
Koebe Quarter Theorem
Let $f: \C \to \C$ be a schlicht function, that is, a univalent complex function such that $\map f 0 = 0$ and $\map {f'} 0 = 1$. Then the image of the unit disk contains the closed disk of radius $\dfrac 1 4$. Hence for any $w \in f \sqbrk {\Bbb D}$ we have that $\cmod w \le \dfrac 1 4$. The constant $\dfrac 1 4$ is sharp and so cannot be improved.
Limit to Infinity of Summation of Euler Phi Function over Square
:$\displaystyle \lim_{n \mathop \to \infty} \dfrac {\map \Phi n} {n^2} = \dfrac 3 {\pi^2}$ where: :$\map \Phi n = \displaystyle \sum_{k \mathop = 1}^n \map \phi k$ :$\map \phi k$ is the Euler $\phi$ function of $k$. Numerically, this evaluates to: :$\dfrac 3 {\pi^2} \approx 0 \cdotp 30396 35509 \ldots$ {{OEIS|A104141}}
Blaschke's Theorem
Let $C$ be a closed convex curve. Let the minimum width $w$ of $C$ be such that $w \ge 1$. Then $C$ can contain a circle whose radius $\dfrac 1 3$.
At Least One Third of Zeros of Riemann Zeta Function on Critical Line
At least $\dfrac 1 3$ of the nontrivial zeroes of the Riemann $\zeta$ function lie on the critical line.
Complex Sine Function is Entire
Let $\sin: \C \to \C$ be the complex sine function.  Then $\sin$ is entire.
Summation of Reciprocal of Zero of Order 1 Bessel Function by Order 0 Bessel Function of it
:$\displaystyle \sum_{n \mathop = 1}^\infty \dfrac 1 {x_n \map {J_0 } {x_n} } = 0 \cdotp 38479 \ldots$ where: :$x_n$ is the $n$th zero of the order $1$ Bessel function of the first kind :$\map {J_0 } {x_n}$ is the order $0$ Bessel function of the first kind of $x_n$.
Bloch's Theorem
Let $f: \C \to \C$ be a holomorphic function in the unit disk $\cmod z \le 1$. Let $\cmod {\map {f'} 0} = 1$. Then there exists: :a disk $D$ of radius $B$ :an analytic function $\phi$ in $D$ such that $\map f {\map \phi z} = z$ for all $z$ in $D$ where $B > \dfrac 1 {72}$ is an absolute constant.
Complex Cosine Function is Entire
Let $\cos: \C \to \C$ be the complex cosine function.  Then $\cos$ is entire.
Arccosecant Logarithmic Formulation
Let $x$ be a real number.  Let $x \in \hointl {-\infty} {-1} \cup \hointr 1 {\infty}$. Then:  :$\displaystyle \arccsc x = -i \map \Ln {\sqrt {1 - \frac 1 {x^2} } + \frac i x}$ where: :$\arccsc$ is the arccosecant function :$\Ln$ is the principal branch of the complex logarithm whose imaginary part lies in $\hointl {-\pi} \pi$.
Landau's Theorem
Let $f: \C \to \C$ be a holomorphic function in the unit disk $\cmod z \le 1$. Let $\cmod {\map {f'} 0} = 1$. Then there exists: :a disk $D$ of radius $B$ :an analytic function $\phi$ in $D$ such that $\map f {\map \phi z} = z$ for all $z$ in $D$ such that $L$ is an absolute constant where: :$L > B$ where $B$ is Bloch's constant.
Lower Bound of Bloch's Constant
Bloch's constant has a lower bound as follows: :$\dfrac {\sqrt 3} 4 + \dfrac 2 {10 \, 000} \le B$
Upper Bound of Bloch's Constant
Bloch's constant has an upper bound as follows: :$B \le \sqrt {\dfrac {\sqrt 3 -1} 2} \times \dfrac {\map \Gamma {\frac 1 3} \map \Gamma {\frac {11} {12} } } {\map \Gamma {\frac 1 4} }$
Limit of Difference between Consecutive Prime Numbers
The Prime Number Theorem indicates that the average value of the difference between two consecutive prime numbers is of the order of $\log p_n$. Let $E = \displaystyle \liminf_{n \mathop \to \infty} \dfrac {p_{n + 1} - p_n} {\log p_n}$. If there are infinitely many twin primes, then $E = 0$. If not, then it is not known what the value of $E$ is.
Distribution of Numbers with More than 2 Prime Factors
For sufficiently large $x$, there always exists an integer with more than $2$ prime factors between $\paren {x - x^\alpha}$ and $x$, where: :$\alpha \ge 0 \cdotp 477 \ldots$
All Nontrivial Zeroes of Riemann Zeta Function are on Critical Strip
All of the nontrivial zeroes of the Riemann $\zeta$ function lie on the critical strip.
Nontrivial Zeroes of Riemann Zeta Function are Symmetrical with respect to Critical Line
The nontrivial zeroes of the Riemann $\zeta$ function are distributed symmetrically with respect to the  critical line. That is, suppose $s_1 = \sigma_1 + i t$ is a nontrivial zero of $\zeta$. Then there exists another nontrivial zero $s_2$ of $\zeta$ such that: :$s_2 = 1 - \sigma_1 + i t$
Critical Line Theorem
There exist an infinite number of nontrivial zeroes of the Riemann $\zeta$ function on the critical line.
Prime-Counting Function in terms of Eulerian Logarithmic Integral
Let $\map \pi x$ denote the prime-counting function of a number $x$. Let $\map \Li x$ denote the Eulerian logarithmic integral of $x$: :$\map \Li x := \displaystyle \int_2^x \dfrac {\d t} {\ln t}$ Then: :$\map \pi x = \map \Li x + \map {\mathcal O} {x \, \map \exp {-c \sqrt {\ln x} } }$ where: :$\mathcal O$ is the big-O notation :$c$ is some constant.
Mertens' Third Theorem
:$\displaystyle \lim_{x \mathop \to \infty} \ln x \prod_{\substack {p \mathop \le x \\ \text {$p$ prime} } } \paren {1 - \dfrac 1 p} = e^{-\gamma}$ where $\gamma$ denotes the Euler-Mascheroni constant.
Abi-Khuzam Inequality
Let $\triangle ABC$ be a triangle. Then: :$\sin A \cdot \sin B \cdot \sin C \le k A \cdot B \cdot C$ where: :$A, B, C$ are measured in radians :$k = \paren {\dfrac {3 \sqrt 3} {2 \pi} }^3 \approx 0 \cdotp 56559 \, 56245 \ldots$
Limit to 1 of Zeta of s minus Reciprocal of s-1
:$\displaystyle \lim_{s \mathop \to 1} \paren {\map \zeta s - \dfrac 1 {s - 1} } = \gamma$ where: :$\zeta$ denotes the Riemann $\zeta$ (zeta) function :$\gamma$ denotes the Euler-Mascheroni constant.
Limit to Infinity of x minus Gamma of Reciprocal of x
:$\displaystyle \lim_{x \mathop \to \infty} \paren {x - \map \Gamma {\dfrac 1 x} } = \gamma$ where: :$\Gamma$ denotes the $\Gamma$ (Gamma) function :$\gamma$ denotes the Euler-Mascheroni constant.
Jung's Theorem
Let $S \subseteq \R^n$ be a compact subspace of an $n$-dimensional Euclidean space. Let $d = \displaystyle \max_{x, y \mathop \in S} \map d {x, y}$ be the diameter of $S$. Then there exists a closed ball ${B_r}^-$ with radius $r$ such that: :$r = d \sqrt {\dfrac n {2 \paren {n + 1} } }$ such that $S \subseteq {B_r}^-$.
Jung's Theorem in the Plane
Let $S \subseteq \R^2$ be a compact region in a Euclidean plane. Let $d$ be the diameter of $S$. Then there exists a circle $C$ with radius $r$ such that: :$r = d \dfrac {\sqrt 3} 3$ such that $S \subseteq C$.
Alternating Sum and Difference of Factorials to Infinity
According to {{AuthorRef|Leonhard Paul Euler}}: {{begin-eqn}} {{eqn | l = \sum_{n \mathop = 0}^\infty \paren {-1}^n n!       | r = \int_0^\infty \dfrac {e^{-u} } {1 + u} \rd u       | c =  }} {{eqn | r = G       | c = the Euler-Gompertz constant }} {{eqn | o = \approx       | r = 0 \cdotp 59634 \, 73623 \, 23194 \, 07434 \, 10784 \, 99369 \, 27937 \, 6074 \ldots       | c =  }} {{end-eqn}} {{explain|Clarify meaning of this equality. Using naive manipulations (carelessly swapping integration/sum and using geometric series results) it's quite straightforward to see why someone might think this is true but we should establish the sense in which it is, since the sum on the LHS diverges.<br/>What we need to do is go back to Euler's original statement of this and see what he meant. He did lots of stuff like this, plugging values into formulae that they weren't applicable to, like e.g. $1 + 2 + 3 + \ldots {{=}} \dfrac 1 {12}$, I presume it's like one of those.}}
Arcsine Function in terms of Gaussian Hypergeometric Function
:$\displaystyle \arcsin x = x \, {}_2 \map {F_1} {\frac 1 2, \frac 1 2; \frac 3 2; x^2}$
Mean Distance between Two Random Points in Cuboid
Let $B$ be a cuboid in the Cartesian $3$-space $\R^3$ as: :$\size x \le a$, $\size y \le b$, $\size z \le c$ Let $E$ denote the mean distance $D$ between $2$ points chosen at random from the interior of $B$. Then: {{begin-eqn}} {{eqn | l = E       | r = \dfrac {2 r} {15} - \dfrac 7 {45} \paren {\paren {r - r_1} \paren {\dfrac {r_1} a}^2 + \paren {r - r_2} \paren {\dfrac {r_2} b}^2 + \paren {r - r_3} \paren {\dfrac {r_3} c}^2}       | c =  }} {{eqn | o =       | ro= +       | r = \dfrac 8 {315 a^2 b^2 c^2} \paren {a^7 + b^7 + c^7 - {r_1}^7 - {r_2}^7 - {r_3}^7 + r^7}       | c =  }} {{eqn | o =       | ro= +       | r = \dfrac 1 {15 a b^2 c^2} \paren {b^6 \sinh^{-1} \dfrac a b + c^6 \sinh^{-1} \dfrac a c - {r_1}^2 \paren { {r_1}^4 - 8 b^2 c^2} \sinh^{-1} \dfrac a {r_1} }       | c =  }} {{eqn | o =       | ro= +       | r = \dfrac 1 {15 a^2 b c^2} \paren {c^6 \sinh^{-1} \dfrac b c + a^6 \sinh^{-1} \dfrac b a - {r_2}^2 \paren { {r_2}^4 - 8 c^2 a^2} \sinh^{-1} \dfrac b {r_2} }       | c =  }} {{eqn | o =       | ro= +       | r = \dfrac 1 {15 a^2 b^2 c} \paren {a^6 \sinh^{-1} \dfrac c a + b^6 \sinh^{-1} \dfrac c b - {r_3}^2 \paren { {r_3}^4 - 8 a^2 b^2} \sinh^{-1} \dfrac c {r_3} }       | c =  }} {{eqn | o =       | ro= -       | r = \dfrac 4 {15 a b c} \paren {a^4 \arcsin \dfrac {b c} {r_2 r_3} + b^4 \arcsin \dfrac {a c} {r_3 r_1} + c^4 \arcsin \dfrac {a b} {r_1 r_2} }       | c =  }} {{end-eqn}} where: {{begin-eqn}} {{eqn | l = r       | r = \sqrt {a^2 + b^2 + c^2} }} {{eqn | l = r_1       | r = \sqrt {b^2 + c^2} }} {{eqn | l = r_2       | r = \sqrt {a^2 + c^2} }} {{eqn | l = r_3       | r = \sqrt {a^2 + b^2} }} {{end-eqn}}
Mean Distance between Two Random Points in Unit Cube
The mean distance $R$ between $2$ points chosen at random from the interior of a unit cube is given by: {{begin-eqn}} {{eqn | l = R       | r = \frac {4 + 17 \sqrt 2 - 6 \sqrt3 - 7 \pi} {105} + \frac {\map \ln {1 + \sqrt 2 } } 5 + \frac {2 \, \map \ln {2 + \sqrt 3} } 5 }} {{eqn | o = \approx       | r = 0 \cdotp 66170 \, 71822 \, 67176 \, 23515 \, 582 \ldots       | c =  }} {{end-eqn}} The value $R$ is known as the Robbins constant.
Uniform Matroid is Matroid
Let $S$ be a finite set of cardinality $n$. Let $0 \le k \le n$. Let $U_{k,n} = \struct{S, \mathscr I}$ be the uniform matroid of rank $k$. Then $U_{k,n}$ is a matroid.
Matroid Induced by Linear Independence in Vector Space is Matroid
Let $V$ be a vector space. Let $S$ be a finite subset of $V$. Let $\struct{S, \mathscr I}$ be the matroid induced on $S$ by linear independence in $V$. That is, $\mathscr I$ is the set of linearly independent subsets of $S$. Then $\struct{S, \mathscr I}$ is a matroid.
Matroid Induced by Algebraic Independence is Matroid
Let $L / K$ be a field extension. Let $S \subseteq L$ be a finite subset of $L$. Let $\struct{S, \mathscr I}$ be the matroid induced by algebraic independence over $K$ on $S$. That is, $\mathscr I$ is the set of algebraically independent subsets of $S$. Then $\struct{S, \mathscr I}$ is a matroid.
Matroid Induced by Affine Independence is Matroid
Let $\R^n$ be the $n$-dimensional real Euclidean space. Let $S = \set{x_1, \dots, x_r}$ be a finite subset of $\R^n$. Let $\struct{S, \mathscr I}$ be the matroid induced by affine independence on $S$. That is, $\mathscr I$ is the set of affinely independent subsets of $S$. Then $\struct{S, \mathscr I}$ is a matroid.
Matroid Induced by Linear Independence in Abelian Group is Matroid
Let $\struct{G, +}$ be a torsion-free Abelian group. Let $\struct{G, +, \times}$ be the $\Z$-module associated with $G$. Let $S$ be a finite subset of $G$. Let $\struct{S, \mathscr I}$ be the matroid induced by linear independence in $G$ on $S$. That is, $\mathscr I$ is the set of linearly independent subsets of $S$. Then $\struct{S, \mathscr I}$ is a matroid.
Largest Mutually Coprime Subset of Initial Segment of Natural Numbers
Let $n \in \N$ be a natural number. Consider the set $\N_n$ defined as: :$\N_n = \closedint 1 n = \set {1, 2, \ldots n}$ Let $Q_n$ be the largest subset of $\N_n$ such that no element of $Q_n$ is the divisor of another element of $Q_n$. Let $\map f n$ be the cardinality of $Q_n$. Then for sufficiently large $n$: :$0 \cdotp 6725 \ldots \le \dfrac {\map f n} n \le 0 \cdotp 6736 \ldots$
Independent Set can be Augmented by Larger Independent Set
Let $M = \struct{S, \mathscr I}$ be a matroid. Let $X, Y \in \mathscr I$ such that: :$\size X < \size Y$ Then there exists non-empty $Z \subseteq Y \setminus X$ such that: :$X \cup Z \in \mathscr I$ :$\size {X \cup Z} = \size Y$
All Bases of Matroid have same Cardinality
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\rho: \powerset S \to \Z$ be the rank function of $M$. Let $B$ be a base of $M$. Then: :$\size B = \map \rho S$ That is, all bases of $M$ have the same cardinality, which is the rank of $M$.
Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom
Let $S$ be a finite set. Let $\mathscr B$ be a non-empty set of subsets of $S$. Then $\mathscr B$ is the set of bases of a matroid on $S$  $\mathscr B$ satisfies the base axiom: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1}}
Number of Integer Partitions into Sum of Consecutive Primes
Let $n$ be a natural number. Let $\map f n$ denote the number of integer partitions of $n$ where the parts are consecutive prime numbers. For example: :$\map f {41} = 3$ because: :$41 = 11 + 13 + 17 = 2 + 3 + 5 + 7 + 11 + 13$ Then: :$\displaystyle \lim_{x \mathop \to \infty} \dfrac 1 x \sum_{n \mathop = 1}^x \map f n = \ln 2$
Dixon's Theorem (Group Theory)
Let $P_1$ and $P_2$ be distinct elements of the symmetric group on $n$ letters. The probability that $\set {P_1, P_2}$ forms a generator of $S_n$ approaches $\dfrac 3 4$ as $n$ tends to infinity.
P-adic Integer is Limit of Unique P-adic Expansion
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ be the $p$-adic integers. Let $x \in \Z_p$. Then $x$ is the limit of a unique $p$-adic expansion of the form: :$\displaystyle \sum_{n \mathop = 0}^\infty d_n p^n$
P-adic Number is Limit of Unique P-adic Expansion
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $x \in \Q_p$. Then $x$ is the limit of a unique $p$-adic expansion.
Number of Natural Numbers Less than x which are Squares or Sums of Two Squares
Let $x$ be a real number. The number of natural numbers smaller than $x$ which are either square or the sum of $2$ squares is given by the expression: :$\map N x \approx \dfrac {k x} {\sqrt {\ln x} }$ where $k$ is given by: :$k = \sqrt {\dfrac 1 2 \displaystyle \prod_{\substack {r \mathop = 4 n \mathop + 3 \\ \text {$r$ prime} } } \paren {1 - \dfrac 1 {r^2} }^{-1} }$ The number $k$ is known as the Landau-Ramanujan constant: {{:Definition:Landau-Ramanujan Constant}}
Finite Set Contains Subset of Smaller Cardinality
Let $S$ be a finite sets. Let :$\size S = n$ where $\size {\, \cdot \,}$ denotes cardinality. Let $0 \le m \le n$. Then there exists a subset $X \subseteq S$ such that: :$\size X = m$
Bounds on Number of Odd Terms in Pascal's Triangle
Let $P_n$ be the number of odd elements in the first $n$ rows of Pascal's triangle. Then: :$0 \cdotp 812 \ldots < \dfrac {P_n} {n^{\lg 3} } < 1$ where $\lg 3$ denotes logarithm base $2$ of $3$. The lower bound $0 \cdotp 812 \ldots$ is known as the Stolarsky-Harborth constant.
Integral from 0 to 1 of Complete Elliptic Integral of First Kind
Let $G$ denote Catalan's constant. Then: :$2 G = \displaystyle \int_0^1 \map K k \rd k$ where $\map K k$ denotes the complete elliptic integral of the first kind: :$\map K k = \displaystyle \int \limits_0^{\pi / 2} \dfrac {\d \phi} {\sqrt {1 - k^2 \sin^2 \phi} }$
Trivial Group is Smallest Group
Let $G = \struct {\set e, \circ}$ be a trivial group. Then $G$ is the smallest group possible, in that there exists no set with lower cardinality which is the underlying set of a group.
Definition:Salem Constant
The '''Salem constant''' is the greatest real root of Lehmer's polynomial: :$x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1 = 0$ Its value is approximately: :$1 \cdotp 17628 \, 08182 \, 599 \ldots$ {{OEIS|A073011}}
Partial Sums of P-adic Expansion forms Coherent Sequence
Let $p$ be a prime number. Let $\displaystyle \sum_{n \mathop = 0}^\infty d_n p^n$ be a $p$-adic expansion. Let $\sequence{\alpha_n}$ be the sequence of partial sums;  that is: :$\forall n \in \N :\alpha_n = \displaystyle \sum_{i \mathop = 0}^n d_i p^i$. Then $\sequence{\alpha_n}$ is a coherent sequence.
Gibbs Phenomenon
The Fourier series overshoots at a jump discontinuity, and adding more terms to the sum does not cause this overshoot to die out. :500px :500px {{finish|Anyone care to flesh this out? I've got bored with it.}}
Mergelyan-Wesler Theorem
Let $P = \sequence {D_1, D_2, \dotsc}$ be an infinite sequence of disjoint open disks whose union is the unit disk $D$ except for a set of measure zero. Let $r_n$ be the radius of $D_n$. Then: :$\displaystyle \sum_{k \mathop = 1}^\infty r_k = +\infty$
Definition:Solid-Packing Constant for Circles
Let $P = \sequence {D_1, D_2, \dotsc}$ be an infinite sequence of disjoint open disks whose union is the unit disk $D$ except for a set of measure zero. Let $r_n$ be the radius of $D_n$. Let $x \in \R_{>0}$ be a (strictly) positive real number. Let $\map {M_x} P$ be defined as: :$\map {M_x} P = \displaystyle \sum_{k \mathop = 1}^\infty {r_k}^x$ For each $P$, there exists a (real) number $\map e P$ such that: :$\map {M_x} P$ is divergent for $x < \map e P$ :$\map {M_x} P$ is convergent for $x > \map e P$ From the Mergelyan-Wesler Theorem: :$1 < \map e P < 2$ for all $P$. The constant $S$ such that: :$S < \map e P$ is known as the '''solid-packing constant for circles'''. It can be interpreted as the fractal dimension of the set of points of $P$ which are not covered by the $D_n$ open disks.
Value of Solid-Packing Constant for Circles
The value of the solid-packing constant for circles is estimated as being: :$S \approx 1 \cdotp 306951 \ldots$
Integer Arbitrarily Close to Rational in Valuation Ring of P-adic Norm
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$. Let $x \in \Q$ such that $\norm x_p \le 1$. Then for all $i \in \N$ there exists $\alpha \in \Z$ such that: :$\norm{x - \alpha}_p \le p^{-i}$
Equivalence Class in P-adic Integers Contains Unique Coherent Sequence
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\mathbf a$ be an equivalence class in $\Q_p$ such that $\norm{\mathbf a}_p \le 1$. Then $\mathbf a$ has exactly one representative that is a coherent sequence.
Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/P-adic Expansion
Then $\mathbf a$ has exactly one representative that is a $p$-adic expansion of the form: :$\displaystyle \sum_{n \mathop = 0}^\infty d_n p^n$
Equivalence Class in P-adic Numbers Contains Unique P-adic Expansion
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\mathbf a$ be an equivalence class in $\Q_p$. Then $\mathbf a$ has exactly one representative that is a $p$-adic expansion.
Value of Plastic Constant
The plastic constant $P$ is evaluated as: {{begin-eqn}} {{eqn | l = P       | r = \sqrt [3] {\frac {9 + \sqrt {69} } {18} } + \sqrt [3] {\frac {9 - \sqrt {69} } {18} }       | c =  }} {{eqn | r = 1 \cdotp 32471 \, 79572 \, 44746 \, 02596 \, 09088 \, 54 \ldots       | c =  }} {{end-eqn}}
Plastic Constant is Smallest Pisot-Vijayaraghavan Number
The plastic constant is the smallest Pisot-Vijayaraghavan number.
Unique Integer Close to Rational in Valuation Ring of P-adic Norm
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$. Let $x \in \Q$ such that $\norm{x}_p \le 1$. Then for all $i \in \N$ there exists a unique $\alpha \in \Z$ such that: :$(1): \quad \norm {x - \alpha}_p \le p^{-i}$ :$(2): \quad  0 \le \alpha \le p^i - 1$
Hermite Constant for Dimension 2
The Hermite constant for dimension $2$ is: :$\gamma_2 = \dfrac 2 {\sqrt 3}$ or, as it is often presented: :$\paren {\gamma_2}^2 = \dfrac 4 3$
Upper Bound of Hermite Constant
Let $\gamma_n$ be the Hermite constant of dimension $n$. Then: :$\gamma_n \le \dfrac {\paren {1 + \epsilon_n} n} {\pi e}$ where $e_n \to 0$
Angle between Straight Lines in Plane
Let $L_1$ and $L_2$ be straight lines embedded in a cartesian plane, given by the equations: {{begin-eqn}} {{eqn | ll= L_1:       | l = y       | r = m_1 x + c_1 }} {{eqn | ll= L_2:       | l = y       | r = m_2 x + c_2 }} {{end-eqn}} Then the angle $\psi$ between $L_1$ and $L_2$ is given by: :$\psi = \arctan \dfrac {m_1 - m_2} {1 + m_1 m_2}$
Parallel Straight Lines have Same Slope
Let $L_1$ and $L_2$ be straight lines in the plane. Let $L_1$ and $L_2$ have slopes of $m_1$ and $m_2$ respectively. Then $L_1$ and $L_2$ are parallel  $m_1 = m_2$.
Product of Slopes of Perpendicular Lines is Minus 1
Let $L_1$ and $L_2$ be straight lines in the plane. Let $L_1$ and $L_2$ have slopes of $m_1$ and $m_2$ respectively. Then $L_1$ and $L_2$ are perpendicular  $m_1 m_2 = -1$.
Slope of Tangent to Lemniscate at Origin
Consider the lemniscate of Bernoulli $M$ embedded in a Cartesian plane such that its foci are at $\tuple {a, 0}$ and $\tuple {-a, 0}$ respectively. Let $O$ denote the origin. The tangents to $M$ at $O$ are at an angle of $45 \degrees = \dfrac \pi 4$ to the $x$-axis. :630px
Area of Lobe of Lemniscate of Bernoulli
Consider the lemniscate of Bernoulli $M$ embedded in a Cartesian plane such that its foci are at $\tuple {a, 0}$ and $\tuple {-a, 0}$ respectively. Let $O$ denote the origin. The area of one lobe of $M$ is $a^2$.