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Borishaase (talk | contribs) (Greatest-prime Criterion and Transcendence of Euler's Constant) |
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= Welcome to MWiki = | = Welcome to MWiki = | ||
== Theorems of the month == | == Theorems of the month == | ||
− | === | + | === Cube Theorem === |
− | + | A sum <math>m \in {}^{\omega }{\mathbb{Z}}</math> consists of three cubes for <math>a, b, c, n \in {}^{\omega }{\mathbb{Z}}</math> if and only if | |
− | = | + | <div style="text-align:center;"><math>m=n^3 + (n + a)^3 + (n - b)^3 = 3n^3 + a - b + 6c \ne \pm 4\mod 9</math></div> |
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+ | is true. This implicitly quadratic equation yields the formula to be satisfied by <math>n.\square</math> | ||
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+ | === Fickett's Theorem === | ||
− | + | For any relative positions of two overlapping congruent rectangular <math>n</math>-prisms <math>Q</math> and <math>R</math> with <math>n \in {}^{\omega }\mathbb{N}_{\ge 2}</math>, it can be stated for the exact standard measure <math>\mu</math>, where <math>\mu</math> for <math>n = 2</math> needs to be replaced by the Euclidean path length <math>L</math>, that: | |
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− | + | <div style="text-align:center;"><math>1/(2n - 1) < r := \mu(\partial Q \cap R)/\mu(\partial R \cap Q) < 2n - 1.</math></div> | |
==== Proof: ==== | ==== Proof: ==== | ||
− | + | Since the underlying extremal problem has its maximum for rectangles with the side lengths <math>s</math> and <math>s + 2d0</math>, min <math>r = s/(3s - 2d0) \le r \le</math> max <math>r = (3s - 2d0)/s</math> holds. The proof for <math>n > 2</math> is analogous.<math>\square</math> | |
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== Recommended reading == | == Recommended reading == |
Revision as of 14:14, 2 October 2020
Welcome to MWiki
Theorems of the month
Cube Theorem
A sum [math]\displaystyle{ m \in {}^{\omega }{\mathbb{Z}} }[/math] consists of three cubes for [math]\displaystyle{ a, b, c, n \in {}^{\omega }{\mathbb{Z}} }[/math] if and only if
is true. This implicitly quadratic equation yields the formula to be satisfied by [math]\displaystyle{ n.\square }[/math]
Fickett's Theorem
For any relative positions of two overlapping congruent rectangular [math]\displaystyle{ n }[/math]-prisms [math]\displaystyle{ Q }[/math] and [math]\displaystyle{ R }[/math] with [math]\displaystyle{ n \in {}^{\omega }\mathbb{N}_{\ge 2} }[/math], it can be stated for the exact standard measure [math]\displaystyle{ \mu }[/math], where [math]\displaystyle{ \mu }[/math] for [math]\displaystyle{ n = 2 }[/math] needs to be replaced by the Euclidean path length [math]\displaystyle{ L }[/math], that:
Proof:
Since the underlying extremal problem has its maximum for rectangles with the side lengths [math]\displaystyle{ s }[/math] and [math]\displaystyle{ s + 2d0 }[/math], min [math]\displaystyle{ r = s/(3s - 2d0) \le r \le }[/math] max [math]\displaystyle{ r = (3s - 2d0)/s }[/math] holds. The proof for [math]\displaystyle{ n > 2 }[/math] is analogous.[math]\displaystyle{ \square }[/math]