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(Counter-directional and Archimedes' theorem)
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= Welcome to MWiki =
 
= Welcome to MWiki =
 
== Theorems of the month ==
 
== Theorems of the month ==
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=== Prime number theorem ===
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=== Counter-directional theorem ===
  
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For <math>\pi(x) := |\{p \in {}^{\omega}{\mathbb{P}} : p \le x \in {}^{\omega}{\mathbb{R}}\}|</math>, it holds that <math>\pi(\omega) = \widetilde{{_e}\omega}\omega + \mathcal{O}({_e}\omega{\omega}^{\tilde{2}})</math>.
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If the path <math>\gamma: [a, b[ \, \cap \, C \rightarrow V</math> with <math>C \subseteq \mathbb{R}</math> passes the edges of every <math>n</math>-cube of side length <math>\iota</math> in the <math>n</math>-volume <math>V \subseteq {}^{(\omega)}\mathbb{R}^{n}</math> with <math>n \in \mathbb{N}_{\ge 2}</math> exactly once, where the opposite edges in all two-dimensional faces of every <math>n</math>-cube are traversed in reverse direction, but uniformly, then, for <math>D \subseteq \mathbb{R}^{2}, B \subseteq {V}^{2}, f = ({f}_{1}, ..., {f}_{n}): V \rightarrow {}^{(\omega)}\mathbb{R}^{n}, \gamma(t) = x, \gamma(\curvearrowright t) = \curvearrowright x</math> and <math>{V}_{\curvearrowright } := \{\curvearrowright x \in V: x \in V, \curvearrowright x \ne \curvearrowleft x\}</math>, it holds that
  
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==== Proof: ====
 
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In the sieve of Eratosthenes, the number of prime numbers decreases almost regularly. From intervals of fix length <math>y \in {}^{\omega}{\mathbb{R}_{&gt;0}}, \hat{y}</math> set-2-tuples of prime numbers are formed such that the first interval has the unchanged representative prime number density and the second interval is empty, then the interval with the second most prime number density is followed by the second least one etc.
 
  
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For induction basis <math>n = 2</math> resp. 3, the induction hypothesis is that the first interval contains <math>x_n/{_e}x_n</math> prime numbers for <math>n \in {}^{\omega}{\mathbb{N}_{\ge2}}</math> and arbitrary <math>x_4 \in [2, 4[</math>. Then the induction step from <math>x_n</math> to <math>x_n^2</math> by considering the prime gaps of prime <math>p\# /q + 1</math> for <math>p, q \in {}^{\omega}\mathbb{P}</math> proves that there are <math>\pi(x_n^2) = \pi(x_n) \check{x}_n</math> prime numbers only from <math>\pi(x_n) = x_n/{_e}x_n</math>. The average distance between the prime numbers is <math>{_e}x_n</math> and the maximal <math>x_n^2</math> to <math>x_n</math> behaves like <math>\omega</math> to <math>{\omega}^{\tilde{2}}.\square</math>
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<div style="text-align:center;"><math>\uparrow_{t \in [a,b[ \; \cap \; C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}(t)\downarrow t}=\uparrow_{\begin{smallmatrix} (x,\curvearrowright x) \\ \in V\times {{V}_{\curvearrowright}} \end{smallmatrix}}{f(x)\downarrow x}=\uparrow_{\begin{smallmatrix} t \in [a,b[ \; \cap \; C, \\ \gamma | {\partial{}^{\acute{n}}} V \end{smallmatrix}}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}(t)\downarrow t}.</math></div>
  
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=== Gelfond-Schneider theorem ===
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==== Proof: ====
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If two arbitrary squares are considered with common edge of length <math>\iota</math> included in one plane, then only the edges of <math>V\times{V}_{\curvearrowright}</math> are not passed in both directions for the same function value. They all, and thus the path to be passed, are exactly contained in <math>{\partial}^{\acute{n}}V.\square</math>
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It holds <math>a^b \in {}^{\omega} \mathbb{T}_\mathbb{C}</math> where <math>a, c \in {}^{\omega} \mathbb{A}_\mathbb{C}^{*} \setminus \{1\}, Q :=  {}^{\omega} \mathbb{R} \setminus {}^{\omega} \mathbb{T}_\mathbb{R}</math> and <math>b, \varepsilon \in {}^{\omega} \mathbb{A}_\mathbb{C} \setminus Q</math>.
 
  
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==== Proof: ====
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=== Archimedes' theorem ===
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Where <math>b \in Q</math> puts the minimal polynomial <math>p(a^b) = p(c^q) = 0</math>, assuming <math>a^b = c^{q+\varepsilon}</math> for maximum <math>q \in Q_{>0}</math> leads to the contradiction <math>0 = (p(a^b) - p(c^q)) / (a^b - c^q) = p^\prime(a^b) = p^\prime(c^q) \ne 0.\square</math>
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There exists <math>m \in {}^{\nu}\mathbb{N}</math> such that <math>a < bm</math> if and only if <math>a < b\nu</math> whenever <math>a > b</math> for <math>a, b \in {\mathbb{R}}_{>0}</math>, since <math>\nu = \max {}^{\nu}\mathbb{N}</math> holds.<math>\square</math>
 
== Recommended reading ==
 
== Recommended reading ==
  

Revision as of 19:45, 31 August 2023

Welcome to MWiki

Theorems of the month

Counter-directional theorem

If the path [math]\displaystyle{ \gamma: [a, b[ \, \cap \, C \rightarrow V }[/math] with [math]\displaystyle{ C \subseteq \mathbb{R} }[/math] passes the edges of every [math]\displaystyle{ n }[/math]-cube of side length [math]\displaystyle{ \iota }[/math] in the [math]\displaystyle{ n }[/math]-volume [math]\displaystyle{ V \subseteq {}^{(\omega)}\mathbb{R}^{n} }[/math] with [math]\displaystyle{ n \in \mathbb{N}_{\ge 2} }[/math] exactly once, where the opposite edges in all two-dimensional faces of every [math]\displaystyle{ n }[/math]-cube are traversed in reverse direction, but uniformly, then, for [math]\displaystyle{ D \subseteq \mathbb{R}^{2}, B \subseteq {V}^{2}, f = ({f}_{1}, ..., {f}_{n}): V \rightarrow {}^{(\omega)}\mathbb{R}^{n}, \gamma(t) = x, \gamma(\curvearrowright t) = \curvearrowright x }[/math] and [math]\displaystyle{ {V}_{\curvearrowright } := \{\curvearrowright x \in V: x \in V, \curvearrowright x \ne \curvearrowleft x\} }[/math], it holds that


[math]\displaystyle{ \uparrow_{t \in [a,b[ \; \cap \; C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}(t)\downarrow t}=\uparrow_{\begin{smallmatrix} (x,\curvearrowright x) \\ \in V\times {{V}_{\curvearrowright}} \end{smallmatrix}}{f(x)\downarrow x}=\uparrow_{\begin{smallmatrix} t \in [a,b[ \; \cap \; C, \\ \gamma | {\partial{}^{\acute{n}}} V \end{smallmatrix}}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}(t)\downarrow t}. }[/math]

Proof:

If two arbitrary squares are considered with common edge of length [math]\displaystyle{ \iota }[/math] included in one plane, then only the edges of [math]\displaystyle{ V\times{V}_{\curvearrowright} }[/math] are not passed in both directions for the same function value. They all, and thus the path to be passed, are exactly contained in [math]\displaystyle{ {\partial}^{\acute{n}}V.\square }[/math]

Archimedes' theorem

There exists [math]\displaystyle{ m \in {}^{\nu}\mathbb{N} }[/math] such that [math]\displaystyle{ a \lt bm }[/math] if and only if [math]\displaystyle{ a \lt b\nu }[/math] whenever [math]\displaystyle{ a \gt b }[/math] for [math]\displaystyle{ a, b \in {\mathbb{R}}_{\gt 0} }[/math], since [math]\displaystyle{ \nu = \max {}^{\nu}\mathbb{N} }[/math] holds.[math]\displaystyle{ \square }[/math]

Recommended reading

Nonstandard Mathematics

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