0,1,2,3,…

IN:={Ø,{Ø},{Ø,{Ø}},{Ø,{Ø},{Ø,{Ø}}},…}

Wolstenholme December 28, 2008

Filed under: Primzahlen,Reihen,Zahlentheorie — abi007 @ 12:34 pm

Wilson: \displaystyle p\in\mathbb{P} \Leftrightarrow (p-1)!\equiv -1\pmod{p}\Rightarrow\forall p\in\mathbb{P},n\in\mathbb{N}:
\displaystyle {{np-1}\choose{p-1}} \equiv 1 \pmod{p}

Charles Babbage bewies 1819, dass für jede Primzahl p>2 diese Kongruenz gilt:

\displaystyle{{2p-1}\choose{p-1}} \equiv 1 \pmod{p^2}

Der Mathematiker Joseph Wolstenholme (1829-1891) bewies dann 1862, dass für jede Primzahl p>3 die folgende Kongruenz gilt:

\displaystyle{{2p-1}\choose{p-1}} \equiv 1 \pmod{p^3}

Satz von Wolstenhome.
Für 5\leq p\in\mathbb{P} gilt
\displaystyle \frac{a}{b}=1+\frac12+\frac13+\frac14+...+\frac{1}{p-1} \Rightarrow a|p^2

Der Satz ist äquivalent dazu:
\displaystyle \frac{a}{b}= 1+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\ldots+\frac1{(p-1)^2} \Rightarrow a|p

 

Dirichlet’s Theorem March 25, 2008

Filed under: Zahlentheorie — abi007 @ 11:52 pm

Dirichlet’s Theorem:
Let U_m=(\mathbb{Z}/m\mathbb{Z})^{\times} be the multiplicative group of units modulo m. Suppose k\in U_m, that is, gcd(k,m)=1. Then there exist infinitely many primes p such that p==k(mod m)

http://aux.planetmath.org/files/lec/11/dirichlet.pdf

http://www.secamlocal.ex.ac.uk/people/staff/rjchapma/etc/dirichlet.pdf

 

Seminar Primzahl February 21, 2008

Filed under: Uncategorized,Zahlentheorie — abi007 @ 5:15 pm

Benjamin Fine, Gerhard Rosenberger – Number Theory An Introduction Via The Distribution of Primes

Section 4.2 Chebychev’s Estimate and Some Consequences. (p.136ff)

Version in the book: \displaystyle \frac{Log(2)}{4}\frac{x}{Log{(x)}}<\pi(x)<32Log(2)\frac{x}{Log(x)}

Section 4.6 The Prime Number Theorem – The Elementary Proof (p.180ff)

Jeffrey Stopple – A Primer of Analytic Number Theory

Section 5.2 Chevyshev’s Estimates

Version in the book: \displaystyle \frac12\frac{x}{Log{(x)}}<\pi(x)<2\frac{x}{Log{(x)}}, (x>15)

http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf

http://en.wikipedia.org/wiki/Prime_number_theorem

Sylvester’s Papers: http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath&idno=AAS8085

 

Zeta September 8, 2007

Filed under: Analysis,Funktion,Primzahlen,Reihen,Zahlentheorie — abi007 @ 7:23 am

\displaystyle\zeta(n)=\sum_{k=1}^{\infty}\frac1{k^n}
\displaystyle\zeta(s) = \prod_{p\ \mathrm{prim}}\frac1{1-1/p^s}=\frac1{(1-1/2^s)(1-1/3^s)(1-1/5^s)\cdots}.
\displaystyle \zeta_K(s)=\sum_{I\subset\mathcal{O}_K} (N^K_{\mathbb{Q}}(I))^{-s}= \prod_{\mathfrak{p}}\frac{1}{1-(N^K_{\mathbb{Q}}(\mathfrak{p}))^{-s}}

\displaystyle\frac{1}{\zeta(s)}=\sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}

\displaystyle \zeta(2)=\frac{\pi^2}6

http://www.mathlinks.ro/viewtopic.php?t=159424&start=20

http://www.secamlocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf

————————————————————————————–

\displaystyle\gamma:=\lim_{n\to\infty} \left(H_n-\ln n\right)= \lim_{n\rightarrow \infty}\; \sum_{i=1}^n \left[\frac{1}{i} - \ln \left( 1 + \frac{1}{i} \right) \right]

\displaystyle\gamma= -\int\limits_0^1 \ln(-\ln x)\, \mathrm{d}x

\displaystyle\gamma = -\int\limits_0^\infty e^{-x}\ln x\, \mathrm{d}x

\displaystyle\gamma = -\int\limits_0^\infty (\frac1{1-e^{-x}}-\frac1x)e^{-x} \mathrm{d}x

\displaystyle\gamma = -\int\limits_0^\infty \frac1x(\frac1{1+x}-e^{-x}) \mathrm{d}x

\displaystyle \gamma\;\;=\;\;\int_{0}^{1}\;\;\frac{1}{x+1}\cdot\left(\sum_{k=1}^{\infty}\;x^{2^{k}-1}\right)\;\textbf dx

\displaystyle \gamma\;\;=\;\; \int_{0}^{1}\;\;\frac{x+2}{x^{2}+x+1}\cdot\left(\sum_{k=1}^{\infty}\;x^{3^{k}-1}\right)\;\textbf dx

————————————————————————————–

\displaystyle \gamma = - \Gamma'(1)

\displaystyle \Gamma(z) =\lim_{n \to \infty} \frac{n! \; n^z}{z \; (z+1)\cdots(z+n)}

\displaystyle\Gamma(z)=\frac{e^{-\gamma z}}{z}\prod_{n=1}^{\infty}(1 + \frac{z}{n})^{-1}e^{\frac{z}{n}}

\displaystyle \Gamma(x)=\int_0^\infty t^{x-1}e^{-t}dt

\displaystyle =[-t^{x-1}e^{-t}]_0^\infty+\int_0^\infty(x-1)t^{x-2}e^{-t}dt

\displaystyle =(x-1)\int_0^\infty t^{x-2}e^{-t}dt

\displaystyle =(x-1)\Gamma(x-1).

\displaystyle\Gamma \left( x \right)\Gamma \left( {1 - x} \right) = \pi \csc \pi x

\displaystyle\int_{0}^{\infty} \frac{x^n}{e^x-1}=\zeta(n+1)\Gamma(n+1)

http://en.wikipedia.org/wiki/Planck_law#Appendix

————————————————————————————–

\displaystyle\int_0^{\frac{1}{2}} \; \;\frac{{\tan ^{ - 1} \,x}}{x}\;\;{\mathbf{d}}x = \sum\limits_{n = 0}^{ + \infty } {\frac{{( - 1)^n }}{{(2n + 1)^2 \cdot2^{2n + 1} }}}

\displaystyle{\rm K} = \beta (2) = \sum\limits_{n = 0}^\infty {\frac{{( - 1)^n }}{{(2n + 1)^2 }}} = \frac{1}{{1^2 }} - \frac{1}{{3^2 }} + \frac{1}{{5^2 }} - \frac{1}{{7^2 }} + \cdots

\displaystyle = - \int_0^1 {\frac{{\ln (t)}}{{1 + t^2 }}} {\text{ d}}t

\displaystyle = \int_0^1 {\int_0^1 {\frac{1}{{1 + x^2 y^2 }}} } dxdy = \frac{1}{2}\int_0^1 {\text{K}} (x)\,dx = \int_0^1 {\frac{{\tan ^{ - 1} x}}{x}} dx

 

Eulerzahl August 22, 2007

Filed under: Zahlentheorie — abi007 @ 12:40 pm

Taylor:

\displaystyle \boxed{e^x = 1 + \frac{x}{1} + \frac{x^2}{2!} + \frac {x^3}{3!} + \cdots }

Euler:

\displaystyle \boxed {e\ =\ \lim_{n\rightarrow\infty}\ \left(1+\frac{1}{n}\right)^{n}}

\displaystyle \boxed{e\;=\;\lim_{n\to\infty}\;\frac{n}{\sqrt[n]{\;n!\;}}}

\displaystyle \boxed {e^a\ =\ \lim_{n\rightarrow\infty}\ \left(1+\frac{a}{n}\right)^{n}}

Beweis.

Stirling:

\displaystyle n! \sim \sqrt{2\pi n}\, \frac{n^n}{e^n}

\displaystyle \lim_{n \rightarrow \infty} {n!\over \sqrt{2 \pi n} \; \left(\frac{n}{e}\right)^{n} } = 1

\displaystyle \lim_{n\rightarrow \infty}\frac{n}{\sqrt[n] n!}= \lim_{n \rightarrow \infty}\frac{n}{\sqrt{2 \pi n}^{\frac{1}{n}}\left( \frac{n}{e}\right)}= \lim_{n\rightarrow \infty}\frac{e}{\sqrt{2 \pi n}^{\frac{1}{n}}}=e

 

Prime June 22, 2007

Filed under: Primzahlen — abi007 @ 1:15 pm
  • \displaystyle p\in\mathbb{P}\rightarrow p\equiv\pm 1 (mod 6)
  • \displaystyle p\in\mathbb{P},p\equiv 3 (mod 4)\rightarrow

\displaystyle \sum_{x=1}^{p-1}x(\frac{x}{p})<0

\displaystyle x^2+a\nmid p

\displaystyle (1-2(2|p))\sum_{r=1}^{\frac{p-1}{2}}r(r|p)=p\frac{1-(2|p)}{2}\sum_{r=1}^{p}(r|p)

\displaystyle ((2|p)-2)\sum_{r=1}^{p-1}r(r|p)=p\sum_{r=1}^{\frac{p-1}{2}}(r|p)

欧拉定理说:

\displaystyle a^\varphi (m)\equiv 1 (mod m)

如果 m 是素数，那么

\displaystyle \varphi (m)=m-1

立即得到:

\displaystyle a^{m-1}\equiv 1 (mod m)

如果n不是素数，也一样,比如说:

计算 \displaystyle 7^m (mod 15)

\displaystyle 7^{\varphi(15)}=7^8\equiv 1 (mod 15)

然后因为:

\displaystyle 7^1\equiv 7 (mod 15)

你也可以算出所有的

\displaystyle 7^m(mod 15)

至于它的证明，也就只不过是欧拉定理罢了。

 

Theoreme June 21, 2007

Filed under: Diophantine,Zahlentheorie — abi007 @ 9:09 pm
  • Catalan’sche Vermutung:

\displaystyle x^p-y^q=1, x,y,p,q>1 \rightarrow 3^2-2^3=1

  • Fermat’s Little Theorem:

\displaystyle a\in\mathbb{Z}, p\in\mathbb{P}\rightarrow
\displaystyle a^p \equiv a \pmod{p}\,\!

  • Prime number theorem:

\displaystyle \pi(x)\sim\frac{x}{\ln x}

\displaystyle Li(n)=\int_2^n\frac{dx}{ln(x)}\sim\pi(n)

  • Bertrand-Chebyshev theorem

\displaystyle \forall n>1\exists p\in\mathbb{P}:n<p<2n

  • Quadratic Reciprocity Law

\displaystyle (\frac{p}{q})(\frac{q}{p})=(-1)^{\frac{(p-1)(q-1)}{4}}

  • Sophie Germain:

\displaystyle a^4+4b^4=(a^2+2b^2+2ab)(a^2+2b^2-2ab)

 

Möbius June 21, 2007

Filed under: Zahlentheorie — abi007 @ 4:14 pm

\displaystyle\sum\limits_{d\left| n \right.} {\mu (d)\varphi (d) = \prod\limits_{p\left| n \right.} {(2 - p)} },n \in \mathbb{N}

\displaystyle\varphi_1(n)=n\sum\limits_{d\left| n \right.} \frac{|\mu(d)|}{d}\Longrightarrow \varphi_1(n)=\sum\limits_{d^2\left| n \right.}\mu(d)\sigma(\frac{n}{d^2})

\displaystyle S(x)=\sum\limits_{d\left.\leq x \right.}\sigma(k) \Longrightarrow \sum\limits_{n\left.\leq x \right.}\varphi_1(n)=\sum\limits_{d\left.\leq \sqrt{x} \right.} \mu(d)S(\frac{x}{d^2})

\displaystyle\mu(n):=\begin{cases}1 & \mbox{wenn } n=1 \\ (-1)^k & \mbox{wenn } n \mbox{ quadratfrei, } k \mbox{ ist die Anzahl der Primfaktoren} \\ 0 & \mbox{sonst} \end{cases}

 

 
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