From: Peter Schaefer Date: Wed, 30 May 2012 10:42:11 +0000 (+0200) Subject: +Template + UE4 Fehler behoben + UEvo3 X-Git-Url: https://git.leopard-lacewing.eu/?a=commitdiff_plain;h=8c8c8c299675f4367c593ace0472f069c8eb8d64;p=zahlenTA.git +Template + UE4 Fehler behoben + UEvo3 --- diff --git a/UE/template.sty b/UE/template.sty new file mode 100644 index 0000000..8b37bc3 --- /dev/null +++ b/UE/template.sty @@ -0,0 +1,57 @@ + +\usepackage[utf8x]{inputenc} +\usepackage{amsmath,amssymb,ulsy,amsthm} +\usepackage{fullpage} +%\usepackage{txfonts} +\usepackage[ngerman]{babel} +\usepackage{fixltx2e} %Deutschsprach Bugs +%\usepackage[T1]{fontenc} +%\usepackage{lmodern} + +\usepackage{graphicx} +\usepackage{fancyhdr} +% \usepackage{emaxima} + +\def\P{\mathbb{P}} +\def\N{\mathbb{N}} +\def\R{\mathbb{R}} +\def\Z{\mathbb{Z}} + +\def\oder{\vee} +\def\und{\wedge} + +\def\kgV{\text{kgV}} +\def\ggT{\text{ggT}} +\def\sgn{\text{sgn}} + +\def\datum{\date} + +\pagestyle{fancy} +\fancyhf{} +\lhead{\bf \large Aufgabe \thesubsection} +\chead{\thesection. Übung ZtuA} +\rhead{\datum} +\cfoot{\thepage} +\setlength{\headheight}{4\baselineskip} + +\newcommand{\uebung}[2]{ +\setcounter{section}{#1} +\renewcommand{\datum}{#2} +\addcontentsline{toc}{section}{#1. am #2} +} + +\newcommand{\aufgabe}[2]{ +\clearpage +\begin{center} + \large \bf Aufgabe #1 +\end{center} + +\setcounter{subsection}{#1} +\addcontentsline{toc}{subsection}{Aufgabe #1} +{\noindent\it + #2 +} +\medskip +\newline +\noindent +} diff --git a/UE/ue4.pdf b/UE/ue4.pdf index eda2ccd..c66c95b 100644 Binary files a/UE/ue4.pdf and b/UE/ue4.pdf differ diff --git a/UE/ue4.tex b/UE/ue4.tex index fa9c876..8e01f87 100644 --- a/UE/ue4.tex +++ b/UE/ue4.tex @@ -10,7 +10,7 @@ \uebung{4}{30. Mai 2012} \aufgabe{19} -{Man zeige, dass für ein beliebiges $a \in Z$ und eine beliebiges $n \in N^*$ gilt $n | \varphi(a^n − 1)$ . +{Man zeige, dass für ein beliebiges $a \in Z$ und eine beliebiges $n \in \N^*$ gilt $n | \varphi(a^n − 1)$ . } Es gilt \begin{subequations}\label{eq:0} @@ -106,7 +106,7 @@ Daher wird der Solovay-Strassen-Test nicht bestanden, d.h. $2$ ist Zeuge für di \label{eq:10} a^{85} \equiv 32 \mod 341, a^{2\cdot 85} \equiv 1 \mod 341 \end{equation} -Daher wird der Miller-Rabin-Test bestanden. +Daher wird der Miller-Rabin-Test nicht bestanden, d.h. $2$ ist Zeuge für die Zusammengesetztheit von $341$. \item[B-W] Suche $D \equiv 1 \mod 4$ mit $\left( \frac{D}{n} \right)=-1$, erstmals bei $D=37 \Rightarrow P=7$. Daraus erhält man \begin{equation} \label{eq:11} @@ -136,7 +136,7 @@ Es gilt $n=437=19 \cdot 23$. Daher Da $p \equiv q \equiv 3 \mod 4$ kann man ``leicht'' Wurzeln ziehen: \begin{equation} \label{eq:13} - x^{2} \equiv a \mod w \Rightarrow x:=\pm a^{\frac{p+1}{4}} \mod w \textsl{ sind Lösungen.} + x^{2} \equiv a \mod w \Rightarrow x:=\pm a^{\frac{w+1}{4}} \mod w \textsl{ sind Lösungen, mit } w \in \P \end{equation} Daher \begin{subequations} diff --git a/UEvo/UE3.mw b/UEvo/UE3.mw new file mode 100644 index 0000000..9b2fbd0 --- /dev/null +++ b/UEvo/UE3.mw @@ -0,0 +1,223 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +with(numtheory); + + 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 + + + + +interface(verboseproc=2); + + +IiIi + + + + +print(GIgcd); + + 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 + + + + +fermattest:=proc(n::posint,a::posint:=2) + evalb(a&^(n-1) mod n = 1) +end: + + + + +fermattest(341); + + +SSV0cnVlRyUqcHJvdGVjdGVkRw== + + + + +ifactor(341); + + +KiYtSSFHNiI2IyIjNiIiIi1GJDYjIiNKRig= + + + + + + + + + +solovaystrassen:=proc(n::posint,a::posint:=2) + local b:=a&^((n-1)/2) mod n; + if b^2<>1 then return false end if; + if b=-1 then return true end if; + evalb(numtheory[jacobi](a,n)=b) +end: + + + + +solovaystrassen(341); + + +SSZmYWxzZUclKnByb3RlY3RlZEc= + + + + +2&^170 mod 341; + + +IiIi + + + + +jacobi(2,341); + + +ISIi + + + + +millerrabin:=proc(n::posint,aa::posint:=2) + local a:=aa,i,s:=n-1,t:=0; + if n=1 or n mod 2 =0 then return "Invalid n" end if; + do + s:=s/2; t:=t-1; + if s mod 2 = 1 then break end if + end do; + a:=mods(a^s,n); + if a=1 then return true end if; + for i to t do + if a =-1 then return true end if; + a:=mods(a^2,n) + end do; + false + end: + + + + +millerrabin(2047); + + +SSV0cnVlRyUqcHJvdGVjdGVkRw== + + + + +spsp:=proc(n::posint) not isprime(n) and millerrabin(n) end; + + +Zio2IydJIm5HNiJJJ3Bvc2ludEclKnByb3RlY3RlZEdGJkYmRiYzNC1JKGlzcHJpbWVHRiY2IzkkLUksbWlsbGVycmFiaW5HRiZGLUYmRiZGJg== + + + + +select(spsp,[seq(2*k+1,k=1..5000)]); + + +NyQiJVo/IiUibyU= + + + + + + + + \ No newline at end of file diff --git a/UEvo/UE3.pdf b/UEvo/UE3.pdf new file mode 100644 index 0000000..a1198cb Binary files /dev/null and b/UEvo/UE3.pdf differ