+Template + UE4 Fehler behoben + UEvo3

diff --git a/UE/template.sty b/UE/template.sty
new file mode 100644 (file)
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 eda2ccddbaff22ac7010221d55974edb4dea357f..c66c95be49e58b2be36bae7f510fc645199238f2 100644 (file)
Binary files a/UE/ue4.pdf and b/UE/ue4.pdf differ

diff --git a/UE/ue4.tex b/UE/ue4.tex
index fa9c8760ce64daf1ba96a88154efd4f2bac5b82d..8e01f87b5258dcf36c91c6f672f490e5033a4925 100644 (file)
--- 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 (file)
index 0000000..9b2fbd0
--- /dev/null
+++ b/UEvo/UE3.mw
@@ -0,0 +1,223 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<Worksheet>
+<Version major="15" minor="1"/>
+<Label-Scheme value="2" prefix=""/>
+<View-Properties presentation="false"></View-Properties>
+<MapleNet-Properties elisiondigitsbefore="100" labelling="true" indentamount="4" elisiontermsthreshold="10000" ansi="false" errorbreak="1" useclientjvm="true" echo="1" imaginaryunit="I" labelwidth="20" contextmenusize="automatic" plotdriver="opengl" elisiondigitsafter="100" plotoutput="terminal" helpbrowser="standard" rtablesize="10" elisiontermsbefore="100" elisiondigitsthreshold="10000" typesetting="standard" plotdevice="inline" verboseproc="2" showassumed="1" errorcursor="false" longdelim="true" plotoptions="" quiet="false" elisiontermsafter="100" screenwidth="79" preplot="" prettyprint="3" displayprecision="-1" screenpixelheight="1024" warnlevel="3" screenheight="25" latexwidth="6.0" postplot="" prompt="&gt; " ShowLabels="true"/>
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+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(numtheory);</Text-field>
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+</Group>
+<Group labelreference="L2" drawlabel="true">
+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal">interface(verboseproc=2);</Text-field>
+</Input>
+<Output>
+<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">IiIi</Equation></Text-field>
+</Output>
+</Group>
+<Group labelreference="L3" drawlabel="true">
+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal">print(GIgcd);</Text-field>
+</Input>
+<Output>
+<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" 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+</Output>
+</Group>
+<Group labelreference="L4" drawlabel="true">
+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal">fermattest:=proc(n::posint,a::posint:=2)
+  evalb(a&amp;^(n-1) mod n = 1)
+end:</Text-field>
+</Input>
+</Group>
+<Group labelreference="L5" drawlabel="true">
+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal">fermattest(341);</Text-field>
+</Input>
+<Output>
+<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">SSV0cnVlRyUqcHJvdGVjdGVkRw==</Equation></Text-field>
+</Output>
+</Group>
+<Group labelreference="L6" drawlabel="true">
+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ifactor(341);</Text-field>
+</Input>
+<Output>
+<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">KiYtSSFHNiI2IyIjNiIiIi1GJDYjIiNKRig=</Equation></Text-field>
+</Output>
+</Group>
+<Group labelreference="L7" drawlabel="true">
+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
+</Input>
+</Group>
+<Group labelreference="L8" drawlabel="true">
+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal">solovaystrassen:=proc(n::posint,a::posint:=2)
+  local b:=a&amp;^((n-1)/2) mod n;
+  if b^2&lt;&gt;1 then return false end if;
+  if b=-1 then return true end if;
+  evalb(numtheory[jacobi](a,n)=b)
+end:</Text-field>
+</Input>
+</Group>
+<Group labelreference="L9" drawlabel="true">
+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal">solovaystrassen(341);</Text-field>
+</Input>
+<Output>
+<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">SSZmYWxzZUclKnByb3RlY3RlZEc=</Equation></Text-field>
+</Output>
+</Group>
+<Group labelreference="L10" drawlabel="true">
+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal">2&amp;^170 mod 341;</Text-field>
+</Input>
+<Output>
+<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">IiIi</Equation></Text-field>
+</Output>
+</Group>
+<Group labelreference="L11" drawlabel="true">
+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal">jacobi(2,341);</Text-field>
+</Input>
+<Output>
+<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">ISIi</Equation></Text-field>
+</Output>
+</Group>
+<Group labelreference="L12" drawlabel="true">
+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal">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 &quot;Invalid n&quot; 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:      </Text-field>
+</Input>
+</Group>
+<Group labelreference="L13" drawlabel="true">
+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal">millerrabin(2047);</Text-field>
+</Input>
+<Output>
+<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">SSV0cnVlRyUqcHJvdGVjdGVkRw==</Equation></Text-field>
+</Output>
+</Group>
+<Group labelreference="L14" drawlabel="true">
+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal">spsp:=proc(n::posint) not isprime(n) and millerrabin(n) end;</Text-field>
+</Input>
+<Output>
+<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">Zio2IydJIm5HNiJJJ3Bvc2ludEclKnByb3RlY3RlZEdGJkYmRiYzNC1JKGlzcHJpbWVHRiY2IzkkLUksbWlsbGVycmFiaW5HRiZGLUYmRiZGJg==</Equation></Text-field>
+</Output>
+</Group>
+<Group labelreference="L15" drawlabel="true">
+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal">select(spsp,[seq(2*k+1,k=1..5000)]);</Text-field>
+</Input>
+<Output>
+<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NyQiJVo/IiUibyU=</Equation></Text-field>
+</Output>
+</Group>
+<Group labelreference="L16" drawlabel="true">
+<Input>
+<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
+</Input>
+</Group>
+</Worksheet>
\ No newline at end of file

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