From 24ce93d7b7bb5f31a90e857ebe17cb5f270a38ef Mon Sep 17 00:00:00 2001 From: Peter Schaefer Date: Sat, 12 May 2012 19:29:36 +0200 Subject: [PATCH] +design template UE4 template Beispiel --- UE/ue1.pdf | Bin 95941 -> 87979 bytes UE/ue1.tex | 30 +++----- UE/ue2.pdf | Bin 178454 -> 171411 bytes UE/ue2.tex | 207 +++++++++++++++++++++----------------------------- UE/ue3.pdf | Bin 97751 -> 191576 bytes UE/ue3.tex | 174 +++++++++++++++++++----------------------- UE/ue4.tex | 26 +++++++ Vorlesung.pdf | Bin 151645 -> 151645 bytes 8 files changed, 201 insertions(+), 236 deletions(-) create mode 100644 UE/ue4.tex diff --git a/UE/ue1.pdf b/UE/ue1.pdf index 1552c27ec3bb2bd9d9150418f3478a2511062911..865d7798c28c35a8eeb9370f17406abb5d56a9e7 100644 GIT binary patch delta 29647 zcmZsh18m^W*Y0a>ZQC}sZnw5=+it)0*0!~^yKT32x3+ED?*097^S(E^lg#AFBy(mm z$!DHP&KbObtgMBMCxu{PG)L_5xsW@jG_JXWX4or-cS%K|@xqZnonfl> zw-1OV#(L`muvx%{S#Qt0&U5V(I^`j7kfDm~8=g<=Pt_JWc(?cgIenB9^1jn|ZOEa# 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Aufgabe} +\uebung{2}{21. März 2012} + +\aufgabe{1}{} \begin{align} S_n &= \frac 1 1 + \frac 1 2 + \cdots + \frac 1 n \in \Z\\ n &= 1 ;\\ & \Rightarrow S_1 \in \Z\\ @@ -31,7 +19,7 @@ S_n &= \frac{\frac a 1 + \frac a 2+ \cdots \frac a n} a\\ &= \frac{ \overbrace{\frac a 1 + \frac a 2 + \cdots}^{\text{gerade}} + \overbrace{\frac a {2^k}}^{\text{ungerade}} + \overbrace{\cdots + \frac a n}^{\text{gerade}}} {\underbrace{a}_{\text{gerade}}} = \frac{\text{ungerade}}{\text{gerade}} \Rightarrow S_n \notin \Z,n \in \{2,3,\dots\} \end{align} -\subsection*{3. Aufgabe} +\aufgabe{3}{} \begin{align} &\begin{array}{ccccccc} r_{i-2} & r_{i-1}& q_i & x_{i-2} & x_{i-1} & y_{i-2} & y_{i-1}\\ @@ -47,7 +35,7 @@ S_n &= \frac{\frac a 1 + \frac a 2+ \cdots \frac a n} a\\ &= x & = 5\cdot\binom{-25}{29}+\cdot \end{align} -\subsection*{4. Aufgabe} +\aufgabe{4}{} \begin{align} v &= \kgV(a,b) = \frac{|ab|}{\ggT(a,b)}\\ &= a\cdot \frac{\sgn(a) |b|}{\ggT(a,b)} = b\cdot \frac{\sgn(b) |a|}{\ggT(a,b)}\\ diff --git a/UE/ue2.pdf b/UE/ue2.pdf index 36bc36433c17021976b0c7298f5f506c4ecf4403..b56c754cf580451db776180ea14183043b1910b0 100644 GIT binary patch delta 92133 zcmb5VV{m4Hwyqo7w)G|H*tR;hZL8zti*4IUr#rT7+qT`YZm+f1I(waS_pMX+$E+GP zYS#QYpLe|P7^!z~N$qg)ET94K9H8aFfrbPU_YaDU?Sv%qvP zEb*Zs+uEUr)ot1R8d5tTeNGLo=eF(fyz}mCmu4MvmPV&?%7i6N$`WiooSSA9 z$xvf`Wg_}uFRY;!Uck4H5t?5-na2(^zj-AmOFxGCxC~a8l3LNC!(p8LI7yqmL;)#f zO9e|-3I!!|RE(`5`)yPluj|!rHM6!qR1huNGjvtPDtTXynRo?89rp@~p=B$*?Pzp-XKh zS|s>0s9=l*l1NHWha)*CmS6zHIvXzt0{}kQS?Z@?AyN#pn?X$!8fu^uFF}2UHJ>WX zCx<3ryoQ@&sDVP=6YL?~IpMvAe#~l2wyq|Q>hK3+X|`&uo(BiCf5Zg2n8+&ui`J4~ zRtmX*qTwh~IK3H<*7L8h#d-5~XIZfB--z$O`QvB;*n>lULZSEIQRIKu&w}2gr9xpe z3@jjvY2*q%v*^R~1F-{AWXi}A-a&`~!YRuUD*_xJsqIJP!MLz&IZj#9AL24_X%aXB z3KIKxs-3%C0HkOEA5YVO@gzCK2NGk%;lv=Yh!iF}vEh8OAy?{T)MQ(Y{Dp$PVc))2 z0VDIWK||CwG5rE5G2gz6R|f;@q}d9+gmK@%K=^=;_DGY8isgRa5up1x8lI`*Z+vOai3P2TneXHse`rt5jO9`kwlaJ_ zB-*;xT@+6gPACLoFTYuqorRaPSUbz!kq-~&m`03uP?(8F6NZchiFw;yiQ0^WBdHFy zjF?qijhx6K4Qbg@pW%%)Q=o$T7DQ3c9-~~9u{y2dL)};Gv#l~6yt8*|M@)PgGU#D7f4d`h1rv#vef|<<-vhcagcA&*08x-g`tF# z96leB-97XtpFKdij9ZXJ4{i0(&>izm>%f8VTe*v^ZS%XbGP3n|#eVTSQ0*;eJgGfk??+&^e=f*!|Pd*4?&c8 z7Zj8dK8Vhxl=d&0Dv=$L^K1JybB{d1Nfn@2Pm&`gdO~$C!#lE=NNhO@AJ3xRH_*h0yF+AMHJ*xGbI|jp3cQ$_z4|<*K7}$E z`B>`asjNh(6v*4<-a2-2+Huiwb%qQMxw?$h6eBv~vjS>7A9o%V@m&4l;o`wO*FOE^ zt;#q?I4+P6c>mrgO=p*g7FCvL6EyjCH8iI5y7fgOXp*;VOgR;Fv}7S#6$aDaR^C($c>;81b0MCtLe#AlHgA)<>27?M& za0|~=;s`Iz?pH%o-9SNLwa%T_L351m3Tq_G!t!q0hQ^xTYJYb#{A+!+r__@Mr#gog9{ICjr(7}y8j{0lsE zokZifwWx@ii+1mWe&=geP!-~mX9U;B2St}Xa+jn!{zIN6Ep{sf-0zTvNFUpbbvvl9 zchgRvK8v~V_x<7Kot7q%s%sMU;Z>cU-h01)F1#ql^}RW(E+|A;mv0n-uqmm8Ga$UN z&ZZ$Y(`K3B-GC6~&0h&~9B3y02sn)uLLblC>4GI58JISi`GrjSeAup~uN$3Ex_kv5 z;H{(!QP_%mY|@WnncsR-4=ER?Nd=53?Mow^4Jha$dI+oUmlHl2&0B{9R8L+Z|`o8`jtb`vpZ zwU69rss%iF5t|8qU0{JX(+0|v2RiyQz@(4Xn38^dVJRTY%WGhZz(&!kT@pS{yeo<3 z)N0gnpGO-dvT*(81;qSSPA1J%J^eJ()%kob>iZa`Itl8)8d|A)OsgnwiRn({1!C3- zm;^;7HzShv@wX$Ngs$*ebgtTbq|pN@`OVZ9`g9Tu9Ql_lL*Vp`tvGGC{DWfZECzh4 zW0zI?8H_f<;H#pxQdhXuq|>1yeO}QXzLA=0dW~~9+vaFim3GEruYl)|bE!S)T7bNM zrMg6egp|V$AXp4Zw`s2-8UCWWjP-x}`YPH{K& zWqCM_U>h5|@L%Cv;1>8dBf>^Mtc|H4aIxjN-7DAWVe9vcie_JJmE)NVNC>Lw=&F&ravp&-_gwv zQT@$^p)N2udKwlTz<(Iy_rTibD>#5ZQWBlmm9qQej|1(cybs@0uqj1>k*5JNE=;Wp zS?$e*_I;Wqbuutu74=IP+I`xnXNrPX++jDT8mRXMkJ%C%4{bOwG9_=PPIl)pc9 zsCCnCgZ!CoVUj&x(j9}~RLPN>R|y$Gi_}tmWjJ0Nxy`doQJ=#rMh5;WZ8X<0Z@s;uN(G|W$%U9|?cvZ~`Qd_DlWn!) zo>2%9-#lF^Rk06U@%6i6G2hd}4m+<4zUot_w(;Gyc`V(}YpJTZBf*L8!hN&R#LDG~ zp*!tpnyWg2(Yz?Q-K9=6dRzOu<^gKjuAx z@pn6-Z{$Nl|2TO*7cnK(D0l^HU4(%rxn#Wp30a91;f)S>
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-\usepackage[utf8x]{inputenc} -\usepackage{amsmath,amssymb,ulsy} -\usepackage{fullpage} -\usepackage[ngerman]{babel} -\usepackage{fixltx2e} %Deutschsprach Bugs -%\usepackage[T1]{fontenc} -%\usepackage{lmodern} -\usepackage{amsthm} -\usepackage{graphicx} -\usepackage{fancyhdr} -\usepackage{color} -%\usepackage{ngerman} +\usepackage{template} +\begin{document} -\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}} -%opening -\title{} -\author{} +% \thispagestyle{plain} +% \tableofcontents -\begin{document} -\maketitle -\section*{2.Übung} -\subsection*{7. Aufgabe} -{\texttt{Man zeige, durch geschicktes Rechnen mit Kongruenzen und unter Berücksichtigung der beiden Darstellungen +\uebung{2}{25. April 2012} +\aufgabe{7}{Man zeige, durch geschicktes Rechnen mit Kongruenzen und unter Berücksichtigung der beiden Darstellungen \begin{equation}\label{darst641} 641=2^{7} \cdot 5 + 1= 5^{4} + 2^{4} \end{equation} -von $641$, dass $641$ ein Teiler der Fermatzahl $F_{5}=2^{2^{5}}+1$ ist. }}\\ -Aus der Gleichung \eqref{darst641} erhält man durch Umformen sofort -\begin{subequations} +von $641$, dass $641$ ein Teiler der Fermatzahl $F_{5}=2^{2^{5}}+1$ ist. } + + Aus der Gleichung \eqref{darst641} erhält man durch Umformen sofort + \begin{subequations} + \begin{align} + 2^{7} \cdot 5 \equiv -1 \mod 641 \\ + \stackrel{\textsl{zur 4. Potenz}} \implies 2^{28}5^{4} \equiv 1 \mod 641 \label{darst1} + \end{align} + \end{subequations} + Weiters erhält man durch Umformen in \eqref{darst641}: + \begin{equation}\label{darst2} + 5^{4} \equiv -2^{4} \mod 641 + \end{equation} + Setzt man nun \eqref{darst2} in \eqref{darst1} ein, so erhält man zunächst + \begin{equation}\label{gl1} + -2^{4}2^{28} = -2^{32} \equiv 1 \mod 641 + \end{equation} + Unter Beachtung von + \begin{equation}\label{gl0} + 1-F_{5}=1-\left( 2^{2^{5}}+1 \right) = 1- 2^{32} - 1 = -2^{32} + \end{equation} + erhält man nun durch Einsetzen der linken Seite von \eqref{gl0} in \eqref{gl1} das Folgende + \begin{subequations} \begin{align} - 2^{7} \cdot 5 \equiv -1 \mod 641 \\ -\stackrel{\textsl{zur 4. Potenz}} \implies 2^{28}5^{4} \equiv 1 \mod 641 \label{darst1} + 1-F_{5} \equiv 1 \mod 641\\ + \implies -F_{5} \equiv 0 \mod 641\\ + \implies 641 \mid F_{5} \end{align} -\end{subequations} -Weiters erhält man durch Umformen in \eqref{darst641}: -\begin{equation}\label{darst2} - 5^{4} \equiv -2^{4} \mod 641 -\end{equation} -Setzt man nun \eqref{darst2} in \eqref{darst1} ein, so erhält man zunächst -\begin{equation}\label{gl1} - -2^{4}2^{28} = -2^{32} \equiv 1 \mod 641 -\end{equation} -Unter Beachtung von -\begin{equation}\label{gl0} - 1-F_{5}=1-\left( 2^{2^{5}}+1 \right) = 1- 2^{32} - 1 = -2^{32} -\end{equation} -erhält man nun durch Einsetzen der linken Seite von \eqref{gl0} in \eqref{gl1} das Folgende -\begin{subequations} -\begin{align} - 1-F_{5} \equiv 1 \mod 641\\ -\implies -F_{5} \equiv 0 \mod 641\\ -\implies 641 \mid F_{5} -\end{align} -\end{subequations} -\subsection*{8. Aufgabe} -{\texttt{Man löse das folgende System von linearen Kongruenzen + \end{subequations} +\aufgabe{8} +{Man löse das folgende System von linearen Kongruenzen \begin{align} x \equiv 2 \mod 3, x \equiv 3 \mod 5, x \equiv 4 \mod 7 \end{align} einmal mit Hilfe der Formel aus dem Chinesischen-Restsatz und einmal, indem man die allgemeine Lösung der ersten Konrguenz in die zweite einsetzt, und dann die allgemeine Lösung der ersten zwei Konrguenzen in die dritte. -}}\\ -System von Gleichungen mit Teilerfremden $m_i$ -\begin{align} - x & \equiv a_i\mod m_i\\ - x & \equiv 2 \mod 3\\ - x & \equiv 3 \mod 5\\ - x & \equiv 4 \mod 7 -\end{align} -Sei $M = \prod_i m_i$ -\begin{align} - x & \equiv u \mod M \\ - x & \equiv u \mod 105 -\end{align} -Sei nun weiterhin $M_i = M / m_i$, dann können $s_i$ und $r_i$ mittels erweitertem Euklid bestimmt werden -\begin{align} -r_i \cdot m_i + s_i \cdot M_i &= 1\\ --12 \cdot 3 + 2 \cdot 35 &= 1\\ --4 \cdot 5 + 1 \cdot 21 &= 1\\ --2 \cdot 7 + 1 \cdot 15 &= 1 -\end{align} -Eine Lösung ist also -\begin{align} - x &= \sum_i a_i \cdot s_i \cdot M_i\\ - x &= 2 \cdot 2 \cdot 35 + 3 \cdot 21 + 4 \cdot 15\\ - x &= 263 -\end{align} -Alle Lösungen sind also: -\begin{align} - x &\equiv u \mod M\\ - 263 &\equiv u \mod 105\\ - x &\equiv 53 \mod 105 -\end{align} -Lösen durch einsetzen: -\begin{align} - x & \equiv 2 \mod 3 &&\Rightarrow x = 3k +2\\ - 3k +2 & \equiv 3 \mod 5 \\ - 3k &\equiv 1 \mod 5&&\Rightarrow k = 5i +2 &&\Rightarrow x = 15i +8\\ - 15i +8 & \equiv 4 \mod 7 \\ - 15i & \equiv 3 \mod 7 &&\Rightarrow i = 7j+3 &&\Rightarrow x = 105j +53 -\end{align} +} + System von Gleichungen mit Teilerfremden $m_i$ + \begin{align} + x & \equiv a_i\mod m_i\\ + x & \equiv 2 \mod 3\\ + x & \equiv 3 \mod 5\\ + x & \equiv 4 \mod 7 + \end{align} + Sei $M = \prod_i m_i$ + \begin{align} + x & \equiv u \mod M \\ + x & \equiv u \mod 105 + \end{align} + Sei nun weiterhin $M_i = M / m_i$, dann können $s_i$ und $r_i$ mittels erweitertem Euklid bestimmt werden + \begin{align} + r_i \cdot m_i + s_i \cdot M_i &= 1\\ + -12 \cdot 3 + 2 \cdot 35 &= 1\\ + -4 \cdot 5 + 1 \cdot 21 &= 1\\ + -2 \cdot 7 + 1 \cdot 15 &= 1 + \end{align} + Eine Lösung ist also + \begin{align} + x &= \sum_i a_i \cdot s_i \cdot M_i\\ + x &= 2 \cdot 2 \cdot 35 + 3 \cdot 21 + 4 \cdot 15\\ + x &= 263 + \end{align} + Alle Lösungen sind also: + \begin{align} + x &\equiv u \mod M\\ + 263 &\equiv u \mod 105\\ + x &\equiv 53 \mod 105 + \end{align} + Lösen durch einsetzen: + \begin{align} + x & \equiv 2 \mod 3 &&\Rightarrow x = 3k +2\\ + 3k +2 & \equiv 3 \mod 5 \\ + 3k &\equiv 1 \mod 5&&\Rightarrow k = 5i +2 &&\Rightarrow x = 15i +8\\ + 15i +8 & \equiv 4 \mod 7 \\ + 15i & \equiv 3 \mod 7 &&\Rightarrow i = 7j+3 &&\Rightarrow x = 105j +53 + \end{align} -\subsection*{9. Aufgabe} -\texttt{Man bestimme alle Lösungen der Kongruenz $f(x)=2x^{3}-3x^{2}+5x+6 \equiv 0 \mod 120$ durch Zurückführen auf die entsprechenden Lösungen mod 8, mod 3 und mod 5 und Anwendung des Chinesischen Restsatzes. }\\ +\aufgabe{9} +{Man bestimme alle Lösungen der Kongruenz $f(x)=2x^{3}-3x^{2}+5x+6 \equiv 0 \mod 120$ durch Zurückführen auf die entsprechenden Lösungen mod 8, mod 3 und mod 5 und Anwendung des Chinesischen Restsatzes. } \begin{align} f(x) &= 2x^3-3x^2+5x+6 \equiv 0 \mod 120 @@ -143,12 +120,15 @@ So ergeben sich die Lösungen: x &= 0 \cdot 40 + 2 \cdot -24 + 3 \cdot -15 & x &= -93 \mod 120 & x &= 27 \mod 120\\ x &= 0 \cdot 40 + 2 \cdot -24 + 6 \cdot -15 & x &= -138 \mod 120 & x &= 102 \mod 120 \end{align} -\subsection*{10. Aufgabe} -{\texttt{Man zeige, dass für jedes $n \in \N^{*}$ gilt + + +\aufgabe{10} +{Man zeige, dass für jedes $n \in \N^{*}$ gilt \begin{equation} \sum \limits_{d \mid n} \varphi(d)=n \end{equation} -(Hinweis: Klasseneinteilung) }} \\ +(Hinweis: Klasseneinteilung) } + Angenommen, man erhält eine Klasseneinteilung $\implies$ dann sind alle Klassen nichtleer und paarweise disjunkt. Kann man nun für jede Klasse zeigen, dass $\vert C_{d} \vert = \varphi(d)$, so ist man fertig. \\ Auf der Menge $N := \lbrace 1,2,3,\ldots, n \rbrace$ definiere eine binäre Relation $\sim$ folgendermaßen: \begin{equation} @@ -171,8 +151,9 @@ Die Anzahl der Elemente von $A_{d}$ ist aber $\varphi(\frac{n}{d})$, daher gilt \lbrace d : d \mid n \rbrace = \lbrace \frac{n}{d} : d \mid n \rbrace, \end{equation} womit die Summe über die gleichen Indizes gebildet wird. -\subsection*{11. Aufgabe} -{\texttt{Sei $p$ eine Primzahl und für jeden positiven Teiler $d$ von $p-1$ sei $A_{d}$ die Menge derjenigen Elemente in $\lbrace 1,2,\ldots, p-1 \rbrace$ mit der Ordnung $d$. Wieviele Elemente hat ein $A_{d}$ unter der Voraussetzung, dass es nichtleer ist? Warum folgt daraus mit Hilfe von Aufgabe $10$, dass keine der Mengen $A_{d}$ leer sein kann, insbesondere also $A_{p-1}$ nicht, d.h. es gibt eine Primitivwurzel mod $p$?}} \\ +\aufgabe{11} +{Sei $p$ eine Primzahl und für jeden positiven Teiler $d$ von $p-1$ sei $A_{d}$ die Menge derjenigen Elemente in $\lbrace 1,2,\ldots, p-1 \rbrace$ mit der Ordnung $d$. Wieviele Elemente hat ein $A_{d}$ unter der Voraussetzung, dass es nichtleer ist? Warum folgt daraus mit Hilfe von Aufgabe $10$, dass keine der Mengen $A_{d}$ leer sein kann, insbesondere also $A_{p-1}$ nicht, d.h. es gibt eine Primitivwurzel mod $p$?} + Es gilt folgende Eigenschaft der Ordnung eines Elements $x$: Angenommen die Menge $A_{d}$ sei nichtleer, und gelte $\mu \in A_{d}$. Dann hat gilt klarerweise $ord(\mu)=d$, d.h. $\mu$ löst \begin{equation}\label{poly} @@ -195,21 +176,9 @@ insbesondere erhält man daraus für die Mächtigkeiten \varphi(d) = \vert A_{d} \vert \end{equation} -\subsection*{12. Aufgabe} -Ansatz von der Tafel: -\begin{align} - \varphi(p^eq^f) = (p^e-p^{e-1})(q^f-q^{f-1}) -\end{align} -Lösgunen für $k=1,2,3,4$: -\begin{enumerate} - \item $\Z_1,\Z_2$ - \item $\Z_3, \Z_4, \Z_6$ - \item $\emptyset$ - \item $\Z_5,\Z_8,\Z_{10},\Z_{12}$ -\end{enumerate} +\aufgabe{12} +{Man zeige, dass die Gleichung $\varphi(x)=k$ für jede positive ganze Zahl nur endlich viele Lösungen haben kann und bestimme diese explizit für $k=1,2,3,4$. } -\subsection*{Bsp 12} -{\texttt{Man zeige, dass die Gleichung $\varphi(x)=k$ für jede positive ganze Zahl nur endlich viele Lösungen haben kann und bestimme diese explizit für $k=1,2,3,4$. }} \newline Man kann die $\varphi$-Funktion für ein $n = \prod \limits_{p \in \P,\nu_{p}(n) \neq 0} p^{\nu_{p}(n)} \in \N$ folgendermaßen berechnen: \begin{equation}\label{phifkt} \varphi(n)=\varphi(\prod \limits_{p \in \P} p^{\nu_{p}(n)})=\prod \limits_{p \in \P,\nu_{p}(n) \neq 0} p^{(\nu_{p}c(n)-1)} \cdot (p-1) diff --git a/UE/ue3.pdf b/UE/ue3.pdf index 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Übung ZtuA} -\rhead{Mi, 9. Mai 2012} +\usepackage{template} -\def\P{\mathbb{P}} -\def\N{\mathbb{N}} -\def\R{\mathbb{R}} -\def\Z{\mathbb{Z}} -\def\oder{\vee} -\def\und{\wedge} +\begin{document} -\def\kgV{\text{kgV}} -\def\ggT{\text{ggT}} -\def\sgn{\text{sgn}} -%opening -\title{$3$. Übung ZtuA} -\author{} +% \thispagestyle{plain} +% \tableofcontents -\begin{document} +\uebung{3}{9. Mai 2012} -%\section*{$3$. Übung} -\subsection*{$13$. Aufgabe} -{\texttt{Man beweise, dass es je unendlich viele Primzahlen der Form a) 4k+3 und b) 4k+1 gibt. (Hinweis: Man verwende dazu jeweils eine geeignete Variante des klassischen Beweises von Euklid über die Unendlichkeit der Menge der Primzahlen, wobei speziell für den Beweisteil b) der erste Ergänzungssatz benötigt wird.)} \newline - \begin{enumerate} - \item[(a)] Angenommen, die Menge aller Primzahlen der Form $4k+3$, d.h. $p_{1}, \ldots, p_{n}$ sind alle. $7 \equiv 3 \mod 4$, daher ist diese Menge nichtleer. Definiere +\aufgabe{13}{Man beweise, dass es je unendlich viele Primzahlen der Form a) 4k+3 und b) 4k+1 gibt. (Hinweis: Man verwende dazu jeweils eine geeignete Variante des klassischen Beweises von Euklid über die Unendlichkeit der Menge der Primzahlen, wobei speziell für den Beweisteil b) der erste Ergänzungssatz benötigt wird.)} + \begin{enumerate} + \item[(a)] Angenommen, die Menge aller Primzahlen der Form $4k+3$, d.h. $p_{1}, \ldots, p_{n}$ sind alle. $7 \equiv 3 \mod 4$, daher ist diese Menge nichtleer. Definiere + \begin{equation} + m:=4p_{1} \ldots p_{n} - 1 \equiv 3 \mod 4 + \end{equation} + insbesondere ist $m$ ungerade. Nun gilt + \begin{equation} + \forall i: p_{i} < 2 p_{i} < 3p_{i} - 1 < 4p_{1} \ldots p_{n} - 1 + \end{equation} + Nach dem Fundamentalsatz der Zahlentheorie hat $m$ mindestens einen Primteiler $p$. Dieses $p$ kann nicht von der Form $4k+1$ sein, da sonst der Rest $-1$ bleiben würde. Daher hat $m$ nur Primteiler der Form $4k+1$, woraus folgt, dass $m \equiv 1 \mod 4$ ist, was ein Widerspruch zur Konstruktion von $m$ ist. \newline + (Anders: $m \equiv 3 \mod 4$, d.h. m kann nicht nur Primfaktoren der Form $4k+1$ haben, sei $p \equiv 3 \mod 4 \land p \mid m \Rightarrow p \mid 1$. WS!) + \item[(b)] Angenommen, es gibt nur endlich viele Primzahlen der Form $4k+1$, diese seien $p_{1}, \ldots, p_{r}$. $5 \equiv 1 \mod 4$, daher $r\geq 1$. Mit \begin{equation} - m:=4p_{1} \ldots p_{n} - 1 \equiv 3 \mod 4 + \alpha \equiv 1 \mod 4 \land \beta \equiv 1 \mod 4 \Rightarrow \alpha \beta \equiv 1 \mod 4, \end{equation} -insbesondere ist $m$ ungerade. Nun gilt -\begin{equation} - \forall i: p_{i} < 2 p_{i} < 3p_{i} - 1 < 4p_{1} \ldots p_{n} - 1 -\end{equation} -Nach dem Fundamentalsatz der Zahlentheorie hat $m$ mindestens einen Primteiler $p$. Dieses $p$ kann nicht von der Form $4k+1$ sein, da sonst der Rest $-1$ bleiben würde. Daher hat $m$ nur Primteiler der Form $4k+1$, woraus folgt, dass $m \equiv 1 \mod 4$ ist, was ein Widerspruch zur Konstruktion von $m$ ist. \newline -(Anders: $m \equiv 3 \mod 4$, d.h. m kann nicht nur Primfaktoren der Form $4k+1$ haben, sei $p \equiv 3 \mod 4 \land p \mid m \Rightarrow p \mid 1$. WS!) -\item[(b)] Angenommen, es gibt nur endlich viele Primzahlen der Form $4k+1$, diese seien $p_{1}, \ldots, p_{r}$. $5 \equiv 1 \mod 4$, daher $r\geq 1$. Mit + erhält man für $n$: \begin{equation} - \alpha \equiv 1 \mod 4 \land \beta \equiv 1 \mod 4 \Rightarrow \alpha \beta \equiv 1 \mod 4, + n:=\left(2p_{1} \cdots p_{r} \right)^{2} +1 = 4 \left( p_{1} \cdots p_{r} \right)^{2} + 1 \Rightarrow n \equiv 1 \mod 4 + \end{equation} + Sei $p \in \P \land p \mid n$ (nach dem Fundamentalsatz der Zahlentheorie): + \begin{equation} + \forall i \in \lbrace 1 , \ldots, r \rbrace: p \neq p_{i} \textsl{ (da Rest 1 bleibt) } \end{equation} -erhält man für $n$: -\begin{equation} -n:=\left(2p_{1} \cdots p_{r} \right)^{2} +1 = 4 \left( p_{1} \cdots p_{r} \right)^{2} + 1 \Rightarrow n \equiv 1 \mod 4 -\end{equation} -Sei $p \in \P \land p \mid n$ (nach dem Fundamentalsatz der Zahlentheorie): -\begin{equation} -\forall i \in \lbrace 1 , \ldots, r \rbrace: p \neq p_{i} \textsl{ (da Rest 1 bleibt) } -\end{equation} -Insbesondere folgt daraus, dass $p \equiv 3 \mod 4$. Man erhält also die folgende Kongruenz: -\begin{equation} -\left( 2p_{1} \cdots p_{r} \right)^{2} \equiv -1 \mod p, -\end{equation} -es ist also $-1$ quadratischer Rest $\mod p$. Dies steht nun im Widersrpuch zum 1. Ergänzungssatz. - \end{enumerate} + Insbesondere folgt daraus, dass $p \equiv 3 \mod 4$. Man erhält also die folgende Kongruenz: + \begin{equation} + \left( 2p_{1} \cdots p_{r} \right)^{2} \equiv -1 \mod p, + \end{equation} + es ist also $-1$ quadratischer Rest $\mod p$. Dies steht nun im Widersrpuch zum 1. Ergänzungssatz. + \end{enumerate} + +\aufgabe{14} +{(``Briefmarkenproblem'') Unter der Annahme, dass man von zwei Briefmarkensorten mit den Werten $a$ und $b$, wobei $a,b>0$ ganz und teilerfremd vorausgesetzt werden, beliebig viele Briefmarken zur Verfügung hat, zeige man, dass ab einer gewissen Schranke $s$ jede ganzzahlige Frankierung damit möglich ist. Was ist die kleinste derartige Schranke? (Hinweis: Man verwende zunächst den Chinesischen Restsatz, um zu zeigen, dass die Menge $S=\lbrace ax+by \mid 0 \leq x < b, 0 \leq y < a \rbrace$ ein volles Restsystem $\mod ab$ ist und betrachte dann die Partition $S=S_{0} \cup S_{1}$ von $S$, wobei $S_{0} = \lbrace x \in S \mid x 0$ ganz und teilerfremd vorausgesetzt werden, beliebig viele Briefmarken zur Verfügung hat, zeige man, dass ab einer gewissen Schranke $s$ jede ganzzahlige Frankierung damit möglich ist. Was ist die kleinste derartige Schranke? (Hinweis: Man verwende zunächst den Chinesischen Restsatz, um zu zeigen, dass die Menge $S=\lbrace ax+by \mid 0 \leq x < b, 0 \leq y < a \rbrace$ ein volles Restsystem $\mod ab$ ist und betrachte dann die Partition $S=S_{0} \cup S_{1}$ von $S$, wobei $S_{0} = \lbrace x \in S \mid x ab$ folgt, dass $S_{0} \neq \emptyset, S_{1} \neq \emptyset$. Sei nun $n \in \N \implies \exists k \in \N: n = k \cdot (ab) + r \land 0 \leq r < ab$. Ist $k \geq 1$, so kann man n sicher mit Hilfe des vollen Restsystems und der Abschätzung \eqref{abschS} darstellen. Die größte nicht darstellbare Zahl $\mod ab$ ist $2ab-a-b-1 \Rightarrow s = (a-1)(b-1)$. -\newpage -\subsection*{$15$. Aufgabe} -{\texttt{Man zeige: Ist $p \in \P$ der Form $4k+3 \Rightarrow x^{2} \equiv -1 \mod p$ ist sicher nicht lösbar, ist $p$ der Form $4k+1$, so ist $x_{0} := \left( \frac{p-1}{2} \right)! \mod p$ eine Lösung. (Hinweis: Für den ersten Teil Primitivwurzel $\mod p$, und über Potenzen von g argumierentieren. Für den zweiten Teil zeige zunächst $x_{0}^{2} \equiv (p-1)! \mod p$ und zeige dann $(p-1)! \equiv -1 \mod p$). }} \newline +\aufgabe{15} +{Man zeige: Ist $p \in \P$ der Form $4k+3 \Rightarrow x^{2} \equiv -1 \mod p$ ist sicher nicht lösbar, ist $p$ der Form $4k+1$, so ist $x_{0} := \left( \frac{p-1}{2} \right)! \mod p$ eine Lösung. (Hinweis: Für den ersten Teil Primitivwurzel $\mod p$, und über Potenzen von g argumierentieren. Für den zweiten Teil zeige zunächst $x_{0}^{2} \equiv (p-1)! \mod p$ und zeige dann $(p-1)! \equiv -1 \mod p$). } + Sei $p \in \P$ und $p \equiv 3 \mod 4$. Nach dem Satz von Gauß existiert eine Primitivwurzel $ g \mod p$. Weiters hat das Polynom $x^{2} - 1 = 0$ genau zwei verschiedene Lösungen in $\Z_{p}$, nämlich $\pm 1$. Nun gilt \begin{equation} @@ -169,17 +143,17 @@ Die Zahlen $1$ und $p-1$ sind klarerweise selbstinvers. Die Zahlen $2,\cdots, p- \end{align} \end{subequations} -\newpage -\subsection*{$16$. Aufgabe} -{\texttt{Man berechne die Legendresymbole (700/769) und (1215/1381) zuerst ohne und dann mit Verwendung von Jacobisymbolen.}} \newline +\aufgabe{16} +{Man berechne die Legendresymbole (700/769) und (1215/1381) zuerst ohne und dann mit Verwendung von Jacobisymbolen.} + Zuest ist Folgendes zu überprüfen, um von Legendre- bzw Jacobisymbolen sprechen zu können: -\begin{maxima} -primep(769); -primep(1381); -\maximaoutput* -\m \mathbf{true} \\ -\m \mathbf{true} \\ -\end{maxima} +% \begin{maxima} +% primep(769); +% primep(1381); +% \maximaoutput* +% \m \mathbf{true} \\ +% \m \mathbf{true} \\ +% \end{maxima} Mit Legendresymbol: \begin{subequations} \begin{align} @@ -210,9 +184,10 @@ Mit Jacobisymbol: \end{align} \end{subequations} -\newpage -\subsection*{$17$. Aufgabe} -{\texttt{Man bestimme alle ungeraden Primzahlen $p$, für welche $10$ quadratischer Rest ist.}} \newline + +\aufgabe{17} +{Man bestimme alle ungeraden Primzahlen $p$, für welche $10$ quadratischer Rest ist.} + Aus $10 = 2 \cdot 5$ erhält man aus Satz 3.2, (4): \begin{equation}\label{starkeMultLegendre} \left( \frac{10}{p} \right) = \left( \frac{2}{p} \right) \cdot \left( \frac{5}{p} \right) @@ -232,17 +207,23 @@ Eine ungerade Primzahl $p \neq 5$ ist kongruent zu $1,2,3,4 \mod p$. Es gilt Daher muss notwendigerweise gelten: $p \equiv \pm 1 \mod 5$. Zusammen erhält man also, dass aus $p \equiv \pm 1 \mod 8 \land p \equiv \pm 1 \mod 5$ folgt, dass $\left( \frac{10}{p} \right) = 1$. Man erhält mit dem Chinesischen Restsatz folgendes System: \begin{subequations} -\begin{cases} p \equiv \pm 1 \mod 40 \\ p \equiv \pm 9 \mod 40 \end{cases} +\begin{align} + \begin{cases} +p \equiv \pm 1 \mod 40 \\ p \equiv \pm 9 \mod 40 +\end{cases} +\end{align} \end{subequations} \item Seien beide Faktoren gleich $-1$. Daher ist $p \equiv \pm 3 \mod 8 \land \left( p \equiv \pm 2 \mod 5 \right)$. Man erhält daher mit dem Chinesischen Restsatz: \begin{subequations} -\begin{cases} p \equiv \pm 3 \mod 40 \\ p \equiv \pm 13 \mod 40 \end{cases} +\begin{align} + \begin{cases} p \equiv \pm 3 \mod 40 \\ p \equiv \pm 13 \mod 40 \end{cases} + \end{align} \end{subequations} \end{itemize} Weiters beachte man $\varphi(40)=\varphi(5 \cdot 8 )=4 \cdot 4 = 16$. -\newpage -\subsection*{$18$. Aufgabe} -{\texttt{Man zeige: Ist $p$ eine Primzahl, sodass auch $q=2p+1$ prim ist, so teilt $q$ entweder $2^{p}-1$ oder $2^{p}+1$ und zwar in Abhängigkeit davon, ob $2$ quadratischer Rest $\mod q$ ist oder nicht. (Für welche Mersenn'sche Zahlen $2^{p}-1$ mit $p<100$ sieht man so sofort, dass sie zusammengesetzt sind?).}} \newline + +\aufgabe{18}{Man zeige: Ist $p$ eine Primzahl, sodass auch $q=2p+1$ prim ist, so teilt $q$ entweder $2^{p}-1$ oder $2^{p}+1$ und zwar in Abhängigkeit davon, ob $2$ quadratischer Rest $\mod q$ ist oder nicht. (Für welche Mersenn'sche Zahlen $2^{p}-1$ mit $p<100$ sieht man so sofort, dass sie zusammengesetzt sind?).} + \begin{enumerate} \item Sei $\left( \frac{2}{q} \right) = 1$, d.h. sei $2$ quadratischer Rest $\mod q$. Daher \begin{equation} @@ -261,19 +242,20 @@ Aus dem kleinen Fermat erhält man nun direkt \underbrace{2^{\frac{q-1}{2}}}_{\equiv -1 \mod q } +1 \equiv -1 + 1 \equiv 0 \mod q \Rightarrow q \mid 2^{p}+1 \end{equation} \end{enumerate} -\begin{maxima} -for p:3 thru 97 step 1 do if primep(p) and primep(2*p+1) and power_mod(2,(p-1)/2,p) = 1 then ldisplay(p); -for i in [23,41,89] do ldisplay(primep(2^i-1)); -618970019642690137449562111-341550071728321; -\maximaoutput* -\t9. p=23 \\ -\t10. p=41 \\ -\t11. p=89 \\ -\m \mathbf{done} \\ -\t12. \mathrm{primep}\left(8388607\right)=\mathbf{false} \\ -\t13. \mathrm{primep}\left(2199023255551\right)=\mathbf{false} \\ -\t14. \mathrm{primep}\left(618970019642690137449562111\right)=\mathbf{true} \\ -\m \mathbf{done} \\ -\m 618970019642348587377833790 \\ -\end{maxima} +% \begin{maxima} +% for p:3 thru 97 step 1 do if primep(p) and primep(2*p+1) and power_mod(2,(p-1)/2,p) = 1 then ldisplay(p); +% for i in [23,41,89] do ldisplay(primep(2^i-1)); +% 618970019642690137449562111-341550071728321; +% \maximaoutput* +% \t9. p=23 \\ +% \t10. p=41 \\ +% \t11. p=89 \\ +% \m \mathbf{done} \\ +% \t12. \mathrm{primep}\left(8388607\right)=\mathbf{false} \\ +% \t13. \mathrm{primep}\left(2199023255551\right)=\mathbf{false} \\ +% \t14. \mathrm{primep}\left(618970019642690137449562111\right)=\mathbf{true} \\ +% \m \mathbf{done} \\ +% \m 618970019642348587377833790 \\ +% \end{maxima} \end{document} + diff --git a/UE/ue4.tex b/UE/ue4.tex new file mode 100644 index 0000000..314b64c --- /dev/null +++ b/UE/ue4.tex @@ -0,0 +1,26 @@ +\documentclass[a4paper,10pt,fleqn]{article} + +\usepackage{template} + +\begin{document} + +% \thispagestyle{plain} +% \tableofcontents + +\uebung{4}{23. Mai 2012} + +\aufgabe{19} +{Man zeige, dass sich jede positive ganze $n$ auf {\bf genau eine} Weise als Summe +\begin{align} + n = \sum_{k=0}^m n_k 2^k \text{ mit } n_k \in \{-1,0,1\} +\end{align} +schreiben läßt, sodass gilt $n_m = 1$ und $n_{k −1} n_k = 0$ für $k=1,2,..,m$ (genannt die NAF- Dar- +stellung von n, von engl. nonadjacent form). +} + +\aufgabe{20} +{} + +\end{document} + + diff --git a/Vorlesung.pdf b/Vorlesung.pdf index 20a521139e837b1d515504dc1f45a7937f82d0f1..aa9ad13c9bdf6f8ff8fc021e964bbf18c7af678c 100644 GIT binary patch delta 109 zcmcb+fb;GG&W0AoEleL>SuKq$3=OA$b7hi)FtS&fWLOpT_0b7hi)Ft