From d3ce4be0785f0d9b9bc78ab09c03a5ba350480ac Mon Sep 17 00:00:00 2001 From: user0 Date: Mon, 7 May 2012 11:11:50 +0200 Subject: [PATCH] =?utf8?q?=20On=20branch=20master=20=09modified:=20=20=20.?= =?utf8?q?gitignore=20(angepasst=20an=20neue=20files)=20=09modified:=20=20?= =?utf8?q?=20UE/.gitignore=20(angepasst=20an=20neue=20files)=20=09new=20fi?= =?utf8?q?le:=20=20=20UE/ue3.tex=20(neue=20=C3=9Cbung=20hinzugef=C3=BCgt)?= =?utf8?q?=20=09modified:=20=20=20Vorlesung.pdf=20(neu=20=C3=BCbersetzt)?= =?utf8?q?=20=09modified:=20=20=20Vorlesung.tex=20(bearbeitet)?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- .gitignore | 2 + UE/.gitignore | 5 + UE/ue3.tex | 108 ++++++++ Vorlesung.pdf | Bin 162245 -> 150254 bytes Vorlesung.tex | 703 +++++++++++++++++++++++++++++++++++++++++++------- 5 files changed, 723 insertions(+), 95 deletions(-) create mode 100644 UE/ue3.tex diff --git a/.gitignore b/.gitignore index 2d6974b..7947a26 100644 --- a/.gitignore +++ b/.gitignore @@ -1,2 +1,4 @@ Vorlesung.aux Vorlesung.log +Vorlesung.dvi +Vorlesung.toc diff --git a/UE/.gitignore b/UE/.gitignore index 69f9a38..2fa199f 100644 --- a/UE/.gitignore +++ b/UE/.gitignore @@ -1,2 +1,7 @@ ue2.aux ue2.log +ue3.log +ue3.aux +ue3.tmp +ue3.dvi +ue3.tex~ diff --git a/UE/ue3.tex b/UE/ue3.tex new file mode 100644 index 0000000..2fb0fa3 --- /dev/null +++ b/UE/ue3.tex @@ -0,0 +1,108 @@ +\documentclass[a4paper,10pt,fleqn]{article} +\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{emaxima} +%\usepackage{ngerman} + + +\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{$3$. Übung ZtuA} +\author{} + +\begin{document} +\maketitle +%\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 + \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. +\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} + \alpha \equiv 1 \mod 4 \land \beta \equiv 1 \mod 4 \Rightarrow \alpha \beta \equiv 1 \mod 4, + \end{equation} +erhält man für $n$: +\begin{equation} +n:=\left(2p_{1} \cdots p_{r} \right)^{2} +1 = 4(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{enumerate} +Insbesondere folgt daraus, dass $p \equiv 3 \mod 4$. Man erhält also die folgende Kongruenz: +\begin{equation} +(2p_{1} \cdots p_{r} \right)^{2} equiv -1 \mod p4, +\end{equation} +es ist also $-1$ quadratischer Rest $\mod p$. Dies steht nun im Widersrpuch zum 1. Ergänzungssatz. + +\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 +Sei $p \in \P$ und $p \equiv 3 \mod 4$. Nach dem Satz von Gauß existiert eine Primitivwurzel $ g \mod p$. +\begin{subequations} +\begin{align} + +\end{align} +\end{subequations} +\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 +\begin{enumerate} +\item Sei $\left( \frac{2}{q} \right) = 1$, d.h. sei $2$ quadratischer Rest $\mod q$. Daher + \begin{equation} + \left( \frac{2}{q} \right) = 1 \implies \exists x \in \Z_{q}: x^{2} \equiv 2 \mod q + \end{equation} +Setzt man diese Tatsache ein, erhält man +\begin{equation} + 2^{p}-1=\left( x^{2} \right)^{p} - 1 = x^{2p} -1 +\end{equation} +Aus dem kleinen Fermat erhält man nun direkt +\begin{equation} + x^{(2p+1)-1} = x^{2p} \equiv 1 \mod 2p+1 \Rightarrow x^{2p}-1 \equiv 0 \mod q \Rightarrow q \mid 2^{p}-1 +\end{equation} +\item Sei $\left( \frac{2}{q} \right) = -1$. Aus dem Euler'schen Kriterium erhält man nun sofort unter Beachtung von $\frac{q-1}{2} = p$, dass + \begin{equation} + \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} +\end{document} diff --git a/Vorlesung.pdf b/Vorlesung.pdf index ff4ef26aa232486c62e24714c52ee23d94b9603d..eabc629dda2a16c041894777f0cb5e1eab316a5c 100644 GIT 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zQ6@a)mh)UaaKV9P1Lng9HfDTOgZv6+A5;b&22dOerP8FhxlfD(sQ{X16N*o)nuE+5L+L8%9XE`z-Pi|&ijC{S4812b`>jlw}`LLfW)7Q3` zsY<)pfWy^fZbtr4}3t4&md45?T0ZrUEE!!R)9DV8Rhg`0UYqY@wo{C@ m2cbwa*5=FhR`Tc~+enmLC0D6M>k+0=DT2iE 0$: \\ +Wähle $q \in \Z$ so, dass: $qb \leq a \land \left( q+1 \right) > a$ (ist in endlich vielen Schritten möglich) (--> Skizze Zahlengerade)\\ +Dann gilt für $r := a - qb$, dass $0 \leq r < b \land a = qb+r$. \\ +Ist aber $b<0$, so gilt nach Obigem: +\begin{equation} + \exists \bar{q},\bar{r} \in \Z: a = \vert b \vert \cdot \bar{q} + \bar{r} \land 0 \leq \bar{r} < \vert b \vert +\end{equation} +Setze $q:=-\bar{q}, r:=\bar{r}$ folgt dann wieder $a = bq +r \land 0 \leq r < b$. Daher ist die Existenz gesichert. +\item[Eindeutigkeit] Angenommen wir haben zwei Darstellungen der folgenden Form: + \begin{equation} + a = q_{1}b+r_{1}=q_{2}+r_{2} \land 0 \leq r_{1} < \vert b \vert \land 0 \leq r_{2} < \vert b \vert + \end{equation} +Daraus erhält man +\begin{subequations} + \begin{align} + \left( q_{1} - q_{2} \right) b = r_{2} - r_{1} \\ +\stackrel{\vert \cdot \vert} \implies \vert q_{1} - q_{2} \vert \cdot \vert b \vert = \underbrace{\vert r_{2} - r_{1} \vert}_{\in [0,\vert b \vert), \textsl{ zumindest } < \vert b \vert} \\ +\Rightarrow \vert q_{1} - q_{2} \vert = 0 \\ +\Rightarrow q_{1} = q_{2} \\ +\Rightarrow r_{1} = r_{2} + \end{align} +\end{subequations} +\end{enumerate} +\end{proof} + +\begin{bem}[@Def 1.4 ggT] + \begin{equation} + a=b=0 \Rightarrow ggT(a,b)=ggT(0,0)=0 + \end{equation} +\end{bem} + +\begin{proof}[Beweis zu Satz $1.6$] +Teilbarkeitsrelation ist invariant unter $\pm$, daher gilt $ggT(a,b)=ggT(\vert a \vert, \vert b \vert)$, insbesondere erhält man dadurch $ggT(a,0)=\vert a \vert$ und $ggT(a,b)=ggT(b,a)$. \\ +oBdA: $a \geq b > 0$. Wende Euklidischen Algorithmus an, $r_{(-1)}:=a, r_{0}:=b$, dann gilt: +\begin{subequations} + \begin{align} + a = q_{1} b+ r_{1}, \quad 0 \leq r_{1} < b \\ +b = q_{2} r_{1} + r_{2}, \quad 0 \leq r_{2} < r_{1} \\ +r_{1} = q_{3} r_{2} + r_{3}, \quad 0 \leq r_{3} < r_{2} \\ +r_{(i-2)} = q_{i} r_{(i-1)} + r_{i}, \quad 0 \leq r_{i} < r_{(i-1)}\\ +\textsl{Folge der Reste ist streng monoton fallend} \rightarrow \textsl{ bricht ab} \\ +r_{(n-1)} = q_{n} r_{n} \label{letzterestgleichung} + \end{align} +\end{subequations} +Insbesondere folgt aus \eqref{letzterestgleichung}, dass $ggT(r_{(n-1)},r_{n})=r_{n}$. +\begin{equation} + ggT(a,b)=ggT(b,r_{1}) = ggT(r_{(n-1)},r_{n})= r_{n} +\end{equation} +Man kann daraus die folgende Behauptung formulieren: +\begin{equation} + \forall k \in \lbrace -1 \rbrace \cup \N: \exists x_{k},y_{k} \in \Z: r_{k} = x_{k}a + y_{k}b +\end{equation} +Beweis mit Induktion nach $k$: \\ +I-Anfang $-1,0$: +\begin{subequations} + \begin{align} + r_{(-1)} = a = 1 \cdot a + 0 \cdot b, \qquad \surd \\ +r_{0} = b = 0 \cdot a + 1 \cdot b, \qquad \surd \\ +\end{align} +\end{subequations} +I-Schritt: $k \rightarrow k+1$: +\begin{subequations} +\begin{align} +r_{k} = r_{(k-2)} - q_{k} r_{(k-1)} = \\ +\stackrel{\textsl{I-VS}} = \left( x_{(k-2)} a + y_{(k-2)} b \right) - q_{k} \left( x_{(k-1)} a + y_{k-1} b \right) = \\ += \underbrace{(x_{(k-2)} - q_{k}x_{(k-1)})}_{x_{k}} a + \underbrace{(y_{(k-2)} - q_{k}y_{(k-1)})}_{y_{k}}b + \end{align} +\end{subequations} +Man beachte, dass für $r_{()}, x_{()},y_{()}$ drei mal die gleiche Rekursionsgleichung mit verschiedenen Startwerten genügt! Diese Methode ist besser um ggT auszurechnen, als im Eukl. Alg. einfach ``hinaufzurechnen''. Es gilt +\begin{subequations} + \begin{align} + r_{-1}=a,r_{0}=b, r_{i} = r_{i-2} - q_{i} r_{i-1}\\ +x_{-1}=1, x_{0}=0, x_{i} = x_{i-2} - q_{i} x_{i-1} \\ +y_{-1}=0, y_{0}=0, y_{i} = y_{i-2} - q_{i} y_{i-1} + \end{align} +\end{subequations} +\end{proof} + +\begin{bsp} +Berechne $ggT(2124, 1764)$ mit Hilfe von Euklidischem Algorithmus, Schema: + +\begin{tabular}{|l|c|c|c|c|c|c|c|}\hline +$r_{i-2}$ & $r_{i-1}$ & $q_{i}$ & $x_{i-2}$ & $x_{i-1}$ & $y_{i-2}$ & $y_{i-1}$ & $i$ \\ +\hline +$2124$ & $1764$ & $1$ & $1$ & $0$ & $0$ & $1$ & $0$\\ +$1764$ & $360$ & $4$ & $0$ & $1$ & $1$ & $-1$ & $1$ \\ +\hline +\end{tabular} +$\Rightarrow ggT(2124,1764)=26=5 \cdot 2124 + (-6) \cdot 1764$. +\end{bsp} + +\begin{bem} +$\sim 40 \%$ der Fälle gilt $q=1$, bzw Einstellig in $90\%$. Statt Divisionen verwendet man interierte Subtraktionen, mit vielleicht Ausnahme einer ersten Division. +\end{bem} + +\begin{bem} +Die Stellenanzahl im dekadischen System einer dekadischen Zahl $n$: $\lfloor \log_{10} n \rfloor +1 \approx \log_{10}$. +\end{bem} + +\begin{bem} +``Normale Multiplikation'' mit ``Schulmathematik'' hat quadratischen Aufwand. +\end{bem} + +\begin{bem}[Umrechnung für $\lambda$] + \begin{subequations} + \begin{align} + \lambda^{\log_{\lambda}(a)} = a \\ +\Rightarrow \log_{10}(\cdot) \\ +\log_{\lambda}(a) \cdot \log_{10}(\lambda)=\log_{10}(a) \\ +\implies \log_{\lambda}(a)=\frac{\log_{10}(a)}{\log_{10}(\lambda)} + \end{align} + \end{subequations} +\end{bem} + +\begin{proof}[Beweis zu Satz $1.8$, Seite $4$] + Betrachte die Fibonacci-Folge + \begin{equation} + 0,1,1,2,3,5,8,13,21,34,55, \ldots + \end{equation} +welche definiert ist durch $F(0)=0,F(1)=1, F(n)=F(n-2)+F(n-1)$ für $n\geq 2$. \\ +Ist $F_{n}$ die nächst kleinere Fibonacci-Zahl zu $a$, so kann der Rechenaufwand zur Berechnung von $ggT(a,b)$ durch den Rechenaufwand von $ggT(F_{n},F_{(n-1)})$ nach oben abgeschätzt werden, z.B. $a=50,b=37 \rightsquigarrow ggT(34,21)$, sind $F_{9}=34, F_{8}=21$ die nächst kleineren Fibonacci-Zahlen und es gilt: +\begin{subequations} + \begin{align} + 34 = 1 \cdot 21 + 13 \\ +21 = 1 \cdot 13 + 8 \\ +13 = 1 \cdot 8 + 5 \\ +8 = 1 \cdot 5 + 3 \\ +5 = 1 \cdot 3 + 2 \\ +3 = 1 \cdot 2 + 1 \\ +2 = 1 \cdot 2 + \end{align} +\end{subequations} +Man benötigt hier $7$, allgemein $n-2$ Divisionen, was eine obere Schranke darstellt. \\ +Es gilt für die $F_{n}$ die folgende explizite Formel: +\begin{equation} + F_{n} = \frac{\lambda_{1}^{n} - \lambda_{2}^{n}}{\sqrt{t}} \textsl{ mit } \lambda_{1,2}=\frac{1 \pm \sqrt{5}}{2} +\end{equation} +Diese Formel zeigt man leicht mit Induktion. \\ +Wenn es eine geometrische Folge gibt, welche die Fibonacci-Folge liefert, so muss diese Folge zwingend erfüllen: +\begin{subequations} +\begin{align} + q^{n} = q^{n-1} + q^{n-2} \\ +\implies q^{2} = q + 1 \\ +\textsl{woraus man die folgenden Lösungen erhält} \\ +F_{n} = \frac{\lambda_{1}^{n} - \lambda_{2}^{2}}{\sqrt{5}} = \frac{1}{\sqrt{5}} \cdot \lambda_{1}^{n} + \underbrace{\left( \frac{-1}{\sqrt{5}} \cdot \lambda_{2}^{n} \right)}_{\stackrel{n \rightarrow \infty}\rightarrow 0} +\end{align} +\end{subequations} +$\lambda_{2}$-Term geht sehr schnell gegen Null. Lässt man den zweiten Teil weg, d.h. man rechnet ``nur'' mit dem Wert $\frac{\lambda_{1}^{n}}{\sqrt{t}}$ und rundet immer auf die nächste ganze Zahl, so erhält man auch wieder die Fibonacci-Folge (Beweis per Derive). \\ +Setzt man $\lambda := \lambda_{1}$, so folgt daraus +\begin{subequations} + \begin{align} + n-2 = \log_{\lambda}(\lambda^{n-2})=\log_{\lambda} \left( \frac{\lambda^{n}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\lambda^{2}} \right) = \\ += \log_{\lambda} \left( \frac{\lambda^{n}}{\sqrt{5}} \right) + \log_{\lambda}\left( \frac{\sqrt{5}}{\lambda^{2}} \right) = \\ += \log_{\lambda}(F_{n}) + \underbrace{ \left( \log_{\lambda}\left( \frac{\lambda^{n}}{\sqrt{5}} - \log_{\lambda}(F_{n}) \right) \right) }_{\approx 0} + \underbrace{\log_{\lambda}\left( \frac{\sqrt{5}}{\lambda^{2}} \right)}_{\approx -0,33} = \\ += \lfloor \log_{\lambda}(F_{n}) \rfloor, \quad n \geq 2 + \end{align} +\end{subequations} +Wegen $F_{n} \leq a$ folgt insgesamt $n-2 \leq \lfloor \log_{\lambda}(a) \rfloor$. +\end{proof} + +\begin{bem}[Bemerkung zu Satz $1.10$] +$ggT(72,108)=36 \cdot ggT(2,3)=36$, ``Distributivgesetz'' gilt. +\end{bem} + +\begin{proof}[Beweis zu Satz $1.10$, Seite $4$] + Zeige zwei Eigenschaften von ggT: + \begin{equation} + ggT(a,b)\mid a \land ggT(a,b)\mid b \Rightarrow \frac{ggT(a,b)}{t} \mid \frac{a}{t} \land \frac{ggT(a,b)}{t} \mid \frac{b}{t} + \end{equation} +Daher ist $\frac{ggT(a,b)}{t}$ ist ein gemeinsamer Teiler von $\frac{a}{t}$ und $\frac{b}{t}$. \\ +Es gilt: $\exists x,y \in \Z: ggT(a,b)=xa+yb$. +\begin{equation} + \implies \frac{ggT(a,b)}{t} = x \frac{a}{t} + y \frac{b}{t} +\end{equation} +Jeder gemeinsame Teiler von $\frac{a}{b}$ und $\frac{b}{t}$ teilt auch rechte Linearkombination, insbesondere daher auch die linke Seite nach Rechenregeln der Teilbarkeitsrelation. +\end{proof} + +\begin{proof}[Beweis zu Satz $1.11$, Seite $4$] +% ein one-liner-beweis + $ggT(a,b)=1 \Rightarrow x,y\in \Z: xa+yb=1$, ``mal c'' liefert + \begin{equation} + xac + ybc = c + \end{equation} +Nun gilt $a \mid xac, a \mid ybc$ nach Voraussetzung, daher teilt $a$ auch Summe, daher $a \mid c$. +\end{proof} + +\begin{proof}[Beweis zu Folgerung $1.14$, Seite $4$] + Unter gegebenen Voraussetzungen gilt: $kgV(a,b)=\frac{\vert ab\vert}{ggT(a,b)} = \vert a b \vert$, weiters gilt $kgV(a,b) \mid c$ weil $c$ ist gemeinsames Vielfaches, daher $ab \mid c$. +\end{proof} + + +\begin{proof}[Beweis zu Satz $1.16$, Seite $5$] \begin{enumerate} - \item Sei $p\in\P$ und es gelte $p|av \und p \nmid a$ , d.h. $\ggT(p,a)=1$. nach dem Lemma von Euklid (Satz 1.11) folgt daher $p|b$. - \item Ist umgekehrt die Bedingung des Satzes erfüllt und gilt $p = a\cdot b$ mit $(a,b \in \N *$, so gilt einerseits $a|p \und b|p$, aber auch $p|a \oder p|b$ nach VS. Daraus folgt aber sofort $p = a \oder p= b,Q$ + \item[``$\Rightarrow$''] Sei $p\in\P$ und es gelte $p \mid ab \und p \nmid a$ , d.h. $ggT(p,a)=1$ (da $p$ eine Primzahl ist), und nach dem Lemma von Euklid (Satz $1.11$) folgt daher $p \mid b$. + \item[``$\Leftarrow$''] Ist umgekehrt die Bedingung des Satzes erfüllt und gilt $p = a\cdot b$ mit $a,b \in \N^{*}$, so gilt einerseits $a \mid p \und b \mid p$, aber auch $p \mid a \oder p \mid b$ nach VS. Daraus folgt aber sofort $p = a \oder p= b$, da alle auftretenden Zahlen nichtnegativ sind. \end{enumerate} -\hfill$\blacksquare$ +\end{proof} -\subsection*{Bew. 1.20} +\begin{bem} +Kein (!) Beweis zum Fundamentalsatz der Zahlentheorie, Satz $1.17$. +\end{bem} + +\begin{proof}[Beweis zu Satz $1.20$, Seite $5$] \begin{eqnarray*} - \ggT(a,b) = \ggT(a,c) = 1 &\Rightarrow& \exists x,y,u,v : xa+yb = ua +vc = 1\\ - &\Rightarrow& \exists x,y,u,v : (xa +yb)(ua +vc) = (xau +xvc +ybu) a +(yv)(bc) = 1\\ - &\Rightarrow& \text{Jeder gem. Teiler von} a \text{ und }c \text{ teilt auch }1 \Rightarrow \ggT(a,c) =1 + ggT(a,b) = ggT(a,c) = 1 &\Rightarrow& \exists x,y,u,v : xa+yb = ua +vc = 1\\ + &\Rightarrow& (xa +yb)(ua +vc) = (xau +xvc +ybu) a +(yv)(bc) = 1\\ + &\Rightarrow& \text{Jeder gemeinsame Teiler von} a \text{ und }c \text{ teilt auch }1 \Rightarrow ggT(a,c) =1 \end{eqnarray*} -\hfill$\blacksquare$ - -\subsection*{Bew. 1.21} -Angenommen, $P=\{p_1,P_2,\cdots,p_r\}$, d.h. endlich. Die Zahl -\begin{displaymath} - N=p_1 p_2\cdots p_r +1 -\end{displaymath} -Zahl dann $> 1$ und daher durch eine Primzahl $p$ teilbar, wobei man -\begin{displaymath} - p := min\{t \in \N * | t | n \und t > 1 \} -\end{displaymath} -Wenn $p| n \und p| n-1 = p_1p_2 \cdots p_r$ folgt daraus +\end{proof} + +\begin{proof}[Beweis zu Satz $1.21$, Seite $5$] +Angenommen $P=\{p_1,p_2,\cdots,p_r\}$, d.h. von endlicher Mächtigkeit. Definiere +\begin{equation} + N: = p_1 p_2 \cdots p_r +1 +\end{equation} +Ist die Menge $\P$ nun endlich oder sogar leer, folgt für diese Zahl $N$ dann $N > 1$ und daher ist $N$ durch eine Primzahl $p$ teilbar, wobei man setzt +\begin{equation}\label{eqminteiler} + p := \min \{ t \in \N^{*} : t \mid n \und t > 1 \} +\end{equation} +Die Menge in \eqref{eqminteiler}, über die das Minimum gebildet wird, ist nichtleer, denn sie enthält $N$, und da die natürlichen Zahlen wohlgeordnet sind, hat sie ein kleinstes Element. Dieses Minimum $p$ ist nun zwingendermaßen prim, da man sonst einen noch kleineren Teiler hätte, der die beiden Bedingungen in \eqref{eqminteiler} erfüllt. Weiters beachte man, dass hier der Fundamentalsatz der Zahlentheorie nicht verwendet wird! +Es gilt +\begin{equation}\label{pteiltN} +p \mid N +\end{equation} +nach Konstruktion von $p$. \\ +$p$ ist prim, und da es nach VS nur endlich viele Primzahlen gibt, folgt, dass $p \in \lbrace p_{1}, \cdots, p_{r} \rbrace$. Daraus erhält man sofort +\begin{equation}\label{pteiltp1bispr} + p \mid p_{1} p_{2} \cdots p_{r} +\end{equation} +Aufgrund der Rechenregeln der Teilbarkeit erhält man nun aus \eqref{pteiltN} und \eqref{pteiltp1bispr}, dass $p$ auch ihre Differenz teilt, und man erhält unter Verwendung von $p_1p_2 \cdots p_r = N-1$, dass gilt: \begin{displaymath} - p|1 = N-1(N-1) \Rightarrow $\blitza$ + p \mid 1 = N-(N-1) \text{Widerspruch!} \end{displaymath} -Also ist $| \P | = \infty$, \hfill$\blacksquare$ -\subsection*{Anmerkung 1.21} +Also ist $\vert \P \vert = \infty$. +\end{proof} + +\begin{bem}[Bemerkung zu Anmerkung $1.21$] \begin{eqnarray*} \sum_{p\in\P} \frac 1 p && \text{divergent (Euler)}\\ -\sum_{k=1}^{\infty} &=& \frac{\pi^2}{6} +\sum_{k=1}^{\infty} &=& \frac{\pi^{2}}{6} \end{eqnarray*} -\subsection*{Anmerkung 1.22} +\end{bem} + +\begin{bem}[Bemerkung zu Satz $1.22$, Seite $6$] \begin{displaymath} \pi(x) = \{ p \in \P | p \leq x\}, x\in \R^+ \end{displaymath} +Der relative Fehler geht für $x \rightarrow \infty$ gegen Null: +\begin{equation} +\lvert \frac{\pi \left( x \right) - \frac{x}{\ln \left( x \right) }}{\pi \left( x \right)} \rvert = 1 - \underbrace{\left(\frac{\frac{x}{\ln \left( x \right)}}{\pi \left( x \right)}\right)}_{\stackrel{x \rightarrow} \longrightarrow 1} \stackrel{x \rightarrow \infty} \longrightarrow 0 +\end{equation} +Man beachte, dass der Absolutfehler beliebig groß werden kann! \\ +Für den Integrallogarithmus betrachtet man eigentlich den Cauchy'schen Hauptwert des des Integrals, d.h. +\begin{equation} + \lim \limits_{\varepsilon \searrow 0^{+}} \left( \int_{0}^{1-\varepsilon} \frac{dt}{\ln \left( t \right)} + \int_{1+\varepsilon}^{x} \frac{dt}{\ln \left( t \right)} \right) +\end{equation} +Oder man kann auch für $x > 1$ das folgende Integral betrachten: +\begin{equation} + \int_{0}^{2} \frac{dt}{\ln \left( t \right)} + \int_{2}^{x} \frac{dt}{\ln \left( t \right)} +\end{equation} Primzahldichte in der Nähe von $x \approx \frac 1 {ln(x)}$\\ -in der Gegend von $10^{100}$ : jede $\approx 230.$ Zahl ist eine Primzahl +In der ``Gegend'' von $10^{100}$ : jede $\approx 230.$ Zahl ist eine Primzahl. Betrachtet man nur ungerade Zahlen, so halbiert sich die WS keine Primzahl zu ``erwischen''. +\end{bem} -\subsection*{Riemansche Vermutung} -\begin{align} - \zeta(s)&= \sum_{n=1}^{\infty}\frac 1{n^s} \hfill (re(s) >1) -\end{align} -Analytisch fortzetzen: +\begin{bem}[Bemerkung zur analytischen Fortsetzung der $\zeta$-Funktion] +Die $\zeta$-Funktion ist folgendermaßen definiert +\begin{displaymath} + \zeta(s)= \sum_{n=1}^{\infty} \frac{1}{n^{s}} \quad (Re(s) >1). +\end{displaymath} +Analytisch fortsetzen: +\begin{subequations} \begin{align} - \zeta(s) & = 1 + \frac 1{2^s} + \frac 1{3^s} + \frac 1{4^s} + \frac 1{5^s} + \cdots\\ -\frac 2 {2^s}\zeta(s) & = \frac 2{2^s} + \frac 2{4^s} + \frac 2{6^s} + \cdots\\ -(1-2^{1-s})\zeta(s) & = 1 - \frac 1{2^s} + \frac 1{3^s} - \frac 1{4^s} + \frac 1{5^s} - \cdots = \eta(s)\\ -\zeta(s) & = \frac{\eta(s)}{(1-2^{1-s})} + \zeta \left( s \right) = 1 + \frac{1}{2^{s}} + \frac{1}{3^{s}} + \frac{1}{4^{s}} + \frac{1}{5^{s}} + \cdots\\ +\frac{2}{2^{s}} \cdot \zeta \left( s \right) = \frac{2}{2^{s}} + \frac{2}{4^{s}} + \frac{2}{6^{s}} + \cdots\\ +\implies \left( 1-2^{1-s} \right) \zeta \left( s \right) = 1 - \frac{1}{2^{s}} + \frac{1}{3^{s}} - \frac{1}{4^{s}} + \frac{1}{5^{s}} - \cdots := \eta \left( s \right) \label{defetafkt} \end{align} - - - +\end{subequations} +Die obige Reihe in \eqref{defetafkt} konvergiert für $Re \left( s \right) > 0$. \\ +Für $s \neq 1$ gilt daher: +\begin{equation} +\zeta \left( s \right) = \frac{\eta \left( s \right)}{1-2^{(1-s)}} +\end{equation} +Die ``Mittelgerade'' des kritischen Streifens hat die Gleichung: $\frac{1}{2} + i \cdot t, t \in \R$. \\ +Riemann-Siegel-Formel erwähnt. \\ +\end{bem} \section*{Vorlesung 28.3.12} Riemann-Siegel-Formel? @@ -90,25 +404,35 @@ Riemann-Siegel-Formel? $\chi : \Z \to \C$ ist ein Charakter $\mod m$, d.h. \begin{enumerate} \item $\chi(ab) = \chi(a)\chi(b) \forall a,b\in\Z$ - \item $ a \equiv b$ und $m \Rightarrow \chi (a) = \chi(b)$ - \item $\chi(a) = 0 \Leftrightarrow \ggT(a,m) \neq 1$ + \item $ a \equiv b \mod m \Rightarrow \chi (a) = \chi(b)$ + \item $\chi(a) = 0 \Leftrightarrow \gcd(a,m) \neq 1$ \end{enumerate} -Für die analytische Fortsetzung von $L_{\lambda}(s)$ auf krit. Streifen gilt Nullstellen die gleiche Aussage. -% + +Für die analytische Fortsetzung von $L_{\lambda}(s)$ auf krit. Streifen gilt Nullstellen die gleiche Aussage.\\ +Eine weitere Möglichkeit der Definition der Riemann'schen Zeta-Funktion ist gegeben durch: +\begin{subequations} \begin{align} - \zeta(s) &= \prod_{p \in \mathbb{P}} \underbrace{\frac 1 {1 - \frac 1 {p^3}}} (Re(s) > 1)\\ - & \sum_{k=0}^{\infty} \frac 1 {p^{ks}} \to \sum_{n=1}^{\infty} \frac 1 {n^s} + \zeta(s) &= \prod_{p \in \P} \underbrace{\frac 1 {1 - \frac 1 {p^{s}}}},\quad (Re(s) > 1)\\ +1-\frac{1}{p^{s}}=\sum \limits_{k=0}^{\infty} \frac{1}{p^{ks}} \\ +% & \sum_{k=0}^{\infty} \frac 1 {p^{ks}} \to \sum_{n=1}^{\infty} \frac 1 {n^s} +\prod \limits_{p \in \P} \left( \frac{1}{\sum (\cdots)} \right) \to \sum \limits_{n=1}^{\infty} \frac{1}{n^{s}} \end{align} -$\pi(x) = li(x) + O\left(\sqrt x \ln x\right) \Leftrightarrow $ Riemansche Vermutung (R. Koch) - - +\end{subequations} +Weiters gilt nach R. Koch: +\begin{equation} +\pi(x) = li(x) + O\left(\sqrt x \ln x\right) \Leftrightarrow \textsl{Riemansche Vermutung} +\end{equation} +Es existiert eine Abschätzung für die maximale Anzahl von Nullstellen, man zeigt, dass die Anzahl der Nullstellen auf der Mittelgeraden mit der maximalen Abschätzung übereinstimmt $\implies$ es gibt keine weiteren. + +\begin{bem}Mögliche Prüfungsfrage: Wozu braucht man Riemann'sche Vermutung? Für Primzahltests und Primzahlverteilung. +\end{bem} \subsection*{Beweis 2.2} -$a \equiv b \mod m \und c \equiv\mod m \Rightarrow m| a-b \und m| c-d$ +$a \equiv b \mod m \land c \equiv d \mod m \Rightarrow m \mid a-b \wedge m\mid c-d$ \begin{align} &\Rightarrow \begin{cases} - m|(a-b)\pm(c-d) = (a\pm x) - (b\pm d)\\ - m| a(b)c+b(c-d) = ac -bd + m|(a-b)\pm(c-d) = (a\pm c) - (b\pm d)\\ + m| (a-b)c+b(c-d) = ac -bd \end{cases}\\ &\Rightarrow \begin{cases} @@ -125,6 +449,14 @@ $a \equiv b \mod m \und c \equiv\mod m \Rightarrow m| a-b \und m| c-d$ &\Rightarrow m|(a-b)+(b-c) = a-c\\ &\Rightarrow a\equiv c\mod m \end{align} +Daher reflexiv, symmetrisch und transitiv, d.h. eine Äquivalenzrelation. Aus der Verträglichkeit mit $+$ und $\cdot$ erhält man sogar eine Kongruenzrelation. \\ +Insbesondere erhält man dadurch auch eine Partition von $\Z$: +\begin{subequations} + \begin{align} + m=0 \Rightarrow \textsl{id-rel} \Rightarrow \Z \\ + m > 0 \Rightarrow \lbrace \bar{0}, \bar{1}, \ldots, \bar{m-1} \rbrace + \end{align} +\end{subequations} \subsection*{Beispiel 2.2} $(\Z_m,f m=z,3,4)$ @@ -142,12 +474,17 @@ m=2:&& \end{matrix}\\ \end{align} +\begin{bem} + @Prf zu Satz 2.5: ``Formulieren und beweisen Sie den Satz, der die Lösbarkeit linearer Kongruenzen beschreibt''. +\end{bem} + \subsection*{Beweis 2.5} -Die Bedingung $d:= \ggT(a,m) | b$ ist notwendig für Lösbarkeit wegen +Die Bedingung $d:= \gcd(a,m) \mid b$ ist notwendig für Lösbarkeit wegen \begin{align} - a\tilde x \equiv b \mod m \Rightarrow \exists k\in \Z : a\tilde x *km = b \Rightarrow d| b ( \text{wegen} d|a ,d|m \text{also auch} d|ax+km) + a\tilde x \equiv b \mod m \Rightarrow \exists k\in \Z : a\tilde x +km = b \Rightarrow d| b \end{align} -Ist umgekehrt die Bedingung $d|m$ efüllt und $d=ra+sm $ mit $r,s\in \Z$ eine Darstellung von $d$ als Linearkombination von $a\ \mod m$. Dann gilt +wegen $d|a ,d|m$, also auch $d|ax+km$. \\ +Ist umgekehrt die Bedingung $d|m$ efüllt und sei $d=ra+sm $ mit $r,s\in \Z$ eine Darstellung von $d$ als Linearkombination von $a\ \mod m$. Dann gilt \begin{align} ra+sm&=d \\ r\frac bd a +s \frac b d m &= b\\ @@ -155,28 +492,27 @@ Ist umgekehrt die Bedingung $d|m$ efüllt und $d=ra+sm $ mit $r,s\in \Z$ eine Da \end{align} Also ist dann $x = r \frac b d $ eine Lösung von $a x \equiv b \mod m$ -Sind ferner $u \und v$ zwei Lösungen von $ax \equiv b\mod m$ , so gilt +Sind ferner $u$ und $v$ zwei Lösungen von $ax \equiv b\mod m$ , so gilt \begin{align} au \equiv av \equiv b \mod m &\Rightarrow m | a(u-v)\\ - &\Rightarrow \frac m d | \frac a d (u-v) \und (\ggT(\frac m d,\frac a d) = 1\\ + &\Rightarrow \frac m d | \frac a d (u-v) \und (\gcd(\frac m d,\frac a d) = 1\\ &\Rightarrow \frac m d | u-v \Rightarrow u\equiv v \mod \frac md \end{align} -Alle $\mod m$ in kongruenten Lösungen sind daher gegeben durch $u, u+ \frac m d, u + \frac{2m}d,\dots, u+(d-1)\frac m d$ \hfill $\blacksquare$ - +Alle $\mod m$ in kongruenten Lösungen, es gibt $d$ Stück, sind daher gegeben durch $u, u+ \frac m d, u + \frac{2m}d,\cdots, u+(d-1)\frac m d$ \hfill $\blacksquare$ \section*{Vorlesung 18.4.12} \subsection*{Erweiterung 2.10} $m \in \P \Leftrightarrow \phi(m) = m-1$\\ -$\gamma = \lim_{n\to\infty}\left( \sum_{k=1}^n \frac 1 k - ln n \right) \approx 0.577\dots$ +$\gamma = \lim_{n\to\infty}\left( \sum_{k=1}^n \frac 1 k - ln n \right) \approx 0.577\cdots$ \subsection*{Ergänzung 2.11} $n = 123$\\ einsetzen Beispiel! \subsection*{Beweis 2.11} -Sei $M_i = \frac {m_1m_2\dots m_r} {m_i} = m_1\dots m_{i-1}m_{i+1}\dots m_r, i = 1,2,\dots,r$ und sie $M_i^*$ Lösung von $M_ix\equiv 1 \mod m_i.$ -( Beachte, dass $\ggT(M_i,m) = 1$ wegen $\ggT(m_j,m_i) = 1 \forall j\neq i$)Es ist dann +Sei $M_i = \frac {m_1m_2\cdots m_r} {m_i} = m_1\cdots m_{i-1}m_{i+1}\cdots m_r, i = 1,2,\cdots,r$ und sie $M_i^*$ Lösung von $M_ix\equiv 1 \mod m_i.$ +( Beachte, dass $\gcd(M_i,m) = 1$ wegen $\gcd(m_j,m_i) = 1 \forall j\neq i$)Es ist dann \begin{align} x&= \sum_{i=1}^r a_iM_i^*M_i \end{align} @@ -185,54 +521,231 @@ Lösung des Kongruenzensystems wegen x = \underbrace{\left( \sum_{k=1}^{i-1} a_k M_k^*\underbrace{M_k}_{\equiv 0 \mod m_i} \right)}_{0 \mod m_i}% + \underbrace{a_i \underbrace{M_i^*M_i}_{\equiv 1 \mod m_i}}_{a_i \mod m_i}% + \underbrace{\left( \sum_{k=i+1}^{r} a_k M_k^*\underbrace{M_k}_{\equiv 0 \mod m_i} \right)}_{0 \mod m_i}% - \equiv a_i \mod m_i,i= 1,2,\dots,r + \equiv a_i \mod m_i,i= 1,2,\cdots,r \end{align} Sind $x_1$ und $x_2$ beides Lösungend des Kongruenzensystems, d.h. \begin{align} - x_1 \equiv x_2 \equiv a_i mod m_i, i = 1,2,\dots,r + x_1 \equiv x_2 \equiv a_i mod m_i, i = 1,2,\cdots,r \end{align} So folgt daraus sofort +\begin{subequations} \begin{align} - m_i|x_1-x_2\forall i = 1,2,\dots,r &\Rightarrow \kgV(m_1,m_2m,\dots,r) = m_1m_2\dots M_r | x_1-x_2 \nonumber \\ - &\Rightarrow x_1 \equiv x_2 \mod m_1,m_2,\dots,m_r + m_i|x_1-x_2\forall i = 1,2,\cdots,r \Rightarrow \lcm (m_1,m_2m,\cdots,r) = m_1m_2\cdots M_r | x_1-x_2 \nonumber \\ + \Rightarrow x_1 \equiv x_2 \mod m_1,m_2,\dots,m_r \hfill \blacksquare \end{align} +\end{subequations} \subsection*{Ergänzung 2.??} \begin{align} - m = m_1m_2\dots m_r , ggT(m_i,m_j)=1 \forall i\neq j \Rightarrow \phi(m_1,m_2,\dots m_r) = \phi(m_1)\phi(m_2)\dots\phi(m_r) + m = m_1m_2\cdots m_r , \gcd(m_i,m_j)=1 \forall i\neq j \Rightarrow \phi(m_1,m_2,\cdots m_r) = \phi(m_1)\phi(m_2)\cdots\phi(m_r) \end{align} $ f:\N \to \N (k\in \N)$ d-h $\phi$-Funktion ist multiplikativ $n\mapsto n^k$ stark Multiplikativ \subsection*{Ergänzung 2.12} -$\phi(p^e) = pe - \# \{ kp | k = 1,2,\dots,p^{e-1} \} = p^e-p^{e-1} = p^e(1- \frac{1}{p})$ +$\phi(p^e) = pe - \# \{ kp | k = 1,2,\cdots,p^{e-1} \} = p^e-p^{e-1} = p^e(1- \frac{1}{p})$ \subsection*{Beweis 2.13} -Sei $\Z_m^* = \{ \bar a_1, \bar a_2, \dots, \bar a_{\phi(m)} \} $ die prime Restklassengurppe $\mod m$. Dann gilt für ein bel. $\bar a \in \Z_m^*$, dass +Sei $\Z_m^* = \{ \bar a_1, \bar a_2, \cdots, \bar a_{\phi(m)} \} $ die prime Restklassengurppe $\mod m$. Dann gilt für ein bel. $\bar a \in \Z_m^*$, dass \begin{align} - \bar a \Z_m^* = \{\bar a \bar a_1,\dots,\bar a \bar a_{\phi(m)}\} = \{ \bar{aa_1},\dots,\bar{aa_{\phi(m)}\} = \{ \bar a_1, \dots, \bar a_{\phi(m)} \} + \bar a \Z_m^* = \lbrace \bar a \bar a_{1},\cdots,\bar a \bar a_{\phi(m)} \rbrace = \{ \bar{aa_{1}},\cdots,\bar{aa_{\phi(m)}} \} = \{ \bar a_{1}, \cdots, \bar a_{\phi(m)} \} \end{align} -(denn wäre $\bar a \bar a_i = \bar a \bar a_j$, für $i\neq j$, so wäre daruas durch Mult mit $\bar a^{-1}$ sofort $\bar a_i = \bar a_j,$ \blitza) +(denn wäre $\bar a \bar a_i = \bar a \bar a_{j}$, für $i\neq j$, so wäre daraus durch Mult mit $\bar a^{-1}$ sofort $\bar a_{i} = \bar a_{j},$ \blitza) \subsection*{Beweis 2.16} Setzen $e:= \ord_m(a)$ \begin{enumerate} -\item Sei $a \equiv 1 \mod m und i = q\cdot e +r $ mit $0 \leq r\leq e$ ( Dann $i$ durch $e $ ist Quotienten $q$ und Rest $r$). +\item Sei $a \equiv 1 \mod m$ und $i = q\cdot e +r $ mit $0 \leq r\leq e$ ( Dann $i$ durch $e $ ist Quotienten $q$ und Rest $r$). Dann gilt: - $a^r\equiv a^{i-q\cdot e} \equiv a^i(a^e)^{-q} \equiv 1 mod m \Rightarrow r = \empty $ (Sonst Wiederspruch zur M eigenschaft von $e = \ord_m(a)$) - Umgekerht folgt aus $e|i$, als $ i= q\cdot e$ für ein $q \in \Z$, dass $a^i = a^{q \cdot e} = (a^e)^q \equiv 1 \mod m$ + $a^r\equiv a^{i-q\cdot e} \equiv a^i(a^e)^{-q} \equiv 1 mod m \Rightarrow r = \empty $ (Sonst Widerspruch zur M eigenschaft von $e = \ord_m(a)$) + Umgekehrt folgt aus $e|i$, als $ i= q\cdot e$ für ein $q \in \Z$, dass $a^i = a^{q \cdot e} = (a^e)^q \equiv 1 \mod m$ \item $a^i \equiv a^j \mod m \Leftrightarrow a^{i-j} \equiv 1 mod m \Leftrightarrow e|i-j \Leftrightarrow i \equiv j \mod e$ \item \end{enumerate} -$\Rightarrow \ord_(a^k) = \frac e {\ggT(k,e)} - - - +$\Rightarrow \ord_(a^k) = \frac e {\gcd(k,e)}$ + +\subsection*{VO 25.4.2012} +Betrachte $ax^{2}+bx+c \equiv 0 \mod m, \gcd(a,m)=1$, m ungerade. Dann gilt +\begin{equation} + x_{1,2}=\frac{-b \pm \sqrt{b^{2}-4ac}}{2a} +\end{equation} +Anzahl der Lösungen $x^{2} \equiv a \mod p$ ist $1+\left( \frac{a}{p} \right)$.\\ +Das Euler'sche Kriterium kann nicht verallgemeinert werden. \\ +@(4) stark multiplikativ im Zähler. \\ + +\begin{proof}[Beweis zu Satz $3.2$, Seite $10$] + \begin{itemize} + \item Folgt unmittelbar aus der Definition des Legendre-Symbols, wegen +\begin{equation} +a \equiv b \mod p \implies \left[ x^{2} \equiv a \mod p \textsl{ ist lösbar } \Leftrightarrow x^{b} \equiv b \mod p \textsl{ ist lösbar} \right] +\end{equation} +\item Ist $g$ eine Primitivwurzel $\mod p$, so ist dann $\lbrace g, g^{2}, \ldots, g^{p-1} \rbrace$ ein volles primes Restsystem $\mod p$ und daher sind + \begin{equation} + \underbrace{g^{2}, g^{4}, \ldots, g^{\left(\frac{p-1}{2}\right)^{2}}=g^{p-1}\equiv 1 \mod p}_{\textsl{versch. quadr. Reste}}, \underbrace{g^{\lbrace \frac{p+1}{2} \right)^{2}}}_{\textsl{Wiederholung}} \equiv g^{2} \mod p, \ldots + \end{equation} +daher sind $g^{2}, g^{4}, g^{p-1}$ alle quadratischen Reste, und die ungeraden Potenzen sind quadratische Nichtreste. +\item \hfill + \begin{itemize} + \item[1. Fall] $a$ ist quadratischer Rest $\implies \exists k \in \N: a \equiv g^{2k} \mod p$, woraus folgt dass + \begin{equation} + a^{\left( \frac{p-1}{2}\right)} \equiv \left( g^{2k} \right)^{\left( \frac{p-1}{2} \right)} = \left( g^{p-1} \right)^{k} \equiv 1 \mod p \equiv \left( \frac{a}{p} \right) \mod p + \end{equation} + \item[2. Fall] $a$ ist quadratischer Nichtrest, d.h. $\exists k \in \N: a \equiv g^{2k+1} \mod p$, dann gilt + \begin{equation} + a^{\left( \frac{p-1}{2} \right)} \equiv \left( g^{2k+1} \right)^{\left( \frac{p-1}{2} \right) } \equiv \underbrace{g^{(p-1)k}}_{\equiv 1 \mod p} \cdot \underbrace{g^{\left( \frac{p-1}{2} \right)}}_{\equiv -1 \mod p} \equiv 1 \equiv \lbrace \frac{a}{p} \rbrace \mod p + \end{equation} +Warum $\equiv -1 \mod p$? da $g^{\frac{p-1}{2}}$ Lösung von $x^{2} \equiv 1 \mod p$ ist $\implies$ + \end{itemize} + \end{itemize} +\end{proof} + +%Vorlesung 2.5.2012 +Zu Lucas-Folgen: sind Verallgemeinerung von Fibonacci-Folgen, sind wichtig für Primzahltests. \\ +\begin{proof}[Beweis zu Satz $4.2$, Seite $13$] + Wegen $(x-\alpha)(x-\beta)=x^{2} - Px + Q = 0$ erhält man durch einen Koeffizientenvergleich + \begin{equation} + P=\alpha+\beta, Q = \alpha \beta + \end{equation} +Durch unmittelbares Einsetzen erhält man: +\begin{subequations} +\begin{align} + U_{m}V_{n} - Q^{n}U_{m-n} = \frac{\alpha^{m}-\beta^{m}}{\alpha-\beta} \left( \alpha^{n} - \beta^{n} \right) - \left( \alpha \beta \right)^{n} \cdot \left( \frac{\alpha^{m-n} - \beta^{m-n}}{\alpha-\beta} \right)= \\ += \left( \frac{1}{\alpha - \beta} \right) \cdot \left( \alpha^{m+n} + \alpha^{m}\beta^{n} - \alpha^{n} \beta^{m} - \beta^{m+n} - \alpha^{m} \beta^{n} + \alpha^{n} \beta^{m} \right) = \\ += \frac{\alpha^{m+n} - \beta^{m+n}}{\alpha-\beta} \stackrel{\textsl{nach Def.}} = U_{m+n} +\end{align} +\end{subequations} +Genauso erhält man für die $V$-Folge +\begin{subequations} + \begin{align} + V_{m}V_{n} - Q^{n} V_{m-n} = \left( \alpha^{m} + \beta^{m}\right) \left( \alpha^{n} + \beta^{n} \right) - \left( \alpha \beta \right)^{n} \left( \alpha^{m-n} + \beta^{m-n} \right) = \\ += \alpha^{n+m} + \alpha^{m} \beta^{n} + \alpha^{n} \beta^{m} + \beta^{m+n} - \alpha^{m} \beta^{n} - \alpha^{n} \beta^{m} = \\ += \alpha^{n+m} + \beta^{m+n} \stackrel{\textsl{nach Def.}} = V_{m+m} + \end{align} +\end{subequations} +\end{proof} + +\begin{proof}[Beweis zu Folgerung $4.3$, Seite $13$] + Durch Einsetzen erhält man direkt + \begin{subequations} +\begin{align} + U_{0} = \frac{\alpha^{0} - \beta^{0}}{\alpha - \beta} = 0, \quad U_{1} = \frac{\alpha^{1} - \beta^{1}}{\alpha - \beta} = 1 \\ + V_{0} = \alpha^{0} + \beta^{0} = 2, \quad V_{1} = \alpha + \beta = P +\end{align} + \end{subequations} +Ferner folgt aus Satz $4.2$ mit $n=1$, dass +\begin{subequations} + \begin{align} + U_{m+1} = U_{m} \underbrace{V_{1}}_{=P} - Q^{1} U_{m-1} = PU_{m} - Q U_{m-1} \\ + V_{m+1} = V_{m} \underbrace{V_{1}}_{=P} - Q^{1} V_{m-1} = PV_{m} - Q V_{m-1}, + \end{align} +\end{subequations} +also dass, was zu zeigen war. +\end{proof} + +\begin{proof}[Beweis zu Folgerung $4.4$, Seite $13$] + Folgt wieder aus Satz $4.2$ wegen + \begin{subequations} + \begin{align} + U_{2n} = U_{n+n} = U_{n}V_{n} - Q^{n}U_{0} = U_{n} V_{n} \\ + U_{2n+1} = U_{(n+1)+n}=U_{n+1}V_{n} - Q^{n} U_{1} = U_{n+1}V_{n} - Q^{n} \\ +V_{2n} = V_{n+n} = V_{n} V_{n} - Q^{n} V_{0} = V_{2}^{2} - 2Q^{n} \\ +V_{2n+1} = V_{(n+1)+n} = V_{n+1}V_{n} - Q^{n} V_{1} = V_{n+1} V_{n} - PQ^{n} + \end{align} + \end{subequations} +$\implies$ leichte Berechenbarkeit der Lucasfolge. +\end{proof} + +\begin{bem} + Bsp zur Berechnung von $V_{100}$ mit Hilfe von $4.4$. +\end{bem} + +\begin{proof}[Beweis von Lemma $4.6$, Seite $13$] + Zur Invertierung von $2 \mod r$ für eine ungerade Zahl $r$: + \begin{equation} + \frac{1}{2} \mod r \equiv \underbrace{\frac{1+r}{2}}_{\in \Z} \mod r + \end{equation} +Man berechne: +\begin{subequations} + \begin{align} + \left( 2 \alpha \right)^{r} = \left( R+\sqrt{D} \right)^{r} = \\ += P^{r} \underbrace{\sum \limits_{k=1}^{r-1} \binom{r}{k} P^{k} \left( \sqrt{D} \right)^{r-k}}_{=:*} + \left( \sqrt(D) \right)^{r} = \\ +\equiv P^{r} + \left( \sqrt{D} \right)^{r} \equiv P^{r} + D^{\frac{r-1}{2}} \sqrt{D} \\ +\stackrel{\textsl{(**)}} \equiv P + \left( \frac{D}{r} \right) \equiv \begin{cases}P + \sqrt{D} \equiv 2 \alpha \mod r, \left( \frac{D}{r} \right) = 1 \\ P - \sqrt{D} \equiv 2 \beta \mod r, \left( \frac{D}{r} \right) = -1 \end{cases} + \end{align} +\end{subequations} +@(*): es gilt +\begin{equation} + \binom{r}{k} = \frac{r(r-1)\cdots(r-k+1)}{1\cdot 2 \cdots k} \equiv 0 \mod r +\end{equation} +denn angenommen $r \mid 1 \cdot 2 \cdots k \Rightarrow r \mid i$ für ein $i \in \lbrace 1,2, \ldots, r-1 \rbrace$, denn wenn eine Primzahl ein Produkt teilt, teilt sie einen Faktor, daher WS zu r ist Primzahl. \\ +@(**): Es gilt nach dem ``kleinen Fermat'': $P^{r} \equiv P \mod r$. Weiters gilt nach dem Euler'schen Kriterium: +\begin{equation} + D^{\frac{r-1}{2}} \equiv \left( \frac{D}{r} \right) \mod r +\end{equation} +Nun gilt aber +\begin{equation} + 2 \alpha^{r} \stackrel{\textsl{kl. Fermat}} \equiv 2^{r} \alpha^{r} = \left( 2 \alpha \right)^{r} \equiv \begin{cases} 2 \alpha \mod r, \left( \frac{D}{r} \right) = 1 \\ 2\beta \mod r, \left( \frac{D}{r} \right) = -1 \end{cases} +\end{equation} +Durch Kürzen durch $2$ ($2^{-1} \mod r$ existiert, da ja $\gcd(2,r)=1$ da $r$ ungerade nach Voraussetzung) folgt die Behauptung für $\alpha$. Beweis für $\beta$ analog. +\end{proof} + +\begin{bem} + Den zwei Fällen im vorigen Lemma liegt folgendes zugrunde: + \begin{itemize} + \item $\alpha, \beta \in \Z_{r} \implies$ ``kleiner Fermat'' $\implies$ fertig. + \item $\alpha \vee \beta \notin \Z_{r} \implies \alpha, \beta \in \Z_{r^{2}} \geq \Z_{r}$. $\Z_{r^{2}}$ hat genau einen nichtrivialen Automorphismus und es gilt + \begin{equation} + \mathcal{F} = \lbrace a + b \sqrt{D} \mid a,b \in \Z_{r} \rbrace + \end{equation} +Es folgt daher, dass die Funktionen $x \mapsto x^{r}$ und $a+b\sqrt{D} \mapsto a - b \sqrt{D}$ der gleiche Automorphismus sind. + \end{itemize} +\end{bem} + +\begin{proof}[Beweis zu Satz $4.7$, Seite $14$] +Es gilt $\left( \frac{D}{r} \right) \in \lbrace \pm 1 \rbrace$ wegen $\gcd(r,QD)=1$. + \begin{itemize} + \item[(1)] Sei zunächst $\lbrace \frac{D}{r} \rbrace = 1$. Dann gilt nach Lemma $4.6$ + \begin{equation} + \alpha^{r} \equiv \alpha \mod r, \beta^{r} \equiv \beta \mod r + \end{equation} +woraus durch Kürzen (beachte $\alpha \nequiv 0 \mod r, \beta \nequiv \mod r$, da sonst $Q = \alpha \beta \equiv 0 \mod r$ wäre, im Widerspruch zu $\gcd(r,QD)=1$) folgt $\alpha^{r-1} \equiv 1 \mod r, \beta^{r-1} \equiv 1 \mod r$. \\ +Es gilt daher +\begin{equation} + \left( \alpha - \beta \right) U_{r-1} = \alpha^{r-1} - \beta^{r-1} \equiv 1 -1 \equiv 0 \mod r +\end{equation} +Nun ist aber $\alpha - \beta = \sqrt{D} \nequiv 0 \mod r$ (weil $\sqrt{D} \mod r \Rightarrow D = (\sqrt{D})^{2} \equiv 0 \mod r \Rightarrow \gcd(r,QD) \neq 1 \blitza$), woraus durch Kürzen tatsächlich $U_{r-1} \equiv 0 \mod r$ folgt. \\ +Sei nun $\left( \frac{D}{r} \right) = -1$. Daher gilt nach Lemma $4.6$ +\begin{subequations} +\begin{align} +\alpha^{r} \equiv \beta \mod r \\ +\beta^{r} \equiv \alpha \mod r +\end{align} +\end{subequations} +Daher gilt +\begin{equation} +(\alpha - \beta)U_{r+1} = (\alpha^{r+1} - \beta^{r+1} )=alpha \alpha^{r} - \beta \beta^{r} \equiv \alpha \beta - \alpha \beta \equiv 0 \mod r, +\end{equation} +woraus wie vorhin durch Kürzen $U_{r+1} \equiv 0 \mod r$ folgt. +\item[(2)] Wegen $U_{2n} = U_{n}V_{n}$ nach $4.4$ gilt + \begin{subequations} + \begin{align} + U_{r-\left( \frac{D}{r} \right)} = U_{s \cdot 2^{t}} = U_{s2^{t-1}}V_{s2^{t-1}} = \cdots = \\ += U_{s}V_{s}V_{2s}V_{4s} \cdots V_{s2^{t-1}} \equiv 0 \mod r \textsl{ nach (1)} + \end{align} + \end{subequations} +$r$ ist Primzahl, daher teilt r einen Faktor, woraus die Behauptung direkt folgt. +\item[(3)] Es gilt + \begin{equation} + (\alpha - \beta)U_{r} \equiv \alpha^{r} - \beta^{r} \equiv \begin{cases}\alpha - \beta \mod r, \left( \frac{D}{r} \right)=1 \\ \beta - \alpha \mod r, \left( \frac{D}{r} \right) = -1 \end{cases} + \end{equation} +Woraus durch Kürzen folgt: $U_{r} \equiv \left( \frac{D}{r} \right) \mod r$. + \end{itemize} +\end{proof} \end{document} -- 2.47.3