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z*m?dG7Y7#?_g~C#@%*)&UN!{#pDKCzFwS490|MFE{)?I4@&~?byFZBlL7bpJ7Z(u3 z#r>!Ff!sjuKM4c5xj6nDkMpIt_|FB*$<4t9`IE>C-M`iX=w&JVIp9m*zn%dvievke zL7o@I|G6-EI9{GU|5l0g|E_fK%P#pZ(Y#3JMeF~MRsFIw|IjD+xdD6ek_N9V{Rm#THmi)I4|8*fsuEh^xfA=y;Hjt?3%PL?K<`Cv!2a2=51M_fk r@d&@;01J};|D))T3ko7TIvF@PxjC4aB7-?W?Ci)iG-3+S;>iCW2ZA0_ diff --git a/UE/ue3.tex b/UE/ue3.tex index 2792b4c..70a8ae2 100644 --- a/UE/ue3.tex +++ b/UE/ue3.tex @@ -45,7 +45,8 @@ 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. +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} \alpha \equiv 1 \mod 4 \land \beta \equiv 1 \mod 4 \Rightarrow \alpha \beta \equiv 1 \mod 4, @@ -94,8 +95,8 @@ Nun erhält man aber aus \eqref{abschS} das Folgende \begin{subequations} \begin{align} u_{a} \cdot inv(b,\Z_{a}) \cdot b + u_{b} \cdot inv(a,\Z_{b}) \cdot a \mod ab \equiv \\ -\equiv \left( u_{a} \cdot inv(b,\Z_{a}) \cdot b \mod ab \right) +\nolimits_{\Z_{ab}} \left( u_{b}\cdot inv(a,\Z_{b}) \cdot a \mod ab \right) \label{gl0} -%\equiv \left( \left( u_{a} \cdot inv(b,\Z_{a}) \mod ab \right) \cdot \nolimits_{\Z_{ab}} \left( b \mod ab \right) + \nolimits_{\Z_{ab}} \left( \left( u_{b} \cdot inv(a,\Z_{b}) \mod ab \right) \cdot \nolimits_{\Z_{ab}} \left( a \mod ab \right) \equiv \label{gl0} \\ +\equiv \left( u_{a} \cdot inv(b,\Z_{a}) \cdot b \mod ab \right) +_{\Z_{ab}} \left( u_{b}\cdot inv(a,\Z_{b}) \cdot a \mod ab \right) \label{gl0} +%\equiv \left( \left( u_{a} \cdot inv(b,\Z_{a}) \mod ab \right) \cdot_{\Z_{ab}} \left( b \mod ab \right) +_{\Z_{ab}} \left( \left( u_{b} \cdot inv(a,\Z_{b}) \mod ab \right) \cdot_{\Z_{ab}} \left( a \mod ab \right) \equiv \label{gl0} \\ \end{align} \end{subequations} Betrachte nun folgende Darstellung für beliebiges $\gamma \in \N$: @@ -113,11 +114,16 @@ Setzt man speziell $\gamma := u_{a} \cdot inv(b,\Z_{a}), \delta := u_{b} \cdot i \end{align} \end{subequations} In \eqref{darst} hat man nun die gesuchte Darstellung als Element von S. Daher ist $S$ ein vollständiges Restsystem $\mod ab$. \\ -Klarerweise sind $S_{0}$ und $S_{1}$ disjunkt. Aus $0 \in S_{0}$, bzw $(b-1,1) \cong a(b-1)+b = ab-a+b>ab$ folgt, dass $S_{0} \neq \emptyset, S_{1} \neq \emptyset$. Daher: -\begin{equation}\label{S1} -\exists \delta \in \Z_{ab}: \left( \exists t \in S_{1}: \delta \equiv t \mod ab \right) \land \left( \forall \sigma \in S_{0}: \delta \nequiv \sigma \mod ab \right) +(Einfacher für Restsystem: betrachte folgendes System von Kongruenzen, welches nach dem Chinesischen Restsatz eine eindeutige Lösung $\mod ab$ hat: +\begin{equation} +\begin{cases}ax+by \equiv u_{a} \mod a \quad \Rightarrow by \equiv u_{a} \mod a \\ ax+by \equiv u_{b} \mod b \quad \Rightarrow ax \equiv u_{b} \mod b \end{cases} +\end{equation} +Aus $\gcd(a,b)=1$ folgt, dass die Inversen von $a$ und $b$ jeweils in $\Z_{b}$, bzw $\Z_{a}$ existieren, wodurch man das folgende äquivalente System von Kongruenzen erhält: +\begin{equation} +\begin{cases} y \equiv b^{-1} u_{b} \mod a \\ x \equiv u_{b} \mod b \end{cases} \end{equation} -Sei nun $n \in \N \implies \exists k \in \N: n = k \cdot (ab) + r \land 0 \leq r < ab$. Ist $k \geq 2$, so kann man n sicher mit Hilfe des vollen Restsystems und der Abschätzung \eqref{abschS} darstellen. Insbesondere folgt aus \eqref{S1}, dass $s=ab$. +Klarerweise sind $S_{0}$ und $S_{1}$ disjunkt. Aus $0 \in S_{0}$, bzw $(b-1,1) \cong a(b-1)+b = ab-a+b>ab$ 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} @@ -177,16 +183,15 @@ primep(1381); Mit Legendresymbol: \begin{subequations} \begin{align} -\left( \frac{700}{769} \right) = \left( \frac{2^{2} 5^{2} 7}{769} \right) = \left( \frac{7}{769} \right) = \\ -= \left( \frac{769}{7} \right) = \left( \frac{6}{7} \right) = \left( \frac{2 \cdot 3}{7} \right) = \left( \frac{2}{7} \right) \cdot \left( \frac{3}{7} \right) = \\ -= - \left( \frac{7}{3} \right) = - \left( \frac{1}{3} \right) = -1 +\left( \frac{700}{769} \right)_{L} = \left( \frac{2^{2} 5^{2} 7}{769} \right)_{L} = \left( \frac{7}{769} \right)_{L} = \\ += \left( \frac{769}{7} \right)_{L} = \left( \frac{6}{7} \right)_{L} = \left( \frac{-1}{7} \right)_{L} = -1 \end{align} \end{subequations} Mit Jacobisymbol: \begin{subequations} \begin{align} \left( \frac{700}{769} \right)_{J} = \left( \frac{2^{2} 175}{769} \right)_{J} = \left( \frac{175}{769} \right)_{J} = \left( \frac{769}{175} \right)_{J} = \left( \frac{69}{175} \right)_{J} = \left( \frac{175}{69} \right)_{J} = \left( \frac{37}{69} \right)_{J} = \\ -\left( \frac{69}{37} \right)_{J} = \left( \frac{32}{37} \right)_{J} = \left( \left( \frac{2}{37} \right)_{J} \right)^{5} = \left( \frac{2}{37} \right)_{J} = -1 +=\left( \frac{69}{37} \right)_{J} = \left( \frac{32}{37} \right)_{J} = \left( \left( \frac{2}{37} \right)_{J} \right)^{5} = \left( \frac{2}{37} \right)_{J} = -1 \end{align} \end{subequations} Mit Legendresymbol @@ -201,7 +206,7 @@ Mit Jacobisymbol: \begin{align} \left( \frac{1215}{1381} \right)_{J} = \left( \frac{1381}{1215} \right)_{J} = \left( \frac{166}{1215} \right)_{J} = \left( \frac{2 \cdot 83}{1215} \right)_{J} = \left( \frac{2}{1215} \right)_{J} \cdot \left( \frac{83}{1215} \right)_{J} = \\ \stackrel{1215 \equiv -1 \mod 8} = \left( \frac{83}{1215} \right)_{J} = \left( \frac{1215}{83} \right)_{J} = \left( \frac{53}{83} \right)_{J} = \left( \frac{83}{53} \right)_{J} = \left( \frac{30}{53} \right)_{J} = \left( \frac{2}{53} \right)_{J} \cdot \left( \frac{15}{53} \right)_{J} = \\ -= \left( \frac{53}{15} \right)_{J} = \left( \frac{8}{15} \right)_{J} = \left( \left( \frac{2}{15} \right)_{J} \right)^{4} = 1 += - \left( \frac{53}{15} \right)_{J} = - \left( \frac{8}{15} \right)_{J} = - \left( \left( \frac{2}{15} \right)_{J} \right)^{4} = -1 \end{align} \end{subequations} @@ -225,10 +230,16 @@ Eine ungerade Primzahl $p \neq 5$ ist kongruent zu $1,2,3,4 \mod p$. Es gilt \left( \frac{1}{5} \right) = 1, \left( \frac{2}{5} \right) = -1, \left( \frac{3}{5} \right) = -1, \left( \frac{4}{5} \right) = 1 \end{equation} 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 $\left( \frac{10}{p} \right) = 1$. -\item Seien beide Faktoren gleich $-1$. Daher ist $p \equiv \pm 3 \mod 8 \land \left( p \equiv 2 \mod 5 \vee p \equiv 3 \mod 5 \right)$. +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} +\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} +\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 -- 2.47.3