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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 $S_{0} \neq \emptyset, S_{1} \neq \emptyset$. 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) \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 $s=ab$. +\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 Sei $p \in \P$ und $p \equiv 3 \mod 4$. Nach dem Satz von Gauß existiert eine Primitivwurzel $ g \mod p$. @@ -134,6 +163,7 @@ 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 Zuest ist Folgendes zu überprüfen, um von Legendre- bzw Jacobisymbolen sprechen zu können: @@ -174,6 +204,8 @@ Mit Jacobisymbol: = \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} + +\newpage \subsection*{$17$. Aufgabe} {\texttt{Man bestimme alle ungeraden Primzahlen $p$, für welche $10$ quadratischer Rest ist.}} \newline Aus $10 = 2 \cdot 5$ erhält man aus Satz 3.2, (4): @@ -197,6 +229,7 @@ Zusammen erhält man also, dass aus $p \equiv \pm 1 \mod 8 \land p \equiv \pm 1 \item Seien beide Faktoren gleich $-1$. Daher ist $p \equiv \pm 3 \mod 8 \land \left( p \equiv 2 \mod 5 \vee p \equiv 3 \mod 5 \right)$. \end{itemize} +\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 \begin{enumerate} -- 2.47.3