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\thispagestyle{plain} % \tableofcontents -\uebung{4}{23. Mai 2012} +\uebung{4}{30. Mai 2012} \aufgabe{19} -{Man zeige, dass sich jede positive ganze $n$ auf {\bf genau eine} Weise als Summe +{Man zeige, dass für ein beliebiges $a \in Z$ und eine beliebiges $n \in N^*$ gilt $n | \varphi(a n − 1)$ . +} + +\aufgabe{20} +{Welche Form haben die Bedingungen 1.-5. aus I, 4.7 für eine ungerade Primzahl $r$ für +die Lucasfolgen $U_n$ bzw. $V_n$ speziell mit $P = 1$ und $Q = -1$? Man überprüfe ferner ihre +Gültigkeit für $r = 41$. +} + +\aufgabe{21} +{Man überprüfe jeweils für die Basis $a=2$, ob $n=341$ den Fermattest, den Solovay- +Strassen-Test oder Miller-Rabin-Test besteht. Was wären ferner die Parameter $P$ und $Q$ in +Hinblick auf den Baillie-Wagstaff-Test? +} + +\aufgabe{22} +{Ist n das Produkt von zwei verschiedenen Primzahlen $p$ und $q$, so hat für jedes u die +Kongruenz $x^2 \equiv u^2 \mod n$ außer $\pm u \mod n$ in der Regel noch ein weiteres Wurzelpaar +$\pm v mod n$ als Lösung. Wie erhält man dieses speziell für $n= 437$ und $u=7$ ? (Hinweis: +Löse obige Kongruenz zunächst mod p bzw. q und „kombiniere“ dann diese Lösungen +mit Hilfe des Chinesischen Restsatzes zu Lösungen $\mod n$.) +} + +\aufgabe{23} +{Man zeige, dass eine Zahl $n$ der Form $n = pqr$, wobei $p,q,r$ Primzahlen der Form \begin{align} - n = \sum_{k=0}^m n_k 2^k \text{ mit } n_k \in \{-1,0,1\} + p = 6m + 1, q = 12m + 1, r = 18m +1 \text{ für ein } m \in N^* \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). +sind, stets eine Carmichaelzahl ist, d.h. die Bedingung $a^{n −1} \equiv 1 \mod n$ für alle zu $n$ +teilerfremden Basen $a \in Z$ erfüllt, obwohl sie zusammengesetzt ist. } -\aufgabe{20} -{} +\aufgabe{24} +{Man zeige: Besteht eine ungerade natürliche Zahl $n>1$ den Fermattest für die Basis $2$, +so besteht dann $2^n − 1$ sogar den Miller-Rabin Test für die Basis $2$. +} \end{document} -- 2.47.3