From d2bc1626a47607ffbd02393f61be5a5965fcc549 Mon Sep 17 00:00:00 2001 From: user0 Date: Tue, 24 Apr 2012 23:35:23 +0200 Subject: [PATCH] On branch master Changes to be committed: MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit modified: ue2.pdf modified: ue2.tex (Bsp 12 hinzugefügt) --- UE/ue2.pdf | Bin 147894 -> 175185 bytes UE/ue2.tex | 109 ++++++++++++++++++++++++++++++++++++++++++++++++++++- 2 files changed, 107 insertions(+), 2 deletions(-) diff --git a/UE/ue2.pdf b/UE/ue2.pdf index fd1038c01691bab2e5b92af89fe33b973c360649..1e47719027cc79ec1c6a577db99c400b268a1217 100644 GIT binary patch delta 142326 zcmb5Vby$^M(>O|ZceCk+O?OClcO#9IgfurLUD6>b4GPk!gp_nQ2uP=-^x5e1e(&@B z&hd)CaX$$QP((HM$6F^^I+hz-lj1xkOn3(Wu=85y{c4q^w+RcACMM$Ru? zTlDD9N99;QR>JJ%>umT&hR=6;P$8z|A+FsZ=3PdZ!b!@g# zgOMhsWww<`bqc9jv8NcBQNez)=I^ETiNsIkVt*ZX<7pbt;o_u$WErUyXQYBq7K*#y zO~fV6drLBrKg^oRhtO_CBENxAh!4(~{9yYiDPgG%yoSj`8K%AWgsVknkk@Zqx1{(< zgLBRz9V=jQYiUPo_3DI#eUNze#SJuPS?J4r+YHku#cvYdpgDJS(NL7rjb3TULoF{i zx#+Ggo?nUpsA~uByx<(Q@lX1+7yoK>aO2)j} zMWhB3mt-PNH)b|{xdv0m6!u+CyabG~s;zqEpxvZ=YJ|yhi>+Ty#t4W>#-a6-E))`y zCB+z`IXCb8?c%mRta%VRGgjAzzNfIrW(qFC1yFBc_cd;Oz!JmTA`}EA%d;|Hk1s`} z&gv%3=K02SUz%2$nFy!E;zWOFy{#lPqPw6xLY_Mpi-^zg)BDm4p9Bd;)x{)<#eq%2 zT@Uo#1^(VJ?85+({yXzNu!w$!r^y2c6t^4m@Hato`md@P*xi)mwr5boR_NSkUtzd! 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zh~H=BKU3y^MaKaIaQ|(X1Nd(}E> 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. \\ +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. \\ Nun gilt aber die Äquivalenz \begin{equation} 1 \leq x \leq n \land \gcd(x,n)=d \Leftrightarrow 1 \leq x/d \leq n \land \gcd(x/d,n/d)=1 @@ -181,6 +181,111 @@ Lösgunen für $k=1,2,3,4$: \item $\Z_5,\Z_8,\Z_{10},\Z_{12}$ \end{enumerate} - +\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) +\end{equation} +@Endlichkeit: \\ +Sei $nextprime(k)$ jene Funktion, die die nächste echt größere Primzahles als $k$ ausgibt. So eine Primzahl gibt es immer, da es erstens unendlich viele gibt und sie zweitens wohlgeordnet sind. Daher kann man Schranken bei den Primzahlen angeben, ab derer die Primzahlen nicht mehr in Formal \eqref{phifkt} vorkommen können. Die Grundlage dafür bildet die folgende Abschätzung +\begin{equation} + nextprime(nextprime(k))^{\nu-1} \cdot (nextprime(nextprime(k))-1) \geq nextprime(k) > k, +\end{equation} +d.h. es können nur endlich viele Primzahlen in diesem Produkt vorkommen. Da die natürlichen Zahlen archimedisch angeordnet sind, und weiters alle Potenzen $x^{k}$ für $x>1$ für $k \rightarrow \infty$ gegen $\infty$ konvergieren (``bestimmt divergieren''), kann man auch für die Exponenten $\nu(p)$ Schranken angeben, sodass nur mehr alle kleineren in Frage kommen. Insgesamt ergibt sich daraus eine endliche Obermenge der Lösungen. \\ +Sei $k\geq 2$: angenommen man hat eine Lösung $x_{0} \in \N$ mit $\varphi(x_{0}) = k$, so erhält man nach \eqref{phifkt} eine Produktdarstellung von $k$. Alle möglichen Produktdarstellungen von $k$ sind gegeben durch assoziatives Umklammern in der Primfaktordarstellung von $k$. Da jede Zahl nur endlich viele Primfaktoren hat, gibt es nur endlich viele Produktdarstellungen von $k$, daher ist das Urbild $\varphi^{-1}(k)$ endlich. \\ +Ist $k=1$, und $y \in \N \land y \geq 2 \land \varphi(y)=1$, so gilt nach \eqref{phifkt} +\begin{equation}\label{fuer1} + \varphi(y)=\prod \limits_{p \in \P, \nu_{p}(y) > 0} p^{\nu_{p}(y)-1} \cdot (p-1)=1 +\end{equation} +Auf der rechten Seite von \eqref{fuer1} steht ein Produkt von natürlichen Zahlen, welches gleich $1$ ist, daher müssen alle Faktoren gleich $1$ sein. Daraus erhält man nun sofort $y=2$. Weiters ist $y=1$ auch eine Lösung. Insgesamt ist hier das vollständige Urbild $\varphi^{-1}(1)$ auch endlich und zweielementig. \\ +Für die Berechnung der restlichen Urbilder sei im Folgenden immer +\begin{equation} + \tilde{P} := \lbrace p \mid p \in \P \land \nu_{p}(x) > 0 \rbrace +\end{equation} +\begin{enumerate} +\item[$k=2$] Es liegt folgende Darstellung vor: + \begin{equation}\label{prod2} + \prod \limits_{\tilde{p} \in \tilde{P}} \tilde{p}^{\nu_{p}-1} \cdot (\tilde{p} -1) = 2. + \end{equation} +Nun gilt +\begin{equation} + \forall p \in \P, p > 3: p^{\nu_{p}-1} \cdot (p-1) \geq p-1 \geq 3, +\end{equation} +daher können Primzahlen echt größer als $3$ in \eqref{prod2} nicht als Index auftreten, man erhält also $\tilde{P} \subseteq \lbrace 2,3 \rbrace$. Man hat hier drei Fälle zu unterscheiden: +\begin{enumerate} +\item[$\tilde{P} = \lbrace 2,3 \rbrace$] Dann gilt + \begin{equation} + 2^{\nu_{2}-1}\cdot3^{\nu_{3}-1}\cdot 2 = 2, + \end{equation} +woraus zwingend folgt, dass $\nu_{2}=1,\nu_{3}=1$. Daher ist $6$ eine Lösung. +\item[$\tilde{P}=\lbrace 3 \rbrace$] Dann gilt + \begin{equation} + 3^{\nu_{3}-1} \cdot 2 = 2, + \end{equation} +woraus zwingend folgt, dass $\nu_{3}=1$. Daher ist $3$ eine Lösung. +\item[$\tilde{P}=\lbrace 2 \rbrace$] Dann gilt + \begin{equation} + 2^{\nu_{2}-1}=2, + \end{equation} +woraus zwingend folgt, dass $\nu_{2}=2$. Daher ist $4$ eine Lösung. +\item Andere Lösungen gibt es nicht. +\end{enumerate} +\item[$k=3$] Es ist also folgende Darstellung vorgelegt + \begin{equation}\label{prod3} +\prod \limits_{\tilde{p} \in \tilde{P}} \tilde{p}^{\nu_{p}-1} \cdot (\tilde{p} -1) = 3. + \end{equation} +Nun gilt aber +\begin{equation} + \forall p \in \P: p > 5: p^{\nu-1}\cdot (p-1) \geq p-1 \geq 5, +\end{equation} +daher können keine Primzahlen $>5$ in \eqref{prod3} vorkommen und es folgt $\tilde{P} \subseteq \lbrace 2,3 \rbrace$, da $5$ auch nicht vorkommen kann. +\begin{enumerate} +\item[$\tilde{P} = \lbrace 2 \rbrace$] Dann gilt + \begin{equation} + 2^{\nu_{2}-1}=3 \Leftrightarrow v_{2} = \log_{2}(3) + 1 \notin \N + \end{equation} +\item[$\tilde{P} = \lbrace 3 \rbrace$] Dann gilt + \begin{equation} + 3^{\nu_{3}-1} \cdot 2 = 3 \Leftrightarrow 3^{\nu_{3}-1} = \frac{3}{2}, \nexists Lsg \in \N + \end{equation} +\item[$\tilde{P}=\lbrace2,3\rbrace$] Dann gilt + \begin{equation} + 2^{\nu_{2}-1}\cdot 3^{\nu_{3}-1}\cdot 2 = 3 \Leftrightarrow 2^{\nu_{2}-1}3^{\nu_{3}-1} = \frac{3}{2}, + \end{equation} +hat keine Lösung in $\N \times \N$ für $\nu_{(\cdot)}$, da Produkt von natürlichen Zahlen wieder eine natürliche Zahl ergibt. \\ +Es gibt keine Lösung. +\end{enumerate} +\item[$k=4$] Es ist also die folgende Darstellung vorgelegt +\begin{equation} +\prod \limits_{\tilde{p} \in \tilde{P}} \tilde{p}^{\nu_{p}-1} \cdot (\tilde{p} -1) = 4. +\end{equation} +Es können keine Primzahlen größer als $7$ auftreten, daher $\tilde{P} \subseteq \lbrace 2,3,5 \rbrace$. +\begin{enumerate} +\item [$\tilde{P} = \lbrace 5 \rbrace$] Es ist $5$ eine Lösung. +\item [$\tilde{P} = \lbrace 3 \rbrace$] Keine Lösung, da $\log_{3}(4) \notin \N$. +\item [$\tilde{P} = \lbrace 2 \rbrace$] Eindeutige Lösung ist $8$. +\item [$\tilde{P} = \lbrace 2,3 \rbrace$] Dann gilt + \begin{equation} + 2^{\nu_{2}-1} 3^{\nu_{3}-1} 2 = 4 \Leftrightarrow 2^{\nu_{2}-1}3^{\nu_{3}-1}=2 + \end{equation} +Da 2 Primzahl ist, ist muss einer der Faktoren trivial sein, daher ist $v_{2}=2,v_{3}=1$ und $12$ eine Lösung. +\item [$\tilde{P} = \lbrace 2,5 \lbrace$] Dann gilt + \begin{equation} + 2^{\nu_{2}-1}5^{\nu_{5}-1}4=4 + \end{equation} +daher gilt $\nu_{2} = 1, \nu_{5}=1$ und $10$ ist eine Lösung. +\item [$\tilde{P}=\lbrace 3,5 \rbrace$] Dann gilt + \begin{equation} + 3^{\nu_{3}-1}\cdot 2 \cdot 5^{\nu_{5}-1}\cdot 4 = 4 + \end{equation} +daher keine Lösung. +\item [$\tilde{P} = \lbrace 2,3,5 \rbrace$] Dann gilt + \begin{equation} + 2^{\nu_{2}-1} 3^{\nu_{3}-1}2 5^{\nu_{5}-1} 4 = 4, + \end{equation} +daher keine Lösung. +\end{enumerate} +\end{enumerate} \end{document} -- 2.47.3