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zP}8nhsNK-Qj~qZ1BTY%-!0Sj~agk)!%T;=8C4hE0euPc=kGvOrU>^P@>`z5K36j`Zpk%mu_=i+12wH3e|3&G~(> zB|NaH|7G(hw4{@L)vBZ!j^(|?=TKjDmCG2Ccxw9504M-}0RKxQQCk9~o*W8f1pMn_bY#T&_XhwF3>Hdc zl|*5K3kib%`6W;g0)jlo0IOhq?$302ubaZbIN*UGVnFa?PoN;sqbLZ01s;SS01y%cKpzo<1s)Ou zfdKG-z5a3~C5SZY@C;&k)z++vQ5d4v#AXo_UAcR1$fZ#u@{+H2U5af|A1c5)) zg+K%!Bqm~R0{%M};Qx0{BGThg(g;9-k31FxfP@|<9OAA&A{KVA1!7eumI@44TS&Kl8DfSz#iOBpdcdt{|pxZf&D*LOfdA}6#G}E z3q2YS3WGezBMgL~5Ku6&TMk9sR0s@{0?EpO5DW;HrDdf-AeaEM)CMzH<3lnD) k{r`#>F^Y@hxVf9Tx_i4?SmFQ$VThFxhmB26RUYU606lzt@Bjb+ diff --git a/UE/ue5.tex b/UE/ue5.tex index ccfbf08..8397bec 100644 --- a/UE/ue5.tex +++ b/UE/ue5.tex @@ -11,7 +11,7 @@ \aufgabe{25} {$p=2^{43112609}-1$ ist die zur Zeit größte bekannte Primzahl. Wie kann man ihre Stellenanzahl am einfachsten berechnen und was sind ihre $3$ Endziffern?} -Die Anzahl der Stellen einer Zahl können mit $\log_{10}$ berechnet werden. Daher ist $\log_{10} 2^{43112609} = 43112609\cdot\log_{10} 2 \approx 12978188.5003$\\ +Die Anzahl der Stellen einer Zahl können mit $\log_{10}$ berechnet werden. Daher ist $\lfloor \log_{10} 2^{43112609} \rfloor +1 = \lfloor 43112609\cdot\log_{10} 2 \rfloor +1 = 12978189$\\ Für die letzten 3 Stellen berechnet man nun: \begin{align} 2^{43112609} \in (\Z_{1000}, * \mod 1000, 1) \text{, d.h. }x=2, k=43112609 @@ -96,11 +96,11 @@ $F_{3}=2^{2^{3}}+1=2^{8}+1=257$. Daher: 3 \\ \rightsquigarrow 9 \mod F_{3}\\ \rightsquigarrow 81 \mod F_{3} \\ -\rightsquigarrow 81\cdot 81=25*257+161 \Rightarrow A=25, B=161 \rightsquigarrow B-A \equiv 161-25=136 \mod F_{3}\\ -\rightsquigarrow 136*136=72*257+64, A=72,B=64 \rightsquigarrow B-A \equiv 64-72 \equiv -8 \equiv 249 \mod F_{3}\\ -\rightsquigarrow 249*249=242*257+49, A=242, B=49 \rightsquigarrow B-A \equiv 49-242 \equiv -193 \equiv 64 \mod F_{3} \\ -\rightsquigarrow 64*64=16*257, A=16, B=0 \rightsquigarrow B-A \equiv -16 \equiv 241 \mod F_{3}\\ -\rightsquigarrow 241*241=226*257+225, A=226, B=225 \rightsquigarrow B-A \equiv 225-226 \equiv -1 \mod F_{3} +\rightsquigarrow 81\cdot 81=25*256+161 \Rightarrow A=25, B=161 \rightsquigarrow B-A \equiv 161-25=136 \mod F_{3}\\ +\rightsquigarrow 136*136=72*256+64, A=72,B=64 \rightsquigarrow B-A \equiv 64-72 \equiv -8 \equiv 249 \mod F_{3}\\ +\rightsquigarrow 249*249=242*256+49, A=242, B=49 \rightsquigarrow B-A \equiv 49-242 \equiv -193 \equiv 64 \mod F_{3} \\ +\rightsquigarrow 64*64=16*256, A=16, B=0 \rightsquigarrow B-A \equiv -16 \equiv 241 \mod F_{3}\\ +\rightsquigarrow 241*241=226*256+225, A=226, B=225 \rightsquigarrow B-A \equiv 225-226 \equiv -1 \mod F_{3} \end{align} \end{subequations} @@ -135,17 +135,24 @@ $Q=1$ zwingend aus Rekursionsformel aus der Theorie der Lucas-Folgen. P=4? Dann \begin{equation} D=P^{2}-4Q=4^{2}-4\cdot 1=16-4=12 \neq 0 \end{equation} -Die sich aus dem Lucas-Lehmer-Test ergebende Folge ist daher die Folge $V_{2j}, j \in \N^{*}$. +Wissen +\begin{subequations} +\begin{align} + V_{1}=s_{1} \\ + k \in \N: k>1: s_{k}=V_{2k} +\end{align} +\end{subequations} + \aufgabe{30} {Man wende den Lucas-Lehmer Test auf die Mersenn'sche Zahl $M_{7}=2^{7}-1=127$ an, wobei insbesondere die Reduktion $\mod M_{7}$ in der Weise durchzuführen sind, dass man die Quotienten und Reste bei der Division durch $2^{7}$ in geeigneter Weise verwendet.} $p=7$ ist eine ungerade Primzahl, daher ist der Lucas-Lehmer-Test anwendbar, setze $s_{1}:=4$. \begin{subequations} \begin{align} s_{2}=16-2=14\\ - s_{3}=14^{2}-2=194=1\cdot 127+66 , A=1, B=66 ,A+B=1+6=7 \equiv 67 \mod M_{7} \\ - s_{4}=67^{2}-2=4487=35\cdot 127+7, A=35, B=7, A+B=35+7=42 \equiv 42 \mod M_{7} \\ -s_{5}=42^{2}-2=1762=13\cdot 127+98 , A=13, B=98 , A+B=13+98=111 \equiv 111 \mod M_{7} \\ -s_{6}=111^{2}-2=12319=96\cdot 127+31, A=96, B=31, A+B=96+31=127 \equiv 0 \mod M_{7} + s_{3}=14^{2}-2=194=1\cdot 128+66 , A=1, B=66 ,A+B=1+6=7 \equiv 67 \mod M_{7} \\ + s_{4}=67^{2}-2=4487=35\cdot 128+7, A=35, B=7, A+B=35+7=42 \equiv 42 \mod M_{7} \\ +s_{5}=42^{2}-2=1762=13\cdot 128+98 , A=13, B=98 , A+B=13+98=111 \equiv 111 \mod M_{7} \\ +s_{6}=111^{2}-2=12319=96\cdot 128+31, A=96, B=31, A+B=96+31=127 \equiv 0 \mod M_{7} \end{align} \end{subequations} \end{document} \ No newline at end of file -- 2.47.3