From 1b954d07b72a1999476cb80458b84bf054587795 Mon Sep 17 00:00:00 2001 From: couchot Date: Tue, 17 Dec 2013 09:35:11 +0100 Subject: [PATCH] ajout de partiels --- partiels/131217S1/partiel.aux | 16 + partiels/131217S1/partiel.log | 1144 +++++++++++++++++++++++++++++ partiels/131217S1/partiel.pdf | Bin 0 -> 105184 bytes partiels/131217S1/partiel.tex | 187 +++++ partiels/131217S1/partiel.tex.bak | 193 +++++ partiels/131217S1/partiel.tex~ | 193 +++++ partiels/131217S1/partiel.thm | 0 partiels/131217S1/symboles.sty | 1 + 8 files changed, 1734 insertions(+) create mode 100644 partiels/131217S1/partiel.aux create mode 100644 partiels/131217S1/partiel.log create mode 100644 partiels/131217S1/partiel.pdf create mode 100644 partiels/131217S1/partiel.tex create mode 100644 partiels/131217S1/partiel.tex.bak create mode 100644 partiels/131217S1/partiel.tex~ create mode 100644 partiels/131217S1/partiel.thm create mode 120000 partiels/131217S1/symboles.sty diff --git a/partiels/131217S1/partiel.aux b/partiels/131217S1/partiel.aux new file mode 100644 index 0000000..b5155d8 --- /dev/null +++ b/partiels/131217S1/partiel.aux @@ -0,0 +1,16 @@ +\relax +\catcode `:\active +\catcode `;\active +\catcode `!\active +\catcode `?\active +\select@language{french} +\@writefile{toc}{\select@language{french}} +\@writefile{lof}{\select@language{french}} +\@writefile{lot}{\select@language{french}} +\@writefile{toc}{\contentsline {section}{\numberline {1}(5 pts) Alg\IeC {\`e}bre de Boole }{1}} +\@writefile{toc}{\contentsline {section}{\numberline {2}(4 pts) Relations binaires entre ensembles}{1}} +\@writefile{toc}{\contentsline {section}{\numberline {3}(3 pts) D\IeC {\'e}monstration par r\IeC {\'e}currence}{1}} +\@writefile{toc}{\contentsline {section}{\numberline {4}(7 pts) Plus Grand Commun Diviseur}{2}} +\newlabel{itm:Q1}{{2}{2}}[None] +\newlabel{itm:Q2}{{3}{2}}[None] +\@writefile{toc}{\contentsline {section}{\numberline {5}(5 pts) Nombres (premiers ou pas) de Fermat}{2}} diff --git 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Partiel de mathématiques discrètes.\\ +Semestre 1 (Décembre 2013). Durée 1h30.} + +\date{} + +\begin{document} +\maketitle + +%\vspace{-3em} +\noindent Seule une fiche manuscrite de format A5 est autorisée. +Tous les moyens de communications sont interdits. +Tous les exercices de ce sujet sont indépendants les uns des autres. +Dans tout ce qui suit, on considère une algèbre de Boole +munie des opérateurs classiques ``+'', ``.'', +``$\overline{\begin{array}{l}~\end{array}}$'' et des variables booléennes +$a$, $b$, $c$, $d$. + + +%\vspace{-1em} +\section{(5 pts) Algèbre de Boole } +On cherche à analyser les codes binaires sur 4 bits. + + +\begin{enumerate} +\item Dans cette question, un code est correct s'il contient au +plus deux 1 consécutifs. Par exemple $1010$ est dit correct +(il ne contient pas de 1 consécutifs) tandis +$0111$ ne l'est pas (il contient trois 1 consécutifs). +Le but de cette question est de concevoir la fonction $F$ qui renvoie 1 +si le code de 4 bits $abcd$ est correct et 0 sinon. +\begin{enumerate} +\item Écrire la table de vérité de cet analyseur dans une table de Karnaugh. +\item Donner l'expression la plus simplifiée de la fonction $F$. +\end{enumerate} + +\item Dans cette question, un code est correct +s'il ne contient pas de 0 consécutifs (par exemple $1011$). +Le but de cette question est de concevoir la fonction $G$ qui renvoie 1 +si le code de 4 bits $abcd$ est correct et 0 sinon. +\begin{enumerate} +\item Écrire la table de vérité de cet analyseur dans une table de Karnaugh. +\item Donner l'expression la plus simplifiée de la fonction $G$. +\end{enumerate} + +\item Dans cette question, un code est correct +s'il vérifie les deux contraintes précédentes. +Le but de cette question est de concevoir la fonction $H$ qui renvoie 1 +si le code de 4 bits $abcd$ est correct et 0 sinon. +\begin{enumerate} +\item Sans utiliser de table de Karnaugh, résoudre algébrique de ce +problème. +\item Exprimer $H$ sous la forme d'une somme de trois monômes. +\item Comment pourrait-on vérifier le résultat sur les tables de Karnaugh +remplies aux deux premières questions. +\end{enumerate} +\end{enumerate} + +%\vspace{-1em} +\section{(4 pts) Relations binaires entre ensembles} + + +\begin{enumerate} +\item +Soit $P^*$ l'ensemble des nombres premiers impairs. On considère +la relation $\mathcal{R}$ entre deux éléments de $P^*$ définie par: +$$ +p \mathcal{R} q \textrm{ si et seulement si } +\dfrac{p+q}{2} \textrm{ appartient à $P^*$.} +$$ +Montrer qu'elle n'est pas transitive. + +\item On considère +la relation $\mathcal{S}$ entre deux éléments de $\Z$ définie par: +$$ +m \mathcal{S} n \textrm{ si et seulement si } +m^2 + m = n^2 + n. +$$ +\begin{enumerate} +\item Montrer que c'est une relation d'équivalence. +\item Construire la classe de 0, nommée $\dot{0}$. +\item Pour tout entier relatif $m$, construire la classe de $m$, + nommée $\dot{m}$. +\end{enumerate} +\end{enumerate} + +%\vspace{-1em} +\section{(3 pts) Démonstration par récurrence} +On considère la suite $(U_n)_{n \in \N}$ définie pour +tout $n \in \N$ par $U_n = 10^n.(9n-1)+1$. +\begin{enumerate} +\item Calculer $U_0$, $U_1$ et $U_2$. +\item Montrer que pour tout $n \in \N$ par +$U_{n+1} = U_n + 9^2.10^n.(n+1)$. +\item Montrer que $U_n$ est divisible par 81 pour tout $n \in \N$. +\end{enumerate} + +%\vspace{-1em} +\section{(7 pts) Plus Grand Commun Diviseur} +Dans ce qui suit, $n$ est un entier naturel strictement positif. +On considère trois suites +$(A_n)_{n \in \N^*}$, $(B_n)_{n \in \N^*}$ et $(d_n)_{n \in \N^*}$ +définies par +$A_n = n^2 - 2n +2$, +$B_n = n² +2n +2$ + et +$d_n= A_n \land B_n$. + +\begin{enumerate} +\item Calculer $A_1$, $A_2$, $A_3$, $A_4$, $B_1$, $B_2$, $B_3$, $B_4$, $d_1$, $d_2$, $d_3$ et $d_4$. Quelles intuitions avez vous à propos de $d_n$? +\item \label{itm:Q1} Montrer que si un entier naturel + $d$ divise à la fois $A_n$ et $n$, alors $d$ divise 2. +\item \label{itm:Q2} Montrer que si un entier naturel + $d$ divise à la fois $A_n$ et $B_n$, alors $d$ divise $4n$. +\item Dans cette question on suppose que $n$ est impair. +\begin{enumerate} +\item Montrer que $A_n$ et $B_n$ sont impairs. + En déduire que $d_n$ est impair. +\item En utilisant entre autre la question~\ref{itm:Q2}, montrer que $d_n$ divise $n$. +\item En utilisant entre autre la question~\ref{itm:Q1}, en déduire que $d_n$ divise 2, + puis que $A_n$ et $B_n$ sont premiers entre eux. +\end{enumerate} +\item On suppose maintenant que $n$ est pair. +\begin{enumerate} +\item Montrer que $d_n$ est pair. +\item Montrer que 4 ne divise pas $A_n$. En déduire la même chose pour $B_n$ +\item Montrer que $d_n$ peut s'écrire sous la forme $d_n = 2.p$, + où $p$ est impair. +\item En utilisant entre autre la question~\ref{itm:Q2}, en déduire + que $p$ divise $n$. +\item En utilisant entre autre la question~\ref{itm:Q1}, en déduire que $d_n=2$. +\end{enumerate} +\end{enumerate} + + + +%\vspace{-1em} +\section{(5 pts) Nombres (premiers ou pas) de Fermat} +Pour $p \in \N$, les nombres de Fermat sont ceux de la forme +$F_p = 2^{2^p}+1$. + + +\begin{enumerate} +\item +Montrer que si $n$ n'est pas une puissance de 2, alors +$2^n+1$ n'est pas un nombre premier. +On pourra se servir de l'égalité +$$ +x^n+ 1 = (x+1)(x^{n-1}-x^{n-2}+\ldots - x + 1) +$$ qui est est consiédérée comme admise +pour tout entier naturel $n$ impair. + +\item Le calcul de $F_5$ donne $F_5 = 4294967297 = 6700417\times641$. Que dire de $F_5$? Est-ce contradictoire avec la question précédente? + +\item Montrer que $F_{n+1} = (F_n - 1)^2 +1$. + +\item Montrer que $F_{n+1} - 2$ est divisible par $F_{n}$. + +\item Montrer $F_{n+1}$ et $F_{n}$ sont premiers entre eux. + +\end{enumerate} + + + +\end{document} diff --git a/partiels/131217S1/partiel.tex.bak b/partiels/131217S1/partiel.tex.bak new file mode 100644 index 0000000..ae07752 --- /dev/null +++ b/partiels/131217S1/partiel.tex.bak @@ -0,0 +1,193 @@ +\documentclass[11pt,a4paper,french]{article} +\usepackage[francais]{babel} +\usepackage[utf8]{inputenc} +\usepackage{a4} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{framed} +\usepackage[amsmath,thmmarks,thref,framed]{ntheorem} +\usepackage[dvips]{graphics} +\usepackage{epsfig} +\usepackage{calc} +\usepackage{tabls} +\usepackage{times} +\usepackage{tabularx} +\usepackage{textcomp} +\usepackage{pst-all} +\usepackage[a4paper]{geometry} + +\input{symboles.sty} +\geometry{hmargin=1cm, tmargin=-0.5cm, bmargin=1.5cm} + +\title{ +DUT d'informatique. Partiel de mathématiques discrètes.\\ +Semestre 1 (Décembre 2013). Durée 1h30.} + +\date{} + +\begin{document} +\maketitle + +\vspace{-3em} +\noindent Seule une fiche manuscrite de format A5 est autorisée. +Tous les moyens de communications sont interdits. +Tous les exercices de ce sujet sont indépendants les uns des autres. +Dans tout ce qui suit, on considère une algèbre de Boole +munie des opérateurs classiques ``+'', ``.'', +``$\overline{\begin{array}{l}~\end{array}}$'' et des variables booléennes +$a$, $b$, $c$, $d$. + + +\vspace{-1em} +\section{( pts) Algèbre de Boole } +On cherche à analyser les codes binaires sur 4 bits. + + +\begin{enumerate} +\item Dans cette question, un code est correct s'il contient au +plus deux 1 consécutifs. Par exemple $1010$ est dit correct +(il ne contient pas de 1 consécutifs) tandis +$0111$ ne l'est pas (il contient trois 1 consécutifs). +Le but de cette question est de concevoir la fonction $F$ qui renvoit 1 +si le code de 4 bits $abcd$ est correct et 0 sinon. +\begin{enumerate} +\item Ecrire la table de vérité de cet analyseur dans une table de Karnaugh. +\item Donner l'expression la plus simplifiée de la fonction $F$. +\end{enumerate} + +\begin{enumerate} +\item Dans cette question, un code est correct +s'il ne contient pas de 0 consécutifs (par exemple $1011$). +Le but de cette question est de concevoir la fonction $G$ qui renvoit 1 +si le code de 4 bits $abcd$ est correct et 0 sinon. +\begin{enumerate} +\item Ecrire la table de vérité de cet analyseur dans une table de Karnaugh. +\item Donner l'expression la plus simplifiée de la fonction $G$. +\end{enumerate} +\item Dans cette question, code est correct +s'il vérifie les deux contraintes précédentes. +Le but de cette question est de concevoir la fonction $H$ qui renvoit 1 +si le code de 4 bits $abcd$ est correct et 0 sinon. +\begin{enumerate} +\item Sans utiliser de table de Karnaugh, résoudre algébrique de ce +problème. +\item Exprimer $H$ sous la forme d'une somme de trois monomes. +\item Comment pourrait-on vérifier le résultat sur les tables de Karnaugh +remplies aux deux premières questions. +\end{enumerate} +\end{enumerate} + +\vspace{-1em} +\section{( pts) Relations binaires entre ensembles} + + +\begin{enumerate} +\item +Soit $P^*$ l'ensemble des nombres premiers supérieurs à 2. On considère +la relation $\mathcal{R}$ entre deux éléments de $P^*$ définie par: +$$ +p \mathcal{R} q \textrm{ si et seulement si } +\dfrac{p+q}{2} \textrm{ appartient à $P^*$.} +$$ +Montrer qu'elle n'est pas transitive. + +\item On considère +la relation $\mathcal{S}$ entre deux éléments de $\mathcal{Z}$ définie par: +$$ +m \mathcal{S} n \textrm{ si et seulement si } +m^2 + m = n^2 + n. +$$ +\begin{enumerate} +\item Montrer que c'est une relation d'équivalence. +\item Construire la classe de 0, nommée $\dot{0}$. +\item Pour tout entier relatif $m$, construire la classe de $m$, + nommée $\dot{m}$. +\end{enumerate} +\end{enumerate} + +\vspace{-1em} +\section{( pts) Démonstration par récurrence} +On considère la suite $(U_n)_{n \in \N}$ définie pour +tout $n \in \N$ par $U_n = 10^n.(9n-1)+1$. +\begin{enumerate} +\item Calculer $U_0$, $U_1$ et $U_2$. +\item Montrer que pour tout $n \in \N$ par +$U_{n+1} = U_n + 9^2.10^n.(n+1)$. +\item Montrer que $U_n$ est divisible par 81 pour tout $n \in \N$. +\end{enumerate} +\end{enumerate} + +\vspace{-1em} +\section{( pts) Plus Grand Commun Diviseur} +Dans ce qui suit, $n$ est un entier naturel strictement positif. +On considère trois suites +$(U_n)_{n \in \N^*}$, $(U_n)_{n \in \N^*}$ et $(U_n)_{n \in \N^*}$ +défines par +$A_n = n^2 - 2n +2$, +$B_n = n² +2n +2$ + et +$d_n= A_n \land B_n$. + +\begin{enumerate} +\item Calculer $A_1$, $A_2$, $A_3$, $A_4$, $B_1$, $B_2$, $B_3$, $B_4$, $d_1$, $d_2$, $d_3$ et $d_4$. Quelles intuitions avez vous à propos de $d_n$? +\item Montrer que si un entier naturel + $d$ divise à la fois $A_n$ et $n$, alors $d$ divise 2. +\item Montrer que si un entier naturel + $d$ divise à la fois $A_n$ et $B_n$, alors $d$ divise $4n$. +\item Dans cette question on suppose que $n$ est impair. +\begin{enumerate} +\item Montrer que $A_n$ et $B_n$ sont impairs. + En déduire que $d_n$ est impair. +\item Montrer que $d_n$ divise $n$. +\item En utilisant la question 1, en déduire que $d_n$ divise 2, + puis que $A_n$ et $B_n$ sont premiers entre eux. +\end{enumerate} +\item On suppose maintenant que $n$ est pair. +\begin{enumerate} +\item Montrer que $d_n$ est pair. +\item Montrer que 4 ne divise pas $A_n$. En déduire la même chose pour $B_n$ +\item Montrer que $d_n$ peut s'écrire sous la forme $d_n = 2.p$, + où $p$ est impair. +\item En utilisant entre autre la question 2., en décuire + que $p$ divise $n$. +\item En utilisant entre autre la question 1., en déduire que $d_n=2$. +\end{enumerate} +\end{enumerate} + + + +\vspace{-1em} +\section{( pts) Nombres (premiers ou pas) de Fermat} +Pour $p \in \N$, les nombres de Fermat sont ceux de la forme +$F_p = 2^{2^p}+1$. + + +\begin{enumerate} +\item Question préliminaire: montrer que les deux égalités suivantes sont établies: +\begin{enumerate} +\item $x^n- 1 = (x-1)(x^{n-1}+x^{n-2}+\ldots+ x + 1)$ pour tout entier naturel $n$ strictement positif. +\end{enumerate} + +\item +Montrer que si $n$ n'est pas une puissance de 2, alors +$2^n+1$ n'est pas un nombre premier. +On pourra se servir de l'égalité +$$ +x^n+ 1 = (x+1)(x^{n-1}-x^{n-2}+\ldots - x + 1) +$$ qui est établie +pour tout entier naturel $n$ impair. + +\item Le calcul de $F_5$ donne $F_5 = 4294967297 = 6700417\times641$. Que dire de $F_5$? Est-ce contradictoire avec la question précédente? + +\item Montrer que, $F_{n+1} = (F_n - 1)^2 +1$. + +\item Montrer que $F_{n+1} - 2$ est divisible par $F_{n}$. + +\item Montrer $F_{n+1}$ et $F_{n}$ sont premiers entre eux. + +\end{enumerate} + + + +\end{document} diff --git a/partiels/131217S1/partiel.tex~ b/partiels/131217S1/partiel.tex~ new file mode 100644 index 0000000..41af89a --- /dev/null +++ b/partiels/131217S1/partiel.tex~ @@ -0,0 +1,193 @@ +\documentclass[11pt,a4paper,french]{article} +\usepackage[francais]{babel} +\usepackage[utf8]{inputenc} +\usepackage{a4} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{framed} +\usepackage[amsmath,thmmarks,thref,framed]{ntheorem} +\usepackage[dvips]{graphics} +\usepackage{epsfig} +\usepackage{calc} +\usepackage{tabls} +\usepackage{times} +\usepackage{tabularx} +\usepackage{textcomp} +\usepackage{pst-all} +\usepackage[a4paper]{geometry} + +\input{symboles.sty} +\geometry{hmargin=1cm, tmargin=-0.5cm, bmargin=1.5cm} + +\title{ +DUT d'informatique. Partiel de mathématiques discrètes.\\ +Semestre 1 (Décembre 2013). Durée 1h30.} + +\date{} + +\begin{document} +\maketitle + +\vspace{-3em} +\noindent Seule une fiche manuscrite de format A5 est autorisée. +Tous les moyens de communications sont interdits. +Tous les exercices de ce sujet sont indépendants les uns des autres. +Dans tout ce qui suit, on considère une algèbre de Boole +munie des opérateurs classiques ``+'', ``.'', +``$\overline{\begin{array}{l}~\end{array}}$'' et des variables booléennes +$a$, $b$, $c$, $d$. + + +\vspace{-1em} +\section{( pts) Algèbre de Boole } +On cherche à analyser les codes binaires sur 4 bits. + + +\begin{enumerate} +\item Dans cette question, un code est correct s'il contient au +plus deux 1 consécutifs. Par exemple $1010$ est dit correct +(il ne contient pas de 1 consécutifs) tandis +$0111$ ne l'est pas (il contient trois 1 consécutifs). +Le but de cette question est de concevoir la fonction $F$ qui renvoie 1 +si le code de 4 bits $abcd$ est correct et 0 sinon. +\begin{enumerate} +\item Écrire la table de vérité de cet analyseur dans une table de Karnaugh. +\item Donner l'expression la plus simplifiée de la fonction $F$. +\end{enumerate} + +\begin{enumerate} +\item Dans cette question, un code est correct +s'il ne contient pas de 0 consécutifs (par exemple $1011$). +Le but de cette question est de concevoir la fonction $G$ qui renvoie 1 +si le code de 4 bits $abcd$ est correct et 0 sinon. +\begin{enumerate} +\item Écrire la table de vérité de cet analyseur dans une table de Karnaugh. +\item Donner l'expression la plus simplifiée de la fonction $G$. +\end{enumerate} +\item Dans cette question, code est correct +s'il vérifie les deux contraintes précédentes. +Le but de cette question est de concevoir la fonction $H$ qui renvoie 1 +si le code de 4 bits $abcd$ est correct et 0 sinon. +\begin{enumerate} +\item Sans utiliser de table de Karnaugh, résoudre algébrique de ce +problème. +\item Exprimer $H$ sous la forme d'une somme de trois monômes. +\item Comment pourrait-on vérifier le résultat sur les tables de Karnaugh +remplies aux deux premières questions. +\end{enumerate} +\end{enumerate} + +\vspace{-1em} +\section{( pts) Relations binaires entre ensembles} + + +\begin{enumerate} +\item +Soit $P^*$ l'ensemble des nombres premiers supérieurs à 2. On considère +la relation $\mathcal{R}$ entre deux éléments de $P^*$ définie par: +$$ +p \mathcal{R} q \textrm{ si et seulement si } +\dfrac{p+q}{2} \textrm{ appartient à $P^*$.} +$$ +Montrer qu'elle n'est pas transitive. + +\item On considère +la relation $\mathcal{S}$ entre deux éléments de $\mathcal{Z}$ définie par: +$$ +m \mathcal{S} n \textrm{ si et seulement si } +m^2 + m = n^2 + n. +$$ +\begin{enumerate} +\item Montrer que c'est une relation d'équivalence. +\item Construire la classe de 0, nommée $\dot{0}$. +\item Pour tout entier relatif $m$, construire la classe de $m$, + nommée $\dot{m}$. +\end{enumerate} +\end{enumerate} + +\vspace{-1em} +\section{( pts) Démonstration par récurrence} +On considère la suite $(U_n)_{n \in \N}$ définie pour +tout $n \in \N$ par $U_n = 10^n.(9n-1)+1$. +\begin{enumerate} +\item Calculer $U_0$, $U_1$ et $U_2$. +\item Montrer que pour tout $n \in \N$ par +$U_{n+1} = U_n + 9^2.10^n.(n+1)$. +\item Montrer que $U_n$ est divisible par 81 pour tout $n \in \N$. +\end{enumerate} +\end{enumerate} + +\vspace{-1em} +\section{( pts) Plus Grand Commun Diviseur} +Dans ce qui suit, $n$ est un entier naturel strictement positif. +On considère trois suites +$(U_n)_{n \in \N^*}$, $(U_n)_{n \in \N^*}$ et $(U_n)_{n \in \N^*}$ +définies par +$A_n = n^2 - 2n +2$, +$B_n = n² +2n +2$ + et +$d_n= A_n \land B_n$. + +\begin{enumerate} +\item Calculer $A_1$, $A_2$, $A_3$, $A_4$, $B_1$, $B_2$, $B_3$, $B_4$, $d_1$, $d_2$, $d_3$ et $d_4$. Quelles intuitions avez vous à propos de $d_n$? +\item Montrer que si un entier naturel + $d$ divise à la fois $A_n$ et $n$, alors $d$ divise 2. +\item Montrer que si un entier naturel + $d$ divise à la fois $A_n$ et $B_n$, alors $d$ divise $4n$. +\item Dans cette question on suppose que $n$ est impair. +\begin{enumerate} +\item Montrer que $A_n$ et $B_n$ sont impairs. + En déduire que $d_n$ est impair. +\item Montrer que $d_n$ divise $n$. +\item En utilisant la question 1, en déduire que $d_n$ divise 2, + puis que $A_n$ et $B_n$ sont premiers entre eux. +\end{enumerate} +\item On suppose maintenant que $n$ est pair. +\begin{enumerate} +\item Montrer que $d_n$ est pair. +\item Montrer que 4 ne divise pas $A_n$. En déduire la même chose pour $B_n$ +\item Montrer que $d_n$ peut s'écrire sous la forme $d_n = 2.p$, + où $p$ est impair. +\item En utilisant entre autre la question 2., en déduire + que $p$ divise $n$. +\item En utilisant entre autre la question 1., en déduire que $d_n=2$. +\end{enumerate} +\end{enumerate} + + + +\vspace{-1em} +\section{( pts) Nombres (premiers ou pas) de Fermat} +Pour $p \in \N$, les nombres de Fermat sont ceux de la forme +$F_p = 2^{2^p}+1$. + + +\begin{enumerate} +\item Question préliminaire: montrer que les deux égalités suivantes sont établies: +\begin{enumerate} +\item $x^n- 1 = (x-1)(x^{n-1}+x^{n-2}+\ldots+ x + 1)$ pour tout entier naturel $n$ strictement positif. +\end{enumerate} + +\item +Montrer que si $n$ n'est pas une puissance de 2, alors +$2^n+1$ n'est pas un nombre premier. +On pourra se servir de l'égalité +$$ +x^n+ 1 = (x+1)(x^{n-1}-x^{n-2}+\ldots - x + 1) +$$ qui est établie +pour tout entier naturel $n$ impair. + +\item Le calcul de $F_5$ donne $F_5 = 4294967297 = 6700417\times641$. Que dire de $F_5$? Est-ce contradictoire avec la question précédente? + +\item Montrer que, $F_{n+1} = (F_n - 1)^2 +1$. + +\item Montrer que $F_{n+1} - 2$ est divisible par $F_{n}$. + +\item Montrer $F_{n+1}$ et $F_{n}$ sont premiers entre eux. + +\end{enumerate} + + + +\end{document} diff --git a/partiels/131217S1/partiel.thm b/partiels/131217S1/partiel.thm new file mode 100644 index 0000000..e69de29 diff --git a/partiels/131217S1/symboles.sty b/partiels/131217S1/symboles.sty new file mode 120000 index 0000000..a4e1afa --- /dev/null +++ b/partiels/131217S1/symboles.sty @@ -0,0 +1 @@ +/home/couchot/enseignement/stats/partiels/111103/symboles.sty \ No newline at end of file -- 2.39.5