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\def\ZZ{{\mathbb{Z}}}
\def\CC{{\mathbb{C}}}
\def\NN{{\mathbb{N}}}
\def\KK{{\mathbb{K}}}
\def\RR{{\mathbb{R}}}
\def\cM{\mathcal{M}}
\def\ben{\begin{enumerate}}
\def\een{\end{enumerate}}
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\def\ZZ{{\mathbb{Z}}}
\def\CC{{\mathbb{C}}}
\def\NN{{\mathbb{N}}}
\def\KK{{\mathbb{K}}}
\def\RR{{\mathbb{R}}}
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\newtheorem{theorem}{Teorema}[section]
\newtheorem{definition}[theorem]{Defini\c{c}\~{a}o}
\newtheorem{lemma}[theorem]{Lema}
\newtheorem{proposition}[theorem]{Proposi\c{c}\~{a}o}
\newtheorem{corollary}[theorem]{Corol\'{a}rio}
\newtheorem{exemplo}[theorem]{Exemplo}
\newtheorem{obs}[theorem]{Observa\c{c}\~{a}o}
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\begin{document}
\pagestyle{empty}
\noindent
{\sf Universidade do Minho}
\hfill {\sf L.E.M., L.M.C.C. -- 4$^\circ$ ano}

\noindent{\small {\sf Departamento de Matem\'{a}tica}}
\hfill {\sf Mat. (Esp. Ensino), L.C.C. -- 3$^\circ$ ano}


\vskip 0.3cm \centerline{{\Large{\sf Teoria de N\'umeros Computacional}}}


\vskip 0.3cm
 \noindent\rule{1.5cm}{0.2mm}
 \raisebox{-0.5ex}[0cm][0cm]{ {\ \small{\sf{folha v}}}} \hrulefill \rule{1.5cm}{0.2mm}
 \raisebox{-0.5ex}[0cm][0cm]{ {\ \small{\sf{$2^\circ$ semestre, 2006/2007}}}} \rule{1.5cm}{0.2mm}

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\ben

\item Mostre que 91 � um pseudoprimo de base 3.

\item Mostre que 45 � um pseudoprimo de bases 17 e 19.

\item Mostre que $n=161038=2\cdot 73 \cdot 1103$ satisfaz a congru�ncoa $2^n\equiv 2 \mod n$. O inteiro $n$ � de facto o menor pseudoprimo par de base 2.

\item Mostre que se $n$ � um pseudoprimo �mpar de base $a$ ent�o $n$ � um pseudoprimo de base $n-a$.

\item Mostre que se $n$ � um pseudoprimo de bases $a$ e $b$ ent�o � um pseudoprimo de base $ab$.

\item Mostre que se $n$ � um pseudoprimo de $a$ mas n�o o � de base $b$, com $(a,n)=(b,n)=1$, ent�o $n$ n�o � pseudoprimo de base $ab$.

\item Mostre que 25 � um pseudoprimo forte de base 7.

\item Mostre, usando o gp, que 1397 � um pseudoprimo de base 2 mas que n�o � um pseudoprimo forte de base 2.

\item Usando o gp, mostre que 1373653 � um pseudoprimo forte de bases 2 e 3.

\item Usando o gp, mostre que 25326001 � um pseudoprimo forte de bases 2, 3 e 5.

\een

\end{document}
