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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}}}
\def\cMmn{\mathcal{M}_{m\times n}}
\def\cMrp{\mathcal{M}_{r\times p}}
\def\cMrn{\mathcal{M}_{r\times n}}
\def\cMm{\mathcal{M}_{m}}
\def\cMn{\mathcal{M}_{n}}
\def\cMp{\mathcal{M}_{p}}
\def\cMq{\mathcal{M}_{q}}
\def\cMk{\mathcal{M}_{k}}
\def\cM{\mathcal{M}}
\def\cLT{\mathcal{LT}}
\def\cLTn{\mathcal{LT}_n}
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\def\parK{\left( \KK \right)}
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\def\iff{\Leftrightarrow}
\def\imply{\Rightarrow}
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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 iii}}}} \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 Pretende-se determinar $\displaystyle\frac{\pi(x)}{\frac{x}{\log x}}$, para alguns valores de $x$.
\ben
\item Escreva uma função que tenha como argumento $n$ e devolva $\pi(n), \frac{x}{\log x}$ e $\displaystyle\frac{\pi(x)}{\frac{x}{\log x}}$.
\item Use o comando \verb#intnum# para aproximar $\pi(x)$ à custa de $\int_2 ^x \frac{dt}{\log t}.$
\item Use o comando \verb#ploth# para esboçar os gráficos de $\pi(x),Li(x)$ e de $\frac{x}{\log x}$.
\een

\item Use a factorização de Fermat para encontrar uma factorização de
\ben
\item 143
\item 43
\item 2279
\item 11413
\item 8051
\item 11021
\item 73
\item 46009
\item 3200399
\item 24681023
\een

\item Implemente uma função que factorize um número segundo o método de Fermat.

\item Escreva uma fun\c c\~ao que resolva a equa\c c\~ao diofantina $ax+by=c$.

\item Mostre que
\ben
\item se $a$ \'e um inteiro par ent\~ao $a^2\equiv 0 \mod 4$;
\item se $a$ \'e um inteiro ímpar ent\~ao $a^2\equiv 1 \mod 4$.
\een

\item Mostre que se $a$ \'e um inteiro ímpar ent\~ao $a^2 \equiv 1 \mod 8$.

\item O que pode concluir se $a^2\equiv b^2\mod p$, onde $a,b\in \ZZ$ e $p$ \'e primo?

\item Encontre as solu\c c\~oes de:
\ben
\item $123456789x\equiv 9876543210 \mod 10000000001$
\item $333333333x\equiv 87543211376 \mod 967454302211$
\item $734342499999 x\equiv 1 \mod 1533331$
\item $499999x\equiv 1 \mod 1533331$
\item $1000001x\equiv 1 \mod 1533331$
\een
\item Use $\rho$-Pollard, com $x_0=2$ e $f(x)=x^2+1$ para encontrar a factorização  de
\ben
\item 133
\item 1189
\item 1927
\item 8131
\item 36287
\item 48227
\een

\item Use $\rho$-Pollard para factorizar 1387, fazendo uso de 
\ben
\item $x_0=2 ;f(x)=x^2+1$
\item $x_0=3 ;f(x)=x^2+1$
\item $x_0=2 ;f(x)=x^2-1$
\item $x_0=2 ;f(x)=x^3+x+1$
\een


\een

\end{document}
