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\title{Elementary Number Theory II, examination I}
\author{David Pierce}
\date{March 24, 2008}
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\address{Mathematics Dept\\
Middle East Technical University\\
Ankara 06531, Turkey}

\email{dpierce@metu.edu.tr}
\urladdr{http://www.math.metu.edu.tr/~dpierce/}
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\newcommand{\stnd}[1]{\mathbb{#1}}
\newcommand{\Z}{\stnd{Z}}         % integers
\newcommand{\Q}{\stnd{Q}}         % rationals

\newcommand{\mi}{\mathrm i}

\newcommand{\norm}[1]{\operatorname{N}(#1)}

\let\oldsqrt\sqrt
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\begin{document}
  \maketitle\thispagestyle{empty}

  \begin{instructions}
    Take at most 90 minutes to write reasonably legible solutions on
    the blank sheets provided.
    You may want to do scratch-work first, on sheets that you will
    keep.  But the sheets that you turn in should show sufficient work to
    justify your answers.
    You may keep this problem-sheet for future study.  \emph{Kolay gelsin.} 
  \end{instructions}

  \begin{problem}
This problem involves the Gaussian integers.  Let $\alpha=40+5\mi$ and
$\beta=39\mi$.  
    \begin{enumerate}
      \item\label{item:a}
Find a greatest common divisor of $\alpha$ and $\beta$.
\item
If $\gamma$ is your answer to \eqref{item:a}, solve
\begin{equation*}
  (40+5\mi)\cdot\xi+39\mi\cdot\eta=\gamma.
\end{equation*}
    \end{enumerate}
  \end{problem}

  \begin{problem}
This problem involves the Diophantine equation
\begin{equation}\label{eqn:D}
  2x^2-3y^2=2.
\end{equation}
    \begin{enumerate}
      \item
Express $\oldsqrt{3/2}$ as a continued fraction.
\item
  Find a positive solution to~\eqref{eqn:D}.
\item
Find a solution $(a,b)$ to~\eqref{eqn:D} in which each of $a$ and $b$
has two digits (in the usual decimal notation).
\item
Find a solution $(a,b)$ to~\eqref{eqn:D} in which each of $a$ and $b$
has three digits.
\end{enumerate}
  \end{problem}

  \begin{problem}
    In class we found the bijection
    \begin{equation*}
      t\longmapsto\Bigl(\frac{1-t^2}{1+t^2},\frac{2t}{1+t^2}\Bigr)
    \end{equation*}
between $\Q$ and the set of rational solutions (other than $(-1,0)$)
to the equation
\begin{equation*}
  x^2+y^2=1.
\end{equation*}
\begin{enumerate}
  \item
Find all rational solutions to the equation
\begin{equation*}
  x^2+3y^2=1.
\end{equation*}
\item
Find $\alpha$ in $\Q(\mi)$ such that $\norm{\alpha}=1$, but $\alpha$ is
not a Gaussian integer.
\item
Find $\beta$ in $\Q(\sqrt{{-3}})$ such that $\norm{\beta}=1$, but
$\beta$ is not an integer (that is, not an Eisenstein integer).
\end{enumerate}
  \end{problem}

  \begin{problem}
    \mbox{}
    \begin{enumerate}
      \item
Find all distinct solutions (from $\Z$) of the Diophantine equation
\begin{equation*}
  x^2+y^2=221.
\end{equation*}
\item
Find a factorization of $27-57{}\mi$ as a product of Gaussian primes.
    \end{enumerate}
  \end{problem}

\end{document}