%\documentclass[a4paper,twoside,draft,12pt]{article}
\documentclass[a4paper,twoside,reqno,12pt,draft]{amsart}
\usepackage[headings]{fullpage}
%\usepackage{multicol}

%\usepackage[fulloldstyle]{fourier}
\title{Elementary Number Theory II, examination I}
\author{David Pierce}
\date{March 24, 2008}
\usepackage{verbatim}  % allows a comment environment:
%\begin{comment}
\address{Mathematics Dept\\
Middle East Technical University\\
Ankara 06531, Turkey}

\email{dpierce@metu.edu.tr}
\urladdr{http://www.math.metu.edu.tr/~dpierce/}
%\end{comment}

\usepackage{amsmath,amsthm,amssymb}
\usepackage{url}
\usepackage{textcomp}  % supposedly useful with \oldstylenums
\usepackage{hfoldsty} % this didn't work until I added missing
		      % brackets to some of the files.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\renewcommand{\theenumi}{\alph{enumi}}
%\renewcommand{\labelenumi}{\textnormal{(\theenumi)}}
\renewcommand{\theenumi}{\roman{enumi}}
\renewcommand{\labelenumi}{\textnormal{(\theenumi)}}

%\renewcommand{\theenumii}{\roman{enumii}}
%\renewcommand{\labelenumii}{\textnormal{(\theenumii)}}

%%%%%%%%%%%%%%%

\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
\renewcommand{\sqrt}[2][1]{\oldsqrt{\vphantom{#1}}#2}
\newcommand{\rft}{\sqrt{14}}
\newcommand{\rtt}{\sqrt{13}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Theorem-like environments  %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\theoremstyle{definition}
\newtheorem{problem}{Problem}

\theoremstyle{remark}

\newtheorem*{instructions}{Instructions}

\newenvironment{solution}%
{\begin{proof}[Solution.]}%
{\end{proof}}

%\numberwithin{equation}{section}
\renewcommand{\theequation}{\fnsymbol{equation}}


\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}

