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\title{Elementary Number Theory II, examination II}
%\author{David Pierce}
\date{April 28, 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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\renewcommand{\theenumi}{\alph{enumi}}
\renewcommand{\labelenumi}{\textnormal{(\theenumi)}}
%\renewcommand{\theenumi}{\roman{enumi}}
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%\renewcommand{\theenumii}{\roman{enumii}}
%\renewcommand{\labelenumii}{\textnormal{(\theenumii)}}

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\newcommand{\included}{\subseteq}      % [the name suggests the meaning here]

\newcommand{\stnd}[1]{\mathbb{#1}}
\newcommand{\Z}{\stnd{Z}}         % integers
\newcommand{\R}{\stnd{R}}         % reals
\newcommand{\Q}{\stnd{Q}}         % rationals

\newcommand{\mi}{\mathrm i}
%\newcommand{\gr}{\upphi}  % golden ratio
\newcommand{\mLambda}{\mathit{\Lambda}}
\newcommand{\mPi}{\mathit{\Pi}}

\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}}

\newcommand{\lat}[1][\alpha,\beta]{\langle#1\rangle} % lattice
\newcommand{\ord}[1][\mLambda]{\mathfrak O_{#1}}


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\begin{document}
  \maketitle\thispagestyle{empty}

  \begin{instructions}
Solve {four} of these five problems in 90 minutes.  \emph{\.Iyi
  \c cal\i\c smalar.} 
  \end{instructions}
\vfill
  \begin{problem}
Assuming $a>0$, prove
\begin{equation*}
\oldsqrt{a^2+1}=[a;\overline{2a}].
\end{equation*}
  \end{problem}
\vfill
  \begin{problem}
    Let $K=\Q(\sqrt 5)$ and $\mLambda=\lat[1,\sqrt 5]$.
    \begin{enumerate}
      \item
Find the order $\ord$ (that is, $\{\xi\in
K\colon\xi\mLambda\included\mLambda\}$). 
\item
Find the elements of $\ord$ having norm $1$.
    \end{enumerate}
  \end{problem}
\vfill
  \begin{problem}
    Solve in $\Z$:
    \begin{equation*}
    x^2+2xy+4y^2=19.
    \end{equation*}
  \end{problem}

  \begin{problem}\mbox{}
    \begin{enumerate}
      \item
    Prove that, for each $n$ in $\Z$, there are $a_n$ and $b_n$ in
    $\Z$ such that 
    \begin{equation*}
      a_n+b_n\sqrt{21}=2\Bigl(\frac{5+\sqrt{21}}2\Bigr)^n.
    \end{equation*}
\item
Find a quadratic form $f(x,y)$ and a rational integer $m$ such that
each $(\pm a_n,\pm b_n)$ is a solution of
\begin{equation}\label{eqn:f}
  f(x,y)=m.
\end{equation}
\item
Prove that each solution of~\eqref{eqn:f} is  $(\pm a_n,\pm b_n)$ for
some $n$.
    \end{enumerate}
  \end{problem}
\vfill
  \begin{problem}
    \mbox{}
    \begin{enumerate}
      \item
Find a quadratic field $K$, a lattice $\lat$ or $\mLambda$ of $K$, and
$m$ in $\Z$ for which the function
\begin{equation*}
  (x,y)\mapsto x\alpha+y\beta
\end{equation*}
is a bijection between the solution-set (in $\Z\times\Z$) of 
\begin{equation}\label{eqn:2}
  2x^2-3y^2=2
\end{equation}
and the solution-set in $\mLambda$ of
  $\norm{\xi}=m$.
\item\label{item:Pi}
Describe a parallelogram $\mPi$ in the plane $\R^2$ such that,
for every solution $(a,b)$ of~\eqref{eqn:2}, there is a solution
$(c,d)$ in $\mPi$ such that
\begin{equation}\label{eqn:frac}
  \frac{a\alpha+b\beta}{c\alpha+d\beta}\in\ord.
\end{equation}
\item
Find $\mPi$ as in~\eqref{item:Pi} with the additional
condition that, if $(a,b)$ and $(c,d)$ are distinct solutions
to~\eqref{eqn:2} in $\mPi$, then~\eqref{eqn:frac} fails.
    \end{enumerate}
  \end{problem}
\vfill
\end{document}