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\title{Number-theory exercises, II.IV}
\author{David Pierce}
\date{\today}

\address{Mathematics Dept\\
Middle East Tech.\ Univ.\\
%Middle East Technical University\\
Ankara 06531, Turkey}

\email{dpierce@metu.edu.tr}
\urladdr{http://www.math.metu.edu.tr/~dpierce/courses/366/}


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\newcommand{\N}{\stnd{N}}         % natural numbers
%\newcommand{\varN}{\omega}        % my usual preference for this
%\newcommand{\Zp}{\Z_{+}}          % positive integers
\newcommand{\Z}{\stnd{Z}}         % integers
\newcommand{\Q}{\stnd{Q}}         % rationals
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\begin{document}
  \maketitle\thispagestyle{empty}

These exercises involve quadratic Diophantine equations.

\begin{xca}
  Solve
  \begin{equation*}
    2x^2+2xy+y^2=25.
  \end{equation*}
\end{xca}

\begin{xca}
  Solve
  \begin{equation*}
    9x^2+6xy+2y^2=17.
  \end{equation*}
\end{xca}

\begin{xca}
  Solve (if you can!)
  \begin{equation*}
    121x^2+304xy+191y^2=37.
  \end{equation*}
(If nothing else works, try letting $3x+4y=u$ and $4x+5y=v$.)
\end{xca}

  \begin{xca}
    Solve
    \begin{equation*}
      4x^2+2xy-y^2=44.
    \end{equation*}
  \end{xca}

  \begin{xca}
    Concerning
    \begin{equation*}
      8x^2+4xy-y^2=m:
    \end{equation*}
    \begin{enumerate}
      \item
solve when $m=8$;
\item
solve when $m=44$;
\item
find all $m$ for which the equation is soluble, where $0<m<44$.
    \end{enumerate}
  \end{xca}

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