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

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

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


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

  \begin{xca}
    Let $S$ be the set of Pythagorean triples $(a,b,c)$ such
    that
$\gcd(a,b,c)=1$;
$a$, $b$, and $c$ are positive; and
$a<b$.
Order $S$ by the rule
\begin{equation*}
  (a,b,c)<(d,e,f)\iff(a+b<d+e)\lor(a+b=d+e\land b<e).
\end{equation*}
Find the first few elements of $S$ with respect to this ordering. 
  \end{xca}
\vfill
  \begin{xca}
Solve $x^2+4y^2=z^2$.
  \end{xca}
\vfill
  \begin{xca}
    Solve $x^4+y^2=z^2$.
  \end{xca}
\vfill
  \begin{xca}\mbox{}
%    Concerning an arbitrary Diophantine equation $f(x,y)=0$:
    \begin{enumerate}
      \item
Show that $f(x,y)=0$ is soluble if and only if $f(3x+2y,4x+3y)=0$ is
soluble.
\item
Find necessary and sufficient conditions on $a$, $b$, $c$, and $d$
such that an arbitrary Diophantine equation $f(x,y)=0$ is soluble if
and only if $f(ax+by,cx+dy)=0$ is soluble.
    \end{enumerate}
  \end{xca}
\vfill
  \begin{xca}\mbox{}
\begin{enumerate}
\item
    Find the expansion of $\sqrt d$ as a continued fraction for
    various $d$, including $7$. 
\item
Solve the Pell equation $x^2-dy^2=1$ for these $d$.
\end{enumerate}
  \end{xca}
\vfill
  \begin{xca}\mbox{}
    \begin{enumerate}
      \item
Show
$[a_0;a_1,\dots,a_k,a_{k+1},\dots,a_n]
=[a_0;a_1,\dots,a_k,[a_{k+1},\dots,a_n]]$.  
\item
Show
$[a_0;a_1,\dots,a_k,a_{k+1},\dots]
=[a_0;a_1,\dots,a_k,[a_{k+1},\dots]]$.  
\item
Compute $[\overline{2;1}]$ (which is $[2;1,\overline{2,1}]$) in terms
of radicals. 
\item
Show that $[a_0;a_1,\dots,a_k,\overline{a_{k+1},\dots,a_n}]$ is always
the root of a quadratic polynomial.
    \end{enumerate}
  \end{xca}

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