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\title{Number-theory exercises, II.II}
\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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\usepackage{hfoldsty}  % gives old-style numerals


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

\newcommand{\defn}[2]{\textbf{#1#2}}

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

\newcommand{\stnd}[1]{\mathbb{#1}}
\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
%\newcommand{\Pri}{\stnd{P}}         % primes
%\newcommand{\R}{\stnd{R}}         % reals
\newcommand{\C}{\stnd{C}}         % complex numbers

\newcommand{\mi}{\mathrm i}
\newcommand{\gi}{\Z[\mi]}

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

  \begin{xca}
If $d$ is a positive non-square rational integer, prove $\sqrt d$ is
irrational.  
  \end{xca}

  \begin{xca}
    Find a greatest common divisor $\alpha$ of the Gaussian integers
    $27+55\mi$ and $20+18\mi$, and solve
    \begin{equation*}
    (27+55\mi)\xi+(20+18\mi)\eta=\alpha.
    \end{equation*}
  \end{xca}

  \begin{xca}
    Find all solutions of the Diophantine equation
    \begin{equation*}
     x^2+y^2=1170.
    \end{equation*}
  \end{xca}

  \begin{xca}
    Assuming $n$ is positive, prove that the number
    of solutions of the Diophantine equation 
    \begin{equation*}
    x^2+y^2=n
    \end{equation*}
is 4 times the
    excess of the number of positive factors of $n$ that are congruent
    to $1$ \emph{modulo} $4$ over the number that are congruent to $3$
    \emph{modulo} $4$.
  \end{xca}

  \begin{xca}\mbox{}
    \begin{enumerate}
      \item
    Characterize (by describing their prime factorizations) those
    Gaussian integers $\alpha$ such that $\size{\alpha}^2$ is square
    as a rational integer.
\item
Use this characterization to solve the Diophantine equation 
\begin{equation*}
x^2+y^2=z^2.
\end{equation*}
    \end{enumerate}
  \end{xca}

  \begin{xca}
    The polynomial $x^2+x+1$ has two conjugate roots.  Let~$\omega$ be
    the root with positive imaginary part.
    \begin{enumerate}
      \item
Write $\omega$ in radicals.
\item
Sketch $\Z[\omega]$ as a subset of the complex plane.
\item
Letting $\norm z=\size z^2$, show that $\norm{\alpha}\in\N$ when
$\alpha\in\Z[\omega]$. 
\item
Express $\norm{x+\omega y}$ in terms of $x$ and $y$.
\item
Show that $\Z[\omega]$ with $z\mapsto\norm z$ is a Euclidean domain.
    \end{enumerate}
(The elements of $\Z[\omega]$ are the \defn{Eisenstein integer}{s.})
  \end{xca}

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