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\title{Math 412  Homework 9}
\date{Due date: Dec 3, 2010}
\author{your name}


\begin{document} 
\maketitle

Solve the following problems. Please remember to use complete sentences
and good grammar. 



\begin{enumerate}

\item Solve the Pell equation:  $x^2-29y^2=1$. 

\item Show that the Gaussian integers $\alpha$ and $\beta$ are associate if $\alpha|\beta$ and $\beta|\alpha$.  

\item Find all Gaussian integers $\alpha, \beta, \gamma$ such that $\alpha\beta\gamma=\alpha+\beta+\gamma=1$. 

\item Let $x,y\in \Z$. Show that if $(x,y)=1$, then $x$ and $y$ are also coprime in $\Z[i]$.

\item Let $\alpha, \beta\in \Z[i]$ and are coprime.  If $\alpha\beta=\gamma^2$ for some $\gamma\in \Z[i]$, then $\alpha$ and $\beta$ can also be written as square, up to a factor of units. 

\item show that if $x$ and $y$ are integers such that $x^2+1=y^3$, then $x-i$ and $x+i$ are relatively prime. 

\end{enumerate}

\end{document}
