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\title{Math 412  Homework 12}
\date{Due date: noon Dec 4, 2015}
\author{your name}


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\maketitle

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



\begin{enumerate}





\item Show that $1+i$ divides a Gaussian integer $a+ib$ if and only if $a$ and $b$ are both even or both odd. 

\item Find all Gaussian primes of the form $\alpha^2+1$, where $\alpha$ is a Gaussian integer. 



\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 Find an inverse of $4$ modulo $5+2i$ and solve the linear congruence in Gaussian integers $$4x\equiv -3+4i\pmod{5+2i}$$ 

\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. 

\item Use the Gaussian integers to find all solutions to the diophantine equation $x^2+y^2=z^3$ in rational integers $x, y$ and $z$. 

\end{enumerate}

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