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\begin{center}{\bf M445/845, Homework 2, due Friday, September 27, 2013}\end{center}

\noindent Instructions: Do any three problems.

\begin{itemize}

\item[(1)] Euclid's proof that the integers have infinitely many primes goes like this.
Consider any finite nonempty list $P=\{p_1,\ldots,p_r\}$ of primes $p_i$.
Then $N=1+p_1\cdots p_r$ is an integer bigger than 1, 
so $N$ has a prime factor $p$. Since none of the primes $p_i$ evenly divide
$N$, we see $p$ is not any of the $p_i$ so $p$ is a prime not on the list $P$.
Thus the list $P$ is not complete; i.e., no finite list of primes contains
all of the primes, so there are infinitely many primes.
Modify this proof to show that there are infinitely primes $p$
leaving a remainder of 3 when divided by 4 (such as 3, 7, 11, 19, etc.).

\vskip\baselineskip

\item[(2)] Look up Dirichlet's theorem on arithmetic progressions. Use its statement 
to prove both that there are infinitely many primes congruent to 3 modulo 4, and that
there are infinitely many primes congruent to 1 modulo 4.  Also explain why Dirichlet's 
theorem on arithmetic progressions doesn't also imply that there are infinitely many 
primes congruent to 2 modulo 4.
(Primes congrunt to 2 modulo 4 would be even, so we know there can't be infinitely 
many such primes, but why does Dirchlet's theorem work for 1 or 3 mod 4 but not 2 mod 4?)

\vskip\baselineskip

\item[(3)] Use the definition of what it means for an element of a domain 
to be prime to show if $m+ni\in{\bf Z}[i]$ is prime in the Gaussian integers ${\bf Z}[i]$,
then $m-ni$ is also prime. 

\vskip\baselineskip

\item[(4)] 
\begin{itemize}
\item[(a)] Factor $102+107i$ as a product of Gaussian primes.
\item[(b)] Find a quotient $q$ and remainder $r$
such that $a=bq+r$ with either $r=0$ or $N(r)<N(b)$ when
$a=102+107i$ and $b=-40+73i$.
\item[(c)] Find a greatest common divisor for $a=102+107i$ and $b=-40+73i$.
\end{itemize}

\vskip\baselineskip

\item[(5)] Show that the ring of integers for the field ${\bf Q}$ of rationals is just the usual integers ${\bf Z}$ (i.e., show that a rational root of a monic polynomial
in ${\bf Z}[x]$ must be an integer). [Hint: think about the rational root test.]

\vskip\baselineskip

\item[(6)] It is an open problem to determine whether or not $n^2+1$ is prime for infinitely
many integers $n$. Here are two easier problems of this sort.

\begin{itemize}
\item[(a)] Show that there is only one prime of the form $n^4+n^2+1$ for $n$ an integer.
\item[(b)] Show that there is only one prime of the form $n^4+4$ for $n$ an integer.
\end{itemize}

\vskip\baselineskip


\item[(7)] For each real $r>0$, let $n_r$ be the number of
Gaussian primes $m+ni$ such that $N(m+ni)=r^2$
(i.e., such that $m+ni$ is contained in the perimeter
of the circle $C_r$ of radius $r$ with center at 0
in the complex plane ${\bf C}$). Show that $n_r$ is either 0, 4 or 8,
and that each value occurs for infinitely many $r$. 
(When doing the case of $n_r=8$,
you will along the way show that there are infinitely many Gaussian primes
of the form $m+ni$ with $m\neq0\neq n$, and thus that there are
infinitely many primes in ${\bf Z}$ congruent to 1 modulo 4.)

\end{itemize}
\end{document}
