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\centerline{\bf (your name goes here)}
\centerline{}
\centerline{\bf Math 437/537---Homework \#%
	6}
\centerline{due
	Friday, November 4, 2016
by 9:59 {\sc am}}

\medskip{\bf Homework policies:} You are permitted to consult one another concerning the homework assignments, but your submitted solutions must be written by you in your own words. I will consider not only correctness and rigor but also clarity when evaluating your work.

Please submit your homework by emailing me a PDF file. Remember to rename the file with your name---for example, filenames like {\tt GregMartin-homework1.pdf}.

\medskip{1. \it
Suppose $p$ and $q=4p+1$ are both prime. Prove that $2$ is a primitive root modulo $q$, while $p$ is not a primitive root modulo $q$.
}

(your solution goes here)

\newpage
\medskip{2. \it
Let $g(x)=(x^2-2)(x^2-17)(x^2-34)$. Show that $g(x)\equiv0\pmod m$ has solutions for every modulus $m$. (Hint: when $m$ equals a power of 2, note that $x=1$ is a solution modulo 8; use Theorem 2.24.)
}

(your solution goes here)

[Context: For a polynomial equation with integer coefficients to have an integer solution (a ``global'' solution), it's clearly {\em necessary} for it to have both a real solution and a solution modulo $m$ for every $m$ (``local'' solutions); this example shows that existence of these local solutions isn't {\em sufficient} in general. (You can delete this context comment when you submit your solutions.)]

\newpage
\medskip{3. \it
Hint: look for short solutions to these problems!
\begin{enumerate}
\item \book{3.2}{141}{14}
\item Suppose that $a$ is an odd positive integer, $b$ is an even integer, and $p=a^2+b^2$ is prime. Prove that $a$ is a quadratic residue$\pmod p$.
\item Let $p$ be an odd prime. Suppose there exist integers $m$ and $n$ such that $p$ divides both $m^2+1$ and $n^2+2$. Prove that there exists an integer $k$ such that $p$ divides $k^4+1$.
\end{enumerate}
}

\begin{enumerate}
\item (your solution goes here)
\item (your solution goes here)
\item (your solution goes here)
\end{enumerate}

\newpage
\medskip{4. \it
\begin{enumerate}
\item Which years so far this millennium (from $2000$ to $2016$ inclusive) can be written as the sum of two squares? Justify your answer; do not use brute force.
\item \book{3.6}{170}{11 and \#12 (they're so related that there's no reason to keep them separate)}
\end{enumerate}
}

\begin{enumerate}
\item (your solution goes here)
\item (your solution goes here)
\end{enumerate}

\newpage
\medskip{5. \it The point of this problem is to parallel our proof of Fermat's result that characterizes the numbers that can be written in the form $x^2+y^2$. Succeeding at this problem will mean that you have mastered that proof!
\begin{enumerate}
\item Let $p$ be a prime congruent to either $5$ or $7$ modulo $8$, and let $a$ and $b$ be integers. If $p\mid(a^2+2b^2)$, show that $p\mid a$ and $p\mid b$.
\item Let $p$ be a prime congruent to either $1$ or $3$ modulo $8$. Show that there exist integers $a$ and $b$ such that $a^2+2b^2=p$. (See \bbook{2.1}{59--60}{56 and \#58} for a sketch of how to do so.)
\item If both $m$ and $n$ are represented by the binary quadratic form $x^2+2y^2$, show that $mn$ is also represented by $x^2+2y^2$. (Hint: expand the product $(a+ib\sqrt 2)(c+id\sqrt2)$ of complex numbers.)
\item Prove that a positive integer $n$ is represented by $x^2+2y^2$ if and only if $v_p(n)$ is even for every prime $p\equiv5,7\pmod 8$ dividing~$n$.
\end{enumerate}
}

\begin{enumerate}
\item (your solution goes here)
\item (your solution goes here)
\item (your solution goes here)
\item (your solution goes here)
\end{enumerate}

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