\documentclass[10pt]{article} % 
\usepackage{fullpage}
\usepackage{graphicx}
\usepackage{graphics}
\usepackage{psfrag}
\usepackage{amsmath,amssymb}
\usepackage{enumerate}

\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9in}

\newcommand{\cP}{\mathcal{P}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}

\title{Math 412  Homework 7}
\date{Due date: Nov 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 equation $x^2+5x+13\equiv 0\pmod{3^4}$. 

\item Let $p$ be an odd prime and $p\not| a$.  Show that there exist integers $u, v$ with $(u, v)=1$ so that $u^2+av^2\equiv 0\pmod{p}$ if and only if $-a$ is a quadratic residue modulo $p$. 


\item Compute the following Legendre symbols: $\left(\frac{13}{47}\right),  \left(\frac{71}{73}\right)$. 

\item Find all prime $p$ so that $3$ is a quadratic nonresidue modulo $p$. 

\item Show that there are infinite many primes of form $8k-1$.  (hint: consider $N=(p_1p_2\ldots p_n)^2-2$.)

\end{enumerate}

\end{document}
