User:Marty/UCSC Math 110 Fall 2008/Homework 4

For homework 4, complete the following problems.

Walking the topograph, 3 points
Draw a "topograph", and walk from $\pm (8,11)$ to $\pm (3, -5)$. Your topograph should display primitive lax vectors corresponding to regions, and lax bases corresponding to edges. You should draw a path from the region labelled by $\pm (8,11)$ to the region $\pm (3,-5)$, by following edges.

This part of the homework should be drawn neatly on a sheet of paper, not typed (unless you really want to).

Typed Homework
For the rest of the homework, please complete the two theorems and proofs, given in the following LaTex template. Remember to compile it to make a PDF, print, and hand in.

% % AMS-LaTeX Paper ************************************************ % **** --- \documentclass{amsart} \usepackage{graphicx} \usepackage{amsfonts} \usepackage{amscd} \usepackage{amssymb} \usepackage{xy} % \vfuzz2pt % Don't report over-full v-boxes if over-edge is small \hfuzz2pt % Don't report over-full h-boxes if over-edge is small % THEOREMS --- \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} \numberwithin{equation}{section} % MATH --- \newcommand{\Matrix}[4]{ \left( \begin{array}{cc} #1 & #2 \\  #3 & #4 \\ \end{array} \right) } \newcommand{\norm}[1]{\left\Vert#1\right\Vert} \newcommand{\abs}[1]{\left\vert#1\right\vert} \newcommand{\set}[1]{\left\{#1\right\}}

\newcommand{\NN}{\mathbb N} \newcommand{\ZZ}{\mathbb Z} \newcommand{\QQ}{\mathbb Q} \newcommand{\RR}{\mathbb R} \newcommand{\CC}{\mathbb C} \newcommand{\isom}{\cong} % \begin{document}

\title[]{Math 110, Homework 4}% \author{Student X}% \address{Dept. of Mathematics, University of California, Santa Cruz, CA 95064}% \email{studentx AT ucsc DOT edu}%

%\date{}% %\dedicatory{}% %\commby{}% % \maketitle % \section{Bounding Solutions}

In this section, we prove the following: \begin{thm} Suppose that $a$, $b$, and $c$ are positive integers. Then the number of solutions to the Diophantine equation $ax^2 + by^2 = c$ is less than $N$. \end{thm} % Replace $N$ by a a formula involving $a$, $b$, and $c$. You have to find a correct formula! There are many possible answers. \proof % Prove the theorem, using the formula you gave. % Just guessing a correct formula is worth one point. Proving it is worth two more points. % A very good bound (as small as possible) is worth one more point. \qed

\section{A dry quadratic form} In this section, we prove the following: \begin{thm} If $x,y \in \ZZ$, and $11x^2 - 5y^2 = 0$, then $x = 0$ and $y = 0$. \end{thm} \proof % Prove the theorem. % Use what you know about rationality and irrationality, e.g., the ``n-q-z'' theorem. % A clear and concise proof is worth three points. \qed

\end{document} %