User:Marty/UCSC Math 110 Fall 2008/Homework 6

This homework assignment has three problems, and your solutions should be typed using LaTex (which you have installed previously. Follow these steps to complete the homework assignment:
 * Copy and paste the template below into your Tex editor.
 * Save and compile the file, and view the PDF (this will help you see what you are trying to do).
 * Edit the template to complete the computations and proofs.
 * Make sure to change the name and e-mail addresses to your own (you are not Student X).
 * Follow all of the directions in the template, which are given on lines that begin with a % symbol in the body of the text.
 * Save and compile the file now and then, to view the PDF results.
 * After you finish, save and compile the PDF, print the file, and turn it in.
 * Make sure to draw the topograph diagrams in the blank spaces after printing out your final draft.
 * There are two sections to complete.
 * The first section is worth 5 points.
 * A nice exposition on Pell's equation can be found here:
 * The second section is worth 5 points.

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\title[]{Math 110, Homework 5}% \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{Pell's Equation} Pell's equation is the following: $$x^2 - d y^2 = 1.$$ In 1768, Lagrange was able to prove the following theorem: \begin{thm} Suppose that $d \in \ZZ$, $d > 0$, and $d$ is not a perfect square. Then, there exist infinitely many pairs $(x,y) \in \ZZ^2$, such that $x^2 - d y^2 = 1$. \end{thm} \proof A method to find such solutions goes back to the work of Brahmagupta (around 628 AD), and Bhaskara II (12th century AD). Lagrange proved this theorem using continued fractions. We will prove this theorem using the method of Conway's topograph. % Write a paragraph or two, explaining how our knowledge about the range topograph implies that there are infinitely many solutions. % Make sure to explain why there is an endless river in the topograph. % You can use the facts we have proven in class about the range topograph. % You might wish to attach a separate piece of paper, with a drawing of the range topograph for a quadratic form x^2 - d y^2. \qed

The conditions on $d$ are essential in the previous theorem. Namely, we can also prove the following theorem: \begin{thm} Suppose that $d \in \ZZ$, and either $d < 0$ or $d$ is a positive perfect square. Then, there exist finitely many pairs $(x,y) \in \ZZ^2$ such that $x^2 - d y^2 = 1$. \end{thm} \proof Suppose first that $d < 0$. % Write a proof in this case. % Observe that the quadratic form x^2 - d y^2 is positive definite, in this case. % Use the climbing principle, and the existence of a unique well to justify your proof. % Your proof can be a short paragraph explanation, using what you know about range topographs.

Next, suppose that $d > 0$, and $d = e^2$ for some integer $e$. % Write a proof in this case. % You can use properties of the topograph, if you would like, to explain why there are only finitely many pairs $(x,y)$ satisfying $x^2 - e^2 y^2 = 1$. % You can also use a purely algebraic argument, by factoring the polynomial $x^2 - e^2 y^2$ (a difference of squares). \qed

\section{Class Number}

Suppose that $\Delta$ is an integer. Let $h(\Delta)$ denote the number of equivalence classes of binary quadratic forms of discriminant $\Delta$. In class, we have proven the following two results: \begin{itemize} \item If $\Delta < 0$, then $h(\Delta)$ is less than or equal to the number of ``wells'' with discriminant $\Delta$. It follows that $h(\Delta)$ is finite. \item If $\Delta > 0$, and $\Delta$ is not a perfect square, then $h(\Delta)$ is less than or equal to the number of ``riverbends'' with discriminant $\Delta$. It follows that $h(\Delta)$ is finite. \end{itemize} % Notice that the "itemize" environment makes a bulleted list. % Each item occurs after an "\item" tag. % The \begin{itemize} and \end{itemize} mark the beginning and end of the bulleted list environment. % Also, notice that to put a word in quotations, you use a two single back-quotes `` to start, and two single forward quotes '' to end. % This is a bit weird, but it looks great in the end!

Now, we consider the case when $\Delta$ is a positive perfect square. \begin{thm} Suppose that $d \in \ZZ$, and $d > 0$. Let $\Delta = d^2$ denote the resulting positive perfect square. Then $h(\Delta)$ is finite. \end{thm} \proof Suppose that $Q$ is an integer-valued binary quadratic form, of discriminant $\Delta$. The range topograph of $Q$ contains either a double-lake or two lakes joined by a river. Furthermore, the equivalence class of $Q$ is determined by one of the following: \begin{itemize} \item The two numbers adjacent to the double lake. \item The two numbers adjacent to both a lake and the river. \end{itemize}

We begin by proving that there are only finitely many double-lakes with discriminant $\Delta$. \vspace*{3in} % The \vspace*{3in} command forces a 3 inch vertical space in your paper. % Draw a cell (a region in the range topograph with four faces), which contains a double lake, in this vertical space. % Two adjacent faces in the range topograph should be labeled by zero (the double lake). % The other two faces should be labelled by variables e and f.

Observe that the discriminant $\Delta$ must be related to $e$ and $f$ by the following formula: % Compute the discriminant, using the variables e and f. Furthermore, the arithmetic progression rule implies that % Finish the above sentence, to determine the possible values for e and f. Since there are only finitely many possible values adjacent to the double-lake, we find that there are finitely many equivalence classes of quadratic forms, of discriminant $\Delta$, which have a double-lake.

Next, we prove that there are only finitely many river-lake intersection points, with discriminant $\Delta$. \vspace*{3in} % Draw a cell, which contains a river-lake intersection point, in this vertical space. % One face should be labelled by zero (the lake). % The other three faces should be labelled by variables e,u,v. % Put u and v next to the lake, and on opposite sides of the river. Observe that the discriminant $\Delta$ must be related to $u$ and $v$ by the following formula: % Compute the discriminant, using the variables u and v. %  Consider integers u and v, one positive and one negative, satisfying the equation above for the discriminant. % Prove that there are only finitely many such integers u and v.  Write down a bound for the number of such pairs (u,v).

Since there are only finitely many possible values at a river-lake intersection point, we find that there are finitely many equivalence classes of quadratic forms of discriminant $\Delta$, which have a river-lake intersection point.

Since every quadratic form of discriminant $\Delta$ has a double-lake or a river-lake intersection point, it follows that $h(\Delta)$ is finite. % If you want, you can replace the phrase $h(\Delta)$ is finite, by an actual bound for $h(\Delta)$. \qed

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