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\title{Math 168: Homework Assignment 4}
\author{William Stein}
\date{\bf Due: Wednesday, Oct 26, 2005}
\begin{document}
\maketitle

\noindent{\em The problems have equal point value, and multi-part
  problems are of the same value.}

\section{Problems}

\begin{enumerate}
\item Use the enumeration of Pythagorean triples $(a,b,c)$ with $a^2 +
  b^2 = c^2$ to create a program that lists all square-free congruent
  numbers.  These will, of course, not be in increasing order!  You do
  not know when a given congruent number will appear on your list.
  

\item In this problem you will create a procedure to output {\em all}
  triangles with rational side lengths and area $219$.
\begin{enumerate}
\item Let $E$ be the elliptic curve $y^2 = x^3 - 219^2 x$. 
      Compute generators for the Mordell-Weil group $E(\Q)$ using
      a computer program (e.g., SAGE, Magma, mwrank, etc.)
      [Hint: $E$ has rank $2$.]
\item Describe how to systematically list {\em all} elements of 
      $E(\Q)$ with $y\neq 0$.
\item Use Prop 6.5.7 from the class notes and your answer to the above
      question to systematically list all triangles with rational
      side lengths and area $219$.
\end{enumerate}
    
\end{enumerate}
\end{document}
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