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\title{Math 480 (Spring 2007): Homework 8}
\author{\bf Due: Monday, May 21}
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\noindent{\bf There are 6 problems.} Each problem is worth 6 points
and parts of multipart problems are worth equal amounts.  You may work
with other people and use a computer, unless otherwise stated.  Acknowledge
those who help you.\\

\begin{enumerate}
\item Write the integer $9000000000000000001053$ as a sum
of two squares.

\item \label{cfrepeate}
Evaluate the infinite continued fraction $[2,\overline{1,3}]$.
Your answer should be an explicit quadratic irrational number. 

\item 
\begin{enumerate}
\item Write down in any way (no proof required)
the infinite continued fraction of the
quadratic irrational number $\frac{1+\sqrt{7}}{2}$. 
(Your answer should look like a finite continued fraction
followed by a repeating part with a bar over it.)
\item Prove that your answer to (a) is correct
 by doing algebra as in problem \ref{cfrepeate}
to show that the value of the continued fraction
you give is really $\frac{1+\sqrt{7}}{2}$.)
\end{enumerate}

\item\label{ex:qf3} Find a positive integer that has at least three
  different representations as the sum of two squares, disregarding
  signs and the order of the summands.

\item Let $E$ be the elliptic curve $y^2 = x^3 - 7x$ over the 
rational numbers.  
\begin{enumerate}
\item There is a point $P=(a,b)$ on $E$ with
$a,b\in\ZZ$ and $|a| < 10$.  Find it.
\item Compute $Q = P+P$ by any method.
\end{enumerate}

\item Let $E$ be the elliptic curve $y^2 = x^3  + 2x$ over the
finite field $\Z/3\Z$. 
\begin{enumerate}
\item Show that $\# E(\Z/3\Z) = 4$, i.e., that there
are $3$ solutions to $y^2 = x^3  + 2x$ with $x,y \in \Z/3\Z$.
(The fourth element of $E(\Z/3\Z)$ is the point at infinity.)
\item Determine the group structure of the group
$E(\Z/3\Z)$ of order $4$.  [Hint: It is either cyclic or
the Klein four group -- which one is it?]
\end{enumerate}


\end{enumerate}

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