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\title{Math 168: Homework Assignment 1}
\author{William Stein}
\date{\bf Due: Wednesday, Oct 5, 2005}
\begin{document}
\maketitle
\noindent{\em The problems have equal point value,
and multi-part problems are of the same value.
You are allowed to use a computer on any problem,
as long as you include the exact code used to
solve the problem with your solution. Any software
systems (e.g., Magma, SAGE, Mathematica, C) are allowed.}
\section{Announcements}
\begin{enumerate}
\item Homework will typically be assigned on Mondays and due
on the following {\bf Wednesday}. Then you have plenty of time to ask
me questions about it.
\item Grading: For undergrads it is exactly as on the syllabus. For
graduate students, make some effort on the homework and do a final
project and you will get an A. I will greatly appreciate if grad
students help the undergraduates in the course.
\end{enumerate}
\section{Problems}
\begin{enumerate}
\item Prove that there are infinitely pairs $(x,y)$ of
rational numbers such that $3x^2 + 4y^2 = 7$.
\item
Find (by brute force) all pairs $(x,y)$ of integers with $0\leq x < 5$
and $0\leq y < 5$ and
$$
y^2 \con x^3 - x \pmod{5}.
$$
\item Find an elliptic curve $E$ over a finite
field $\F_p$ such that the group $E(\F_p)$
is not cyclic.
\item I'm giving you an account on my ``super-fast'' dual-opteron
server {\tt modular.ucsd.edu}, where you'll be able to run
SAGE and Magma. Please select a login name.
See also {\tt http://modular.ucsd.edu/calc/}.
\item Find 10 distinct solutions $(x,y)$, with $x,y\in\Q$
to the equation $$y^2 + y = x^3 - 7x + 6.$$
\end{enumerate}
\end{document}