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