Author: William A. Stein
1\documentclass[11pt]{article}
2\voffset=-0.05\textheight
3\textheight=1.1\textheight
4\input{macros}
5
6\title{Math 168: Homework Assignment 1}
7\author{William Stein}
8\date{\bf Due: Wednesday, Oct 5, 2005}
9\begin{document}
10\maketitle
11
12\noindent{\em The problems have equal point value,
13and multi-part problems are of the same value.
14You are allowed to use a computer on any problem,
15as long as you include the exact code used to
16solve the problem with your solution.  Any software
17systems (e.g., Magma, SAGE, Mathematica, C) are allowed.}
18
19\section{Announcements}
20\begin{enumerate}
21\item Homework will typically be assigned on Mondays and due
22on the following {\bf Wednesday}.  Then you have plenty of time to ask
24\item Grading: For undergrads it is exactly as on the syllabus.  For
25  graduate students, make some effort on the homework and do a final
26  project and you will get an A.  I will greatly appreciate if grad
27  students help the undergraduates in the course.
28\end{enumerate}
29
30\section{Problems}
31
32\begin{enumerate}
33
34\item Prove that there are infinitely pairs $(x,y)$ of
35rational numbers such that $3x^2 + 4y^2 = 7$.
36
37\item
38Find (by brute force) all pairs $(x,y)$ of integers with $0\leq x < 5$
39and $0\leq y < 5$ and
40$$41 y^2 \con x^3 - x \pmod{5}. 42$$
43
44\item Find an elliptic curve $E$ over a finite
45field $\F_p$ such that the group $E(\F_p)$
46is not cyclic.
47
48\item I'm giving you an account on my super-fast'' dual-opteron
49server {\tt modular.ucsd.edu}, where you'll be able to run
52
53\item Find 10 distinct solutions $(x,y)$, with $x,y\in\Q$
54to the equation $$y^2 + y = x^3 - 7x + 6.$$
55
56
57
58
59\end{enumerate}
60\end{document}
61%%% Local Variables:
62%%% mode: latex
63%%% TeX-master: t
64%%% End:
65