CoCalc Public Fileswww / 168 / homework / 1 / 1.tex
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,
13
and multi-part problems are of the same value.
14
You are allowed to use a computer on any problem,
15
as long as you include the exact code used to
16
solve the problem with your solution. Any software
17
systems (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
22
on the following {\bf Wednesday}. Then you have plenty of time to ask
23
me questions about it.
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
35
rational numbers such that $3x^2 + 4y^2 = 7$.
36
37
\item
38
Find (by brute force) all pairs $(x,y)$ of integers with $0\leq x < 5$
39
and $0\leq y < 5$ and
40
$$41 y^2 \con x^3 - x \pmod{5}. 42$$
43
44
\item Find an elliptic curve $E$ over a finite
45
field $\F_p$ such that the group $E(\F_p)$
46
is not cyclic.
47
48
\item I'm giving you an account on my super-fast'' dual-opteron
49
server {\tt modular.ucsd.edu}, where you'll be able to run
50
SAGE and Magma. Please select a login name.
51
52
53
\item Find 10 distinct solutions $(x,y)$, with $x,y\in\Q$
54
to 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