Experimental Number Theory, Fernando Rodriguez Villegas

Experimental Number Theory

Fall 2005

Instructor: Fernando Rodriguez Villegas
Address: Department of Math, UT Austin, Austin, TX 78712
Phone: (512) 471-1137
Office: RLM 9.164
Fax: 512-471-9038
E-mail: villegas@math.utexas.edu

Historically, a non-trivial amount of Number Theory arose from numerical experimentation; the Birch-Swinnerton-Dyer conjecture, one of the central problems in Number Theory, is perhaps the most notorious example but there are many. This course is an introduction to the use of the computer as an experimental tool in Number Theory. We will discuss this in many different contexts: number fields, elliptic curves, finite fields, etc. The emphasis will be on practicality and concrete use rather than the purely theoretical aspects. One of the goals is to develop a sense of what is feasible in a reasonable amount of time (both programming and running the necessary algorithm). Along the way we will cover a lot of Number Theory, mostly from a computational point of view, as well as the standard algorithms of the subject. No previous programming or Number Theory background is required.

Discussion Group

Check out the discussion group for the class.

Class Text

The main text for the class is the ongoing GP book by yours truly.

Code

The main computer package we will use is PARI-GP

Code for the class.

You may want to check out these GP scripts for further examples.

Homework

There will be 12 homework assignments (labelled by week number, not necessarily matching exactly the corresponding calendar week number in the semester). Each homework assignment will consist of writing a GP script to calculate some specific thing (except if your are the weeks scribe/editor see below).

For each homework assignment you will be pairing with another student who will run your program and make comments, solicit changes, etc. Hopefully this will help make the programming readable by (other) human beings. After a few iterations of this you should e-mail the program to me together with any relevant comments from you or your partner.

Please use the following heading at the top of your assignment to help me organize them. Also, please name your file name-hmwk#.gp. Thanks.

Class GP logs

Logs of the GP sessions for each class.

Scanned notes

Scans of what I wrote in class.

Class notes

Each week on the pairs of students duties will be to by scribe/editor of notes of the lectures.

To keep some uniformity in the notes please use this heading (or similar) for your latex file. Please write one file per lecture with name month_name-day.tex (and please also don't forget to include the date of the lecture in the heading). For example, the notes for the lecture on September 1 would be in the file sep-1.tex (let's agree on, say, sep, oct, nov, dic , for the abbreviation of month_name)

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