MATH 47a (Introduction to Mathematical Research)

This is the web site for Math 47a, taught by Ira Gessel at Brandeis University in the spring of 2000. The topic of the course that year was Fibonacci numbers.

The main assignment for students was a final paper. Here is a list of the final papers of the students, all but one of which can be downloaded (in pdf format).

Elin Barts, Patterns in some Zeckendorf representations
Arudra Burra, Periods of Fibonacci sequences modulo m
Nathan Davis, Which values of a Fibonacci function on Z² completely characterize that function?
Paul Keselman, Continued fractions and Fibonacci numbers
Igor Levit, Generating functions of F_an+b and L_an+b
Vlad Liberzon, On divisibility properties of Fibonacci and Lucas numbers
Andrew See, Lucas representations of F_n, F_mL_n, and L_mL_n
Scott Seltzer Generalized Fibonacci sequences modulo powers of a prime
Naomi Utgoff, A generalization of Zeckendorf's theorem

Course documents

Here are copies of some the class handouts in pdf format. Some of these have been revised from the versions given out in class.

Files to download

Here are some files that may be useful.

Maple files: These are in "Maple Text" format. To use them, open them in Maple as Maple Text.

Functions useful for Fibonacci and Zeckendorf computations. Zfunctions.txt
Functions for getting TeX output from Maple. ZTeX.txt
Examples of the use of these functions. Zexamples.txt

Here's an archive of all three files as Maple worksheets. You will need Stuffit Expander to decode it.

Zeckendorf.sea.bin

LaTeX files:

example.tex An example of a short mathematical paper written in LaTeX
template.tex A "template" that you can use for writing a mathematical paper.
reftemplate.tex Another sample paper that shows the format to use for including references in your paper.

TeX and LaTeX links

Here are some links to sites with information about TeX and LaTeX. You can find many further links through them.

Penn State Macintosh TeX/LaTeX web site (highly recommended)
The TeX Users' Group
LaTeX Project Page
AMS-LaTeX Web Page
On-line LaTeX reference
LaTeX: from quick and dirty to style and finesse by Tony Roberts An online tutorial.

Here are some documents that you can download

The AMS-LaTeX User's Guide. This tells you everything you might want to know about the features of AMS-LaTeX for typesetting mathematics that you won't find in a guide to generic LaTeX. This is a dvi file, so you will need OzTeX (or some other version of TeX) to view and print it.
George Gratzer's Math into LaTeX. This is a pdf file, so you will need Adobe Acrobat reader (a free program which is probably already on your computer) to read or print it. This document is the first part, together with some appendices of a book (by the same title) that is on reserve in Gerstenzang.
A short introduction to LaTeX . Also a pdf file.

The free program BBEdit Lite is very helpful in editing text files. Another Macintosh text editor is Alpha, which is more powerful than BBEdit, but more complicated. Alpha is shareware.

If you want your own copy of OzTeX, you can get it from the OzTeX home page. OzTeX is shareware. If you want to see the OzTeX documentation, you can find it in a file called ozuser.dvi on the Brandeis Macintosh file server; more precisely, it's in the folder
Macintosh File Server:Applications:OZtex 3.1:LaTeX-docs:

If you don't have the free program Stuffit Expander, which you will need to expand compressed and encoded files, you can download it from Aladdin Systems.

Fibonacci numbers

Here are some on-line references to the Fibonacci numbers:

Maple

Here is some online documentation on Maple:

To Ira Gessel's Home Page

To the Brandeis Mathematics Department Home Page