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 Z^{2} 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_{m}L_{n}, and L_{m}L_{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.
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.
Here are some documents that you can download
 The AMSLaTeX User's Guide. This
tells you everything you might want to know about the features of
AMSLaTeX 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:LaTeXdocs:
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 online 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