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
final papers of the students, all but one of which can be downloaded
(in pdf format).
- Elin Barts, Patterns in some
- Arudra Burra, Periods of Fibonacci
sequences modulo m
- Nathan Davis, Which values of a
Fibonacci function on Z2 completely characterize that
- Paul Keselman, Continued fractions
and Fibonacci numbers
- Igor Levit, Generating functions of
Fan+b and Lan+b
- Vlad Liberzon, On divisibility
properties of Fibonacci and Lucas numbers
- Andrew See, Lucas representations of Fn,
FmLn, and LmLn
- Scott Seltzer Generalized Fibonacci
sequences modulo powers of a prime
- Naomi Utgoff, A generalization of
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.
Functions for getting TeX output from Maple.
- Examples of the use of these functions.
Here's an archive of all three files as Maple worksheets.
You will need Stuffit Expander to decode it.
An example of a short mathematical paper written in LaTeX
A "template" that you can use for writing a mathematical
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 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
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
page. OzTeX is shareware. If you want to see the OzTeX
documentation, you can find it in a file called
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
Here are some on-line references to the Fibonacci numbers:
Here is some online documentation on Maple:
To Ira Gessel's Home Page
To the Brandeis Mathematics Department Home Page