Your paper topic should have some solid mathematical content, and your paper should involve a certain amount of detailed mathematical explanations and/or proofs of results. It is also fine to discuss the context of the topic, including possibly applications, or an overview of the area, in addition to the more focused mathematical content.

Audience: You are explaining your topic to someone with a general background in mathematics, similar to your own. Your reader has probably never seen this particular topic before, however.

Strive for clarity, and then for brevity. To get an idea of what I am after, imagine there is a national emergency and the President has asked you to bring him up to speed on your topic, including technical details. Your report should be clear and straightforward --- don't confuse him or waste his time. The paper doesn't have to be dry, however. Especially in the introduction, feel free to "sell" the subject a little.

Bibliography: If you used a source when learning about your topic, including websites, please include it in your bibliography. Any reasonable format for the bibliography is fine. Do not copy (or paraphrase) passages from any source unless they are short and clearly marked as quotations, with the source indicated. Otherwise, this is plagiarism, which counts as academic misconduct (ie. cheating).

Length: the paper should be 10-15 pages, double spaced, with a reasonable font and margins. With any good topic, you will find that there is plenty to say if you delve in a little.

Presentation: this should be 20 minutes or so, given in class. It is probably easiest to plan on using the blackboard.

Here are some areas where you might find interesting paper topics. Some of these may be a little ambitious.

• the Banach-Tarski paradox
• transcendental numbers
• the prime number theorem
• the four-color theorem
• primality testing
• cryptography
• knot invariants
• topology of surfaces
• voting paradoxes
• game theory
• continued fractions
• Conway tangles
• and more ...

Here are some titles of past Capstone papers. Some were very good and others were not-so-great, but I'm not saying which! This list is just to give you an idea of what other students have done.

• AES/Rijndael and Symmetric Key Cryptography
• Application of the Laplace Transform to a RLC circuit
• The 17 Crystallographic groups
• Numerical Weather Prediction
• A Herder's Nightmare: Diophantine Analysis and Archimedes' Problem
• Logic behind the Game of Hex
• An Exploration into the methods for finding Egyptian Fraction Representations
• The Elgamal Cryptosystem
• A Simple Derivation of the Non-Recursive Definition of Two-Term Linear Recursive Integer Sequences
• No End to the Happy End: Convex Graphs of Erdosian Circuits
• The Shamir Threshold Scheme
• The Pappus Hexagon Theorem
• Mathematical Algorithms for the Computation of Integers in Electronic Devices
• Mersenne Primes and Perfect Numbers
• Paradox
• Nim
• Winning the game of Nim using Strong Induction
• Fermat's Little Theorem and Carmichael Numbers
• On Sets with Distinct Subset Sums

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