number theory on physics

MTH 425/525 Written Project and Oral

Presentation

As part of your grade in this course, you are required to do a written project on a topic in Number Theory. Undergraduates taking the course as MTH 425 are also required to give an oral presentation on this topic. Written projects are due Tuesday, April 17 for MTH 425 students and Tuesday, April 24 for MTH 525 students. Oral presentations will be delivered on April 17, April 19, April 24, and April 26. I will give more details on the specifics of the schedule as time progresses.

MTH 425 students must work on this project in pairs, submitting a single paper for the written portion and delivering a joint oral presentation. MTH 525 students are to work on their own, each student turning a separate paper.

Selecting a topic

Before you begin work on this assignment, you must select a topic. You (and your partner, if applicable) must have the topic approved by me by Friday, April 5. If you don’t do this, you will automatically forfeit 25% of your grade for the written project. You should turn in a written proposal, consisting of a title and a brief (1-2 paragraph) outline of what you propose to write/talk about. The topic may come from anything within number theory; however, certain guidelines must be observed:

• There must be significant mathematical content to the project. For example, I will not approve a project about the life of Gauss (even though he did a lot of number theory). However, I may approve a project about Gauss’ contributions to number theory. Whatever it is that you decide to write/talk about, most of it should involve the explanation of some aspect of number theory not covered in class.

• The topic should be somewhat focused – for example, I won’t approve a topic like “cryptography”; however, “a survey of RSA public key cryptography” could make a nice project.

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• For MTH 425 papers, the written project should be no more than six pages long and no less than four pages long, single-spaced (not includ- ing title page and bibliography). For MTH 525 papers, the written project should be no more than twelve pages long and no less than six pages, single-spaced. It must be typeset; however, you may handwrite certain mathematical symbols if you have trouble typesetting them. The project should be written in a manner that is understandable by your peers.

• Your project should involve the use of at least one non-internet based source beyond the course textbook, and ideally should integrate in- formation from various sources. Reading a research paper from the literature and consulting various other sources to “fill in the blanks” could make for an interesting project.

• The oral presentation should be 15-20 minutes long; in addition, you should be prepared to answer questions posed by me and by your peers. The talk should be pitched at a level understandable by your peers.

• No two MTH 425 projects may be on the same topic, and likewise no two MTH 525 projects may be on the same topic. First come, first served – the sooner you get your project approved by me, the less likely your choice will be taken already.

The web is a convenient place to look for possible topics; however, be ad- vised that web content is overall less reliable than that of books or journals. (Wikipedia is, surprisingly, one of the better electronic sources for informa- tion about mathematics.) There are many math-related sites around; you can find these by using any search engine. If you are having trouble finding a topic, please come and talk to me for possible directions.

Examples of reasonable projects:

1. Pick a research paper and write an expository paper on it, presenting all of the content and writing out the details carefully. 2. A survey of RSA public key cryptography (or other methods). 3. The Riemann zeta-function and its role in number theory. 4. Modular arithmetic and music. 5. The cubic reciprocity law. 6. Pseudoprimes.

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7. The abc conjecture. 8. Fermat’s Last Theorem: from inception to solution. 9. Solving Pell’s Equation. 10. Reed-Solomon codes. 11. A proof of Bertand’s Postulate. 12. Paul Erdös and the role of probabilistic reasoning in number theory.

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