Syllabus M312 Number Theory

When I lately had the pleasure of attending the new Slade Professor’s inaugural discourse, I heard him promise to make his pupils participators in his work, by painting pictures in the presence of his class. I aspire to do more than this—not only to paint before the members of my class, but to induce them to take the palette and brush and contribute with their own hands to the work to be done upon the canvas.

–James Joseph Sylvester

Instructor: Steve Kennedy
Office Hrs: MW 12:30-2:00, Tu 3:00-4:30
Office:  CMC 225, ext. 4333
E-mail:  skennedy

Course Content: Classical Number Theory began with the ancient Greeks; there is number theory in Euclid’s Elements. It reached it’s ancient pinnacle (maybe I should say apotheosis, there’s a good Greek word for you) with Diophantus who lived in Alexandria sometime around 250 AD. Diophantus was interested in positive integral solutions to, what are for us, algebra problems. A typical Diophantine problem is illustrated by this fourth-century riddle about D himself: his boyhood lasted 1/6 of his life, his beard grew after 1/12 more, after 1/7 more he married, his son was born 5 years later, that son lived to half his father’s total, and the father died 4 years after the son. It was in his copy of a translation of Diophantus’s Arithmetica that Fermat scrawled his famous marginal note. Classical number theory uses mostly elementary arguments to prove cool facts about the positive integers. Elementary in this usage is not the same as easy, often the argments are incredibly complex and ingenious. Elementary, here, means without the use of sophisticated theories from other branches of mathematics. Modern number theorists must have deep knowledge of modern algebra, analysis, combinatorics,and even probability. We won’t go there.

Course Structure: Mathematics is not a spectator sport–the only way to learn it is by doing. Thus, I will lecture only rarely. Instead I have drawn up a list of problems through which we will work together. You will take turns presenting the solutions to these problems and writing them up. At the end of the term we will gather all the write-ups together into our own number theory book of which you will be the authors.

More precisely, when you have solved one of the problems on the list and wish to present your solution to the class send me e-mail telling me which problem you wish to present and approximately how much time you will need. I will schedule your presentation for the next available time. Someone else in the class will be the official note-taker for your presentation, the scribe. That person will write up your solution very carefully, it is recommended that the scribe consult with the presenter about the write-up. This write-up is due at the next class meeting. I will assign two referees to read this write-up and check it for correctness and readability. Both referees will bring to the next class meeting their commentaries on the write-up. The referees may consult with the presenter and/or the scribe, or they may remain anonymous if they desire.

Expectations: The presenter will give a clear, well-rehearsed talk. He or she should find a volunteer or two to practice on in the day or two before the class presentation. It is not necessary that the presenter solve the problem in isolation. The presenter may consult with anyone in the class while working on the problem and its presentation. Presenters are expected to give proper credit to those who contribute substantively to their solutions. Up to two such collaborators can be named. No presentation credit is awarded to collaborators. Frequently named collaborators will be given some scribing and/or refereeing credit.

The scribe will type up in LaTeX, a word processing program specifically designed to typeset mathematics, the problem and its solution. It is expected that this write-up equal or exceed in clarity that of a typical proof in one of your textbooks. On second thought make that–should definitely exceed the clarity of a typical textbook! The scribe will receive the referees’ reports and make appropriate changes to the write-up in response. A copy of this revised write-up will be distributed to the entire class.

The referees will carefully read the scribe’s write-up. They will eliminate (well, at least identify) all errors—mathematical, grammatical, stylistic, and typographical. They will write a report evaluating the correctness of the mathematics and the style and clarity of the presentation. They will suggest changes to improve any of these. They may be asked to read the scribe’s revised version.

Evaluation: We will have one midterm exam, very tentatively scheduled for October 16. Our final exam is scheduled for Thursday, November 19 at 7:00 p.m. The final exam will most likely be some sort of group presentation. Thus, you must be present at the scheduled time. Your final grade will depend upon the quality and quantity of your presenting, scribing, and refereeing and your performance on the midterm and final. Each of these five activities will contribute equally.