Pete  L. Clark
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  Math 4400/6400 -- Number Theory: MWF 11:15-12:05, 222 Boyd

Instructor: Assistant Professor Pete L. Clark, pete (at) math (at) uga (at) edu

Course webpage: http://math.uga.edu/~pete/teaching.html (i.e., right here)

Office Hours: Boyd 502, Monday 9-10 am, Tuesday 9-10 am; by appointment.
I am quite amenable to booking ``extra'' office hours. The ground rules are: (i) please give me at least 24 hours notice. (ii) Please send me an email the night before a morning appointment or the morning of a later appointment to remind me that we are meeting. (iii) If I do not show for an appointment (empirically, the chance that I will fail to show seems to be about 5-10%), feel free to call me on my cell phone. Probably I'm not too far away. (iv) If we do book an appointment, please do show up or call or email to let me know you're not coming!

Course text: The two recommended texts are Joseph Silverman's A Friendly Introduction to Number theory and William LeVeque's Fundamentals of Number Theory . Neither is required: instead there are online lecture notes: see below.

Discussion Section: I would like to have an extra discussion section, one hour a week, to discuss problems and also for students to present short presentations. I will send out email trying to find a minimally inconvenient time for this.

How Your Grade is Computed: Will you permit me to be a bit vague about this for now? I'm thinking roughly
50% homework
30% final (conceivably the final could be an exam, a project or both)
20% in class (participation, tests and quizzes, if any).

Course Content: This course offers an introduction to number theory, suitable for undergraduates majoring in mathematics. The main prerequisite is some modern algebra as in Math 4000 (as well as calculus and some proficiency with mathematical proofs).

Note that I said introduction but not elementary or recreational. In fact Math 4000 treats a fair amount of the truly elementary number-theoretic material: e.g. unique factorization, gcd's, divisibility, and so forth. I feel strongly that this material should be covered in an introductory course on number theory, so there could be a certain amount of duplication of material at the beginning. The flip side of this is that it seems fair and even advisable to cover some of this material more rapidly and/or in greater depth than what was presented in Math 4000.

In point of fact the treatment will be relatively elementary. I will try to be flexible about the amount of algebra that gets used and will, where possible, discuss one proof of a result using some algebraic ideas and another proof which avoids them.

Graduate credit: The course is also listed as Math 6400, i.e., a graduate level course. The distinction here will be most prominent in the problem sets: in order to get graduate credit, students must do more starred problems and graduate problems. Graduate problems have so far tended to be more abstract and/or more explicitly algebraic.

History track: One traditional component of the leisurely "recreational" number theory course that I admire is the attention to the history; number theory does indeed have an unusually long and rich history (for instance, unlike calculus and its descendants, all the great civilizations of the world contributed some interesting number theory). Number theory also lends itself better to a historical approach than some other subjects because of the many interesting conjectures which remain unproved, or only partially proved. Students with an interest in this aspect of the material will be encouraged to undertake history projects, perhaps even as an alternative to some of the more difficult theoretical material covered in the course. I will suggest topics for short (under three pages, I should think) papers. An especially good paper could result in an in-class presentation; those with interests in math education should find this especially appealing.

Course credo: I hope to create a course in which you will learn a lot and in which you will have a lot of fun. I took a summer course in number theory way back in 1992 (I was in high school at the time) and an undergraduate course in 1996. I loved both of these courses, but I would have had even more fun if we had covered certain topics more deeply. Now I am myself a number theorist, so I have in many cases the luxury of choosing to present material in ways that seem more conceptual, more powerful, and more modern than standard textbook presentations. Of course as a research mathematician, this is my idea of fun (more so than, say, the Fibonacci numbers and the golden ratio). But in case you are not planning to go on to graduate study in number theory, the course doesn't serve as a prerequisite to anything else, so you should be taking it for fun too. If you're not having fun, let me know, and I will do what I can to help, for instance, by allowing you to take a more historical approach.


Algebra Handouts: Because I am not sure exactly what algebra you know and what you don't, I am taking the liberty of collecting some results in algebra handouts. Don't be scared -- at the 4400 level very little of this material will be required of you in any formal sense. However, if you know some algebra, you might as well learn a little more: it will make things easier, not harder. (Students taking the 6400 class should look at the algebra handouts in detail.)

  • Handout A1: Rings, Fields and (Mostly Commutative) Groups. (10 pages) (pdf)

  • Handout A2: Ideals and Quotients. (6 pages) (pdf)

  • Handout A2.5: More on Groups. (10 pages) (pdf)

  • Handout A3: Integral Elements and Extensions. (4 pages) (pdf)


  • Handout 1: Introduction; The Fundamental Theorem and Some Appplications. (17 pages) (pdf)

  • Handout 2: Some Irrationality Proofs. (6 pages) (pdf)

  • Handout 3: The Gaussian Integers. (6 pages) (pdf)

  • Handout 4: Sums of Two Squares. (7 pages) (pdf)

  • Handout 5: The Fermat Equation in Exponent 2: Pythagorean Triples. (4 pages) (pdf)

  • Handout 6: The Fermat Equation in Exponent 4. (4 pages) (pdf)

  • Handout 7: Arithmetical Functions I: Multiplicative Functions. (6 pages) (pdf)

  • Handout 8: Arithmetical Functions II: Convolution and Inversion. (6 pages) (pdf)

  • Handout 9: Arithmetical Functions III: Orders of Magnitude. (8 pages) (pdf)

  • Handout 9.5: A Word About Primitive Roots. (3 pages) (pdf)

  • Handout 10: Gauss' Circle Problem. (6 pages) (pdf)

  • Handout 11: More on Average Values. (4 pages) (pdf)

  • Handout 12: The Primes: Infinitude, density and substance. (10 pages) (pdf)

  • Handout 13: The Prime Number Theorem and the Riemann Hypothesis. (8 pages) (pdf)

  • Handout 14: The Pell Equation. (10 pages) (pdf)

  • Handout 15: Quadratic Reciprocity I. (8 pages) (pdf)

  • Handout 16: Quadratic Reciprocity II: The Proof. (6 pages) (pdf)

  • Handout 17: Rational Quadratic Forms and the Local-Global Principle. (pdf) (13 pages)

  • Handout 18: Repesentations of Integers by Quadratic Forms. (8 pages) (pdf)
  • Bonus Handout I: The Quadratic Reciprocity Law of Duke-Hopkins. (5 pages) (pdf)

  • Bonus Handout II: A Theorem of Minkowski; The Four Squares Theorem. (14 pages) (pdf)

  • Bonus Handout III: The Chevalley-Warning Theorem (Featuring...the Erdos-Ginzburg-Ziv Theorem). (14 pages) (pdf)

    HOMEWORK PAGE Click here to access the homework assignments and their due dates. Do so at least once a week. Ignorance that a problem set is due is not an excuse!