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
30% final (conceivably the final could be an exam, a project or both)
20% in class (participation, tests and quizzes, if any).
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.)
Bonus Handout VI: The Coprime Powers Trick (Featuring...The Bachet-Mordell Equation).
Total Pages as of March, 2009: 229
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!
WHAT LINKS HERE
You (no matter who you are) are more than welcome to read, copy, print, or link to any part of
these notes. I ask only the following:
(i) If you wish to make use of any material from these notes, you do need to cite them
explicitly. Failure to do so may constitute plagiarism.
(ii) If you create any paper, note or webpage which uses and cites these notes, please let me
know about it!
So far, internet searches have turned up the following citations: