Pete L.
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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) Please understand that I may be coming into campus specifically or primarily to meet you. I am more than happy to do this -- unless you don't show up. If you can't make it, please let me know ASAP.

50% homework

20% final exam

20% final paper

20% in class (includes participation plus performance on a midterm exam).

In other words, Math 4400 is a course for math majors towards the end of their undergraduate career (or with equivalent knowledge and experience). It is possible -- and even somewhat traditional -- to teach a very elementary undergraduate number theory course, which would be equally accessible to bright high school students. This is not such a course. I assume for instance that you have had prior exposure not only to congruence modulo n, but to the notion of congruence modulo an ideal in a ring. Not that you have to have mastered such things seamlessly (whatever that might mean): rather there are handouts describing every algebraic concept and theorem that we will use, starting with the definition of a ring. In other words, I will be more than happy to remind and help you with abstract algebra, but it's much harder to remind someone of something that they never knew!

1) 6400 students are be expected to have a more solid and mature command of their background knowledge. This does not necessarily mean that they will be required to know more material, but they will have a graduate level thoroughness and flexibility with regards to the material that they do know.

2) 6400 students are expected to solve more of the difficult homework problems, including some which are more abstract and may involve some outside reading.

3) 6400 students will write longer and more substantial final papers. The papers will either be concerned with deeper mathematical topics or demonstrate a more sophisticated level of scholarship -- i.e., will synthesize material from multiple sources, including some primary sources.

Part I: Introduction: Fundamental Theorem, Linear Diophantine Equations, Some Irrationality Proofs

Part II: Rational and Integral Points on Curves

Part III: Algebraic Number Theory

Part IV: Arithmetical Functions

Part V: The distribution of primes

Part VI: Geometry of Numbers

Part VII: The Chevalley-Warning Theorem

Part VIII: Dirichlet series and Dirichlet's theorem It is rather likely that we will not get to this material.

HOMEWORK

Homework Assignment 0: Due Wednesday, 1/21/09, further problems due Wednesday 1/28/09 (pdf)

Homework Assignment 1: Due Monday, 2/9/09 (pdf)

Homework Assignment 2: Due Wednesday, 2/18/09 (pdf)

Homework Assignment 3: Due Friday, 2/27/09 (pdf)

Homework Assignment 4: Due Monday, 3/23/09 (pdf)

Homework Assignment 5: Due Monday, 3/30/09 (pdf)

Homework Assignment 6: Due Wednesday, 4/8/09 (pdf)

Homework Assignment 7: Due Friday, 4/24/09 (pdf)

FINAL PROJECTS

A list of 30 final project ideas, described in varying amounts of detail: (pdf)

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:

Quadratic Reciprocity, Jordan Schettler, University of Arizona

Quaternion algebras and quadratic forms, Master's thesis, Zi Yang Sham, University of Waterloo

Ungelöste Rätsel, Ein kleiner mathematischer Rundgang, Jörg Resag

Davenport-Cassels Theorem, French Wikipedia

Group Theory: an introduction and an application, Nathan Hatch

VIGRE Seminar: Introducing the p-adic numbers, Jim Stankewicz, UGA (click on Writings/Talks and scroll to the bottom)

Minkowski's Convex Body Theorem, Isabelle Bensimon, Boston University

Project IV (MATH4072) 2013-14 The Polynomial Method, Dan Evans, Durham University