## Syllabus for Number Theory (Math 5653), Fall 2016

### TuTh 2-3:15 pm, Chemistry 305.

• Instructor: Mikhail Ershov
• Office: Kerchof 302
• e-mail: ershov at virginia dot edu
• Office hours: 3 hrs TBA and by APPOINTMENT
• Course webpage:   http://people.virginia.edu/~mve2x/5653_Fall2016
Text: Elementary Number Theory, Gareth Jones and Josephine Jones, corrected edition.

### Prerequisites

• 1. You should have taken MATH 3354 and MATH 1320 prior to this class (note that MATH 3354 is an official prerequisite for MATH 5653, while MATH 1320 is a prerequisite for MATH 3354).
• 2. You should be comfortable with basic analytic techniques (mainly convergence and divergence tests for series and improper integrals) at the level of MATH 1320.
• 3. You should be comfortable with basic group theory and you should have seen the very basic notions of ring and field theory (fields, rings, ideals).
• 4. You should have substantial prior exposure to proofs. You should be completely comfortable with the basic proof techniques (induction, contradiction, case exhaustion etc.) and have experience in both reading (and understanding) proofs and writing your own proofs.
If you are not comfortable with proofs, you will likely find this course very challenging. Prerequisites 3 and 4 should normally be satisfied if you took MATH 3354.

### Course content

Below is a tentative list of topics I plan to cover and approximate amount of time we will spend on each topic.
• Solving polynomial congruences (5 lectures). This will contain a brief review of relevant 3354 material. Chapters 3,4 and parts of 1.
• The Euler function and the structure of the group of units of Zn (5 lectures). Chapters: 5 and 6.
• Quadratic reciprocity and applications (3 lectures). Chapter 7.
• Euclidean Rings and representations of integers by sums of squares (3 lectures). Chapter 10; some material not in the book.
• Pell's equation and continued fractions (3 lectures). Not in the book.
• Fermat's theorem (2 lectures). Chapter 11.
• Arithmetic functions (2 lectures). Chapter 8.
• Riemann zeta function and distribution of primes (4 lectures). Chapter 9; some material not in the book.

### Homework

Weekly homework will be assigned, but you will not be formally evaluated on it. Nevertheless, you are encouraged to write up your solutions and turn them in (for feedback). I may not have time to grade all the problems, but I will try to grade as many as possible.

### Evaluation

The course grade will be based on four tests and the final, with weights distributed as follows:
• tests: 65% (only three of the four will be counted)
• final: 35%
The final will be given in class on Sat, Dec 10th, 9am-12pm. One of the four tests (either the second or the third one) will be given in class; the rest will be take-home. The tentative test dates (due dates in the case of take-homes) are Sep 22, Oct 13, Oct 27 and Nov 17 (all Thursdays).

### Problem session

I plan to run a problem session (most likely on Fridays), starting in week 3. If we are able to find a time slot that works for everyone, extra credit will be given for (active) participation in the problem session. If not, you are still strongly encouraged to attend the problem session if it fits your schedule.

### Make-up policy

• Make-ups or extensions (for take-home exams) will be given only under extreme circumstances (such as serious illness). Except for emergencies, you must obtain my permission for a make-up (resp. extension) before the exam (resp. due date).
• If you miss an in-class exam or fail to submit a take-home exam by the due date without a compelling reason, you will be assigned the score of 0 on that exam.
• University regulations specifically prohibit early make-ups.

### Collaboration policy (on tests).

You MAY NOT discuss problems with other people or use any resources (including web) except for the class textbook and your class notes. You may ask me questions about the exam problems, but normally I will only provide brief hints.

### Announcements

Major announcements will be made in class and also posted on the course webpage. Some other announcements may only be made by e-mail, so check your e-mail account regularly.