### Number Theory 346 / 377, McGill, Winter 2017

### Dr. Steve Lester

### Schedule: MW 4:05-5:25 Trottier Building 1100

(to be posted on MyCourses)

### Course Outline

This course serves as an introduction to elementary number theory. The course topics include:
- The Euclidean algorithm, unique factorization and linear diophantine equations
- The ring of integers modulo n, Wilson and Fermat's little theorem, the Chinese Remainder Theorem
- Primitive roots
- Quadratic residues, Legendre symbol, Gauss sums, quadratic reciprocity
- Roots
of polynomial congruences, Hensel's lemma
- Primality testing, public key encryption
- Quadratic equations, sums of two squares
- Continued fractions, Pell's equation
- Diophantine approximation, Liouville's theorem on rational approximations (optional)
- The prime number theorem, Dirichlet's theorem on primes in arithmetic progressions (optional)
- Pythagorean triples and Fermat's Last Theorem (optional)

A detailed syllabus can be found here.

### Grading

Math 346: 25 % homework, 25 % midterm, and 50 % final

Math 377: 20 % homework, 20 % midterm, 40 % final, and 20 % course project

### Midterm

There will be an in class midterm on Wednesday, February 22.
In the case you are unable to take the midterm (valid reason and documentation required), your final exam will be worth 75 % of your grade (for Math 346), or, 60 % of your grade (for math 377).

### Course Project

Students enrolled in Math 377 are assigned a course project which consists of an 8-12 page paper (written in Latex). Students are free to choose any topic relevant to the course, some
possible topics for the project may be found here.
The deadlines for the course projects are as follows:

January 30: Email me the topic of your choice (suggested). These will be assigned on a first-come, first-served basis to avoid duplicates. You are free to change the topic before March 6.

March 6: Submit a one-page proposal for your project (your topic must be approved prior to the proposal). The proposal will be worth 2% of the total 20% that the project is worth.

April 7 (midnight): Final projects are due. Please email me a PDF copy, the project will be graded on quality of exposition and mathematical rigor.

### Textbook

There is no required textbook for the course. The following texts will complement the lectures:
"Fundamentals of Number Theory", by W. J. LeVeque, Dover 1996.
"Elementary Number Theory: Primes, Congruences and Secrets", by William Stein, Springer 2008. Also available online.

### Contact information

Office hours: MW 2:00-3:00, BH 1243

Email: sjlester (at) gmail.com