Introduction to Number Theory

Fall 2013

A 200x200 grid on which the primes have been mapped in a spiral.

**Official description:**Examines divisibility properties of integers, congruencies [sic], diophantine equations, arithmetic functions, quadratic residues, distribution of primes, and algebraic number fields. Prereq., MATH 3140. Undergraduates must have approval of the instructor.**My unofficial description:**This course aims to introduce students, with a minimum of background (i.e. first-year graduate students), to the main themes of number theory. It is not just an undergraduate `elementary number theory' course on steroids: instead of focussing on elementary methods, we will introduce modern abstract viewpoints via very concrete examples, and motivated by some of the central questions of the subject.**Themes:****Algebraic number theory via quadratic number fields:**unique factorisation, ideals, lattices and quadratic forms.**Analytic number theory via the distribution of primes:**arithmetic functions and the zeta function.**The local perspective:**congruences, finite fields, quadratic reciprocity and*p*-adic numbers.**Computational and algorithmic number theory:**factorization and the search for primes, Sage Mathematical Software.**Applying everything above to the solution of Diophantine equations.**

- For those comparing to last year's course, I will require less algebra background. In particular, since we will restrict to quadratic number fields, I won't expect you to know Galois Theory, and we won't do Dedekind domains in the abstract. Restricting to quadratic fields unfortunately misses some of the interesting complexity of number fields in general, but it allows us to see a lot of the main themes without as much background. Students wishing to see the more general theory will have a brief overview and reading options. Sadly, I'm also relegating Continued fractions and Diophantine approximation to the "if we have time" category so I can slow the course down slightly.

- Grading breakdown: Homework 60%, Takehome Midterm 20%, Takehome Final 20%.
- If you like, you can replace some homework with a small project on a topic of special interest to you. This generally (but not necessarily) means describing a general theory in your own words and either a) working out novel examples in detail or b) implementing algorithms on the computer. If you choose this option, the project is due on the last day of class. Talk to me for details. This is especially appropriate if you have taken number theory before and wish to do something in place of a repetitious part of the course, but it is open to anyone with a special interest in any topic.
- Students who are repeating the course may do a more significant number theory project in place of homework and exams.
- Textbook: None. There will be recommended reading from a variety of sources, as well as course notes.

- Course Notes - will be updated frequently.

**Due Wednesday, September 4th:**Your first homework assignment is here. For more on Sage, see below.**Due Friday, September 20th:**Your second homework assignment is here and solutions are here.**Due Friday, October 11th:**Your third homework assignment is here and solutions are here.**Due Monday, December 9th:**Your fourth homework assignment is here and solutions are here.**Takehome Midterm Due Monday, October 28th:**Now up and solutions are here.**Takehome Final Due Thursday, December 19th:**Now posted and solutions are here.

**Sage:**Use Sage at Colorado. For information on sage, see sagemath.org. For some number theory functions, see here including the quick reference sheet.**Sage Worksheets Designed for Our Class:**

Introductory worksheet

Arithmetic functions worksheet- A few fun visuals of the zeta function here and here. You can get involved in computing zeroes at ZetaGrid.
- Joe Silverman's Arizona Winter School Lecture Notes on Arithmetic Dynamics
- Mandelbrot set explorer