In Math 322 we will gain a broad understanding of number theory, a mathematical topic that has been of great interest since the ancient Greeks. The two major topics we will study are the distribution of prime numbers and solution sets to Diophantine equations. We will explore connections with modular arithmetic and with cryptography and learn Gauss’s law of quadratic reciprocity. We will also practice good mathematical writing and clear communication of mathematical and technical ideas.
The course syllabus is available here.
- Introduction: Pythagorean Triplets
- Section 1: Integers and Divisibility
- Section 2: Prime Numbers (Updated 9/20)
- Section 3: Modular Arithmetic (Finished Oct 11)
- Section 4: Exponential Congruences (Updated 10/23)
- Section 5: Multiplicative Functions (Updated 11/9)
- Section 6: Primitive Roots (Updated 11/21)
- Section 7: Quadratic Reciprocity (Updated 12/4)
- Homework 1, due on Friday September 9
- Homework 2, due on Friday September 16
- Homework 3, due on Friday September 23
- Homework 4, due on Friday September 30 (Updated 9/25)
- Homework 5, due on Friday October 14
- Homework 6, due on Friday October 21
- Homework 7, due on Friday October 28 (Updated Sunday 10/23)
- Homework 8, due on Friday November 4
- Homework 9, due on Friday November 11
- Homework 10, due on Friday November 18
- Homework 11, due on Wednesday November 30
- Final Exam practice problems
All students in this course will be required to submit an approximately 3 page paper on some number theory related topic. I will hand out a separate sheet with topic suggestions. (If you are still looking for a Senior Comps topic, some of these would make good topics for that as well, although your senior comps paper will of course be longer and more in-depth; you may not submit identical papers for both assignments).
A partial reference for project ideas is available here. There is also a rubric for the paper, which should guide you on what to do and include.
Here is a relatively clean sample document and the LaTeX source code for it, along with a sample bibliography file.
A first draft of the paper will be due on the Monday before Thanksgiving, November 21. Submitting this draft (at all) will be worth 5% of your final grade. I will give feedback and a preliminary grade; your final submission, worth 10% of your final grade, will be due Monday, December 5. (Note: by college policy I can not give extensions on this date).
There is no mandatory textbook for this course. I will post complete lecture notes and homework assignments on this page.
The following references may be helpful to you if you’re looking for extra reading; I will try to include pointers to specific chapters in the notes.
- Elementary Number Theory & its applications by Kenneth H Rosen is the main reference I will be (loosely) following. We have used it for this course in previous years. It is currently in the sixth edition.
- Number Theory: A Lively Introduction with Proofs, Applications, and Stories by Pommersheim, Marks, and Flapan.
- Elementary Number Theory: Primes, Congruences, and Secrets by William Stein is available free online. It is targeted at a slightly higher
level than this course, but you still may find it helpful. It focuses on computational applications and cryptography.
- A Computational Introduction to Number Theory and Algebra by Victor Shoup is also free online. Unsurprisingly, it is also focused on
computational applications. Like the previous entry, it is slightly more advanced than I intend for this course.