## Course Goals

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.

## Course Notes

- Section 1: Integers and Divisibility
- Section 2: Prime Numbers
- Section 3: Modular Arithmetic
- Section 4: Exponents and Pseudoprimes
- Section 5: Multiplicative Functions
- Section 6: Primitive Roots
- Section 7: Quadratic Reciprocity

## Homework

- Homework 1, due Friday September 8
- Homework 2, due Friday September 15
- Homework 3, due Friday September 22
- Homework 4, due Friday September 29
- Homework 5, due
**Wednesday October 4** - Homework 6, due Friday October 20
- Homework 7, due Friday October 27
- Homework 8, due Friday November 3
- Homework 9, due Friday November 17
- Homework 10, due
**Wednesday**November 29

## Tests

- Test 1 (Tentatively October 4)
- Test 2 (Tentatively November 8)
- Final Exam (December 11, 8:30 – 11:30 AM)

## Paper

All students in this course will be required to submit a roughly 4-5 page paper on some number theory related topic. I will hand out a separate sheet with topic suggestions.

A partial reference for project ideas is available here. There is a rubric for the paper, which should guide you on what to do and include.

You should write your papers in LaTeX. If you are not familiar with LaTeX, I have a guide to getting started. Feel free to come to me with any questions you have about LaTeX, whether for this class or not.

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).

## References

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.