This course serves two distinct purposes:
Number Theory is the study of whole numbers, especially related to the notion of divisibility of numbers. It evolves around two key notions: On one side the decomposition of numbers into `prime’ components, on the other side modular arithmetic, which is essentially a generalization of the arithmetic that would take place in a 12-hour clock. Number theory has been an object of study since the ancient times, and still contains numerous simple to state but quite intractable questions.
Number Theory also has important applications to Cryptography. In essence, the security of every internet transaction is based on some fundamental number theory facts, the most common amongst them being that if a number is the product of two large prime numbers, then there is no efficient way to recover those prime numbers if all you know is their product. All current cryptographic techniques make heavy use of the tools we will learn in this class. We will explore some of these connections along the way.
As an advanced course in mathematics, this course will push your knowledge regarding the various research and proof techniques that mathematicians employ. We will spend the majority of each class discussing theorems and their proofs that the students will be providing. You will be asked to think about the validity of mathematical statements, to produce proofs for them as well as to confirm the veracity of proofs provided by others.
Here are some of the concrete methods of mathematical thought, proof and analysis that you will practice in this course:
In the class schedule page you will find, for each class day, a list of links to reading assignments. Your homework will require you to have a solid understanding of the material covered there, so I strongly encourage you not to get behind.
You are expected to attend every class meeting. As most of the class period will revolve around presenting your work and discussing other people’s work, your presence is crucial and a lot of the learning in the class will occur in those discussions.
There will be roughly three different kinds of assignments:
Presentations will be a daily occurence in the class. Writeups will likely occur about once a week, while programming tasks will be less frequent.
There will be three exams tentatively schedule for: Friday 10/4, Friday 11/8 and during exam week. The exams you do better on will count slightly more towards your final grade.
Your final grade depends on all the above components, as follows:
| Component | Percent |
|---|---|
| Class Presentations | 25% |
| Written Assignments | 10% |
| Coding Assignments | 5% |
| Worst Exam | 15% |
| Middle Exam | 20% |
| Best Exam | 25% |
This gives a number up to 100, which is then converted to a letter grade based roughly on the following correspondence:
| Letter grade | Percentage Range |
|---|---|
| A, A- | 90%-100% |
| B+, B, B- | 80%-90% |
| C+, C, C- | 70%-80% |
| D+, D, D- | 60%-70% |
| F | 0%-60% |