Final Project

The goal for the final project is for you to learn about some piece of number theory that we didn’t talk about in class.

Instructions

1. Form a group of about 2–4 and choose a topic that’s related to course material but that we did not cover. There are many possibilities here! Here are some:

   • Quadratic reciprocity (eg, chapter 4 in Ste17)
   • Continued fractions (eg, chapter 5 in Ste17)
   • Elliptic curves (eg, chapter 6 in Ste17)
   • Algorithms for testing primality
   • Algorithms for factoring numbers
   • Algorithms for finding all prime numbers up to a given point
   • Algorithms for finding big prime numbers
   • Cardinality of sets (eg, chapters 11–14 in Ham18)

   Feel free to choose something not on this list. During week 3, you’ll be asked to submit a proposal. All you have to do for this is tell me who you’ll be working with and what you’re planning on doing. Submit it through gradescope as a group submission by following these instructions.

2. Write up a document in which you give a brief exposition of the topic and also do “something interesting” with your topic. Here are some thoughts about this:

   • It should be written in LaTeX. The length is flexible (somewhere in the whereabouts of 3 pages might be reasonable).

   • Make sure to give clear definitions of any new terms. If there’s a theorem involved whose proof is too long or hard, you can omit the proof and just provide the statement. Write in a way that a fellow classmate would be able to understand what you’ve written, and make sure to cite your sources.

   • If you’re talking about an algorithm, you should include a description of the algorithm (eg, in pseudocode). Also include any proofs of correctness or of algorithmic complexity that are accessible to you.

   • “Something interesting” is intentionally vague. For example, you could write some code: write a bit about what your program does and what outputs you got, and email me the code you wrote as a separate file. Alternatively, if you’re studying a topic discussed in the book, you could pick out a few exercises from the relevant sections of the book and solve them.

     If you have an idea for something interesting you might want to do but aren’t sure about it, feel free to run your idea by me!

   Submit the document through gradescope as a group submission by following these instructions.

3. Prepare a brief and informal (10ish minutes) presentation about your topic for your class. It can take any format you like: just make sure everyone in your group is involved in presenting.

Grading

Your project will be graded out of 15 points, as follows.

• (5 points) Mathematical content. For full credit, make sure that there’s some substantial mathematical content to whatever you do.
• (3 points) Exposition. For full credit, write up your exposition clearly and precisely, in a way that’s easy to follow. Make sure to cite your references too.
• (3 points) “Something interesting.” If you do exercises from the book, you’ll get full credit if your solutions are correct and you pick out some exercises that are not too easy. If you do some programming, you’ll get full credit if your code works like you think it should.
• (4 points) Presentation. For full credit, your presentation should clearly express the main ideas of what you worked on.