Projects

Presentations:

Monday, 12/7/15: Hakim, Collin, Rose

Wednesday, 12/9/15: Andrew, Matan, Josh, Nico

Paper due date: 12/11/15

On this page, I'll list potential projects for students to pursue for their final project and presentation for the class.

Projects:

Collin - Continued Fractions

Rose - Mathematics of Lagrange

Matan - Bertrand's Postulate

Andrew - Gaussian Integers

Hakim - The pythagorean theorem, pythagorean triples, and Fermat's last theorem

Josh - Partitions of integers

Nico - Pseudorandom Numbers

Details of the project: Students will be expected to write a short 8-10 page paper on a topic of their choice (pending approval by the instructor) on a topic related to number theory of their choice. Students will also give a short 10-15 minute presentation on the approved topic during the last week of classes. Projects must be done individually, and no two students may work on the same project. Please use this as an opportunity to explore a topic in Number Theory you are interested in! The paper should include some short theorems and proofs as well, but I will not require you to prove the difficult theorems (since it would put you over the 10 page limit :) ). You should think of this like writing a chapter of a book (and NOT a book report) where you fully explain concepts and ideas.

Sample Projects:

1. The pythagorean theorem, pythagorean triples, and Fermat's last theorem: Proofs that x^3 + y^3 = z^3 and x^4 + y^4 = z^4 have no integer solutions. The first was proved by Fermat, the second was proved by Euler. A brief history of attempts to solve this problem.

2. Partitions of integers: given a positive integer n, we can write it as a sum of smaller positive integers, where order doesn't matter. For example, 5 = 3 + 2 and 5 = 3 + 1 + 1. An important question to ask in Number Theory is how many partitions a given number n has. Moreover, what if we put restrictions on the parts? For example, how many partitions does an integer n have where the parts are all distinct? The study of integer partitions was initiated by Euler and continues to this day, and is an incredibly rich area of study.

3. The Gaussian Integers Z[i]: This is the set of complex numbers of the form a+bi, where a,b are both integers. This set, maybe surprisingly, behaves a lot like the integers in a lot of respects. For example, we have a division algorithm, primes, etc. But, many things change. For example, 5 is no longer a prime in Z[i], since we can now write 5 = (1+2i)(1-2i), while 3 and 7 are still prime in Z[i]. This is a fascinating set of numbers that often pops up in abstract algebra and number theory.

4. Cryptography: In class, we'll touch on how number theory is applied to encrypt and decrypt messages. There are many, many other ways we didn't touch on to encrypt and decrypt messages, often involving primes (AKS cryptosystem, El Gamal Cryptosystem, Symmetric Key Cryptography, etc.) The goal of this project would be to learn and explain how these techniques are used to encrypt and decrypt messages to secure communications between parties. Another topic of interest here is to determine how to decrypt messages using things like statistical attacks.

5. The proof of Bertrand's postulate: Paul Erdos gave an elementary proof that, given a positive integer n > 1, there is always a prime between n and 2n. This project would explore and explain this proof and related ideas.

6. Factoring and Primality Testing: An important question in number theory and especially in cryptography is: "Can we factor numbers quickly, or determine quickly if numbers are prime?" This project would explore various tests for primality (Miller-Rabin, AKS primality test) and factorization techniques.

7. Quaternions: We're all familiar with the complex numbers, numbers of the form a+bi where a,b are real numbers. We'll see in class that these numbers are strongly tied to integers which can be expressed as a^2+b^2, where a,b are both integers. This project explores the quaternions, a number system that extend the complex numbers, and its ties to number theoretic problems (for example, that integers can be expressed as sums of 4 squares). The quaternions are numbers of the for a+bi+cj+dk, where a,b,c,d are real, and i^2 = j^2 = k^2 = -1, and ij = k, jk = i, and ki =j.

8. Continued Fractions: Continued fractions are fractions of the form

In this project, explore the idea of continued fraction, both finite and infinite. There are many, many interesting things that can be proved about these. For example,

9. Dirichlet's Theorem and primes in arithmetic progressions: In class, we discussed Dirichlet's theorem, which states that there are infinitely many primes of the form an+b if (a,b)=1. This project would examine what we know about this theorem (including its proof), for example: How many consecutive primes occur in an arithmetic progression? What are the largest known primes that occur as consecutive terms in an arithmetic progression? Students should also explore the recently proved Green-Tao theorem and its consequences.

10. Pseudorandom Numbers: Random numbers are useful in many applications (computer simulations, computer games, generating random samples, etc). In this project, students would study techniques for generating numbers we call pseudorandom numbers: numbers that are chosen in a methodical manner but appear to be random. This project would involve exploring techniques for generating such numbers, as well as their applications in other fields.