# The final paper is on a topic of your choice relating to the course material

The final paper is on a topic of your choice relating to the course material, 5-7 pages long, in LaTex or Word, 12 point font, single spaced, submitted via Canvas, . If you are having trouble picking a topic, here some Final Paper Ideas.

The goal of your final paper is to communicate clearly information about the topic you’ve chosen. You should focus on why the topic is interesting you, how it relates to the material we’ve seen in our class, and why the reader of the paper should care about the topic.

It should be mathematically and historically correct, and you should carefully cite any materials which you use.

There will be an element of peer grading for the final paper: you will be asked to comment on the papers of the rest of your Active Learning Group. These will be short comments addressing whether you found the essay compelling, and why.

Here are some ideas for your final paper. Remember that the final paper is on a topic of your choice relating to the course material, 5-7 pages long, in LaTex or Word, 12 point font, single spaced, submitted via Canvas, due on December 14th at 4:30pm . I am also happy for you to suggest ideas as well.Conjectures and Theorems:Pick a famous conjecture or theorem in number theory, and write a brief intellectual history of the problem and attempts to solve it.

Some examples:(1) The Collatz Conjecture: start with an integer x. If it is even, divide by 2. If it is odd, multiply by 3 and add 1. Does this process always end in the cycle 1–>4–>2–>1?(2) Fermat’s Last Theorem:

There are no non-trivial integer solutions (x,y, z) to the equationxn+yn=znfor n > 2.(3) The Twin Prime conjecture: there are infinitely many pairs of primes of the form (p, p+2).(4) Goldbach’s conjecture: every even integer bigger than 2 is the sum of two primes.

Equations and Geometry:Describe the geometry, number theory, and history behind a famous Diophantine equations, for example:(1) Pythagorean triples: solutions to x^2 + y^2 = z^2, where x, y, and z are all integers.

(2) Markoff triples: solutions to x^2 + y^2 + z^2 = 3xyz, where x, y and z are all integers.(3) Apollonian Quadruples: solutions to (x + y + z + w)^2 = 2 (x^2 + y^2 + z^2 + w^2), where x,y,z, w are integers.History and Concepts:

Describe the history of an important concept, for example:(1) Cryptography: a brief history of famous cryptosystems through history, including a discussion of RSA cryptography, and for bonus points, elliptic curves.(2) Prime numbers: when was the concept first understood?

Who have been some of the important contributors to understanding them?People:summarize the mathematical contributions of your favorite mathematician(s).