**Possible undergraduate research projects: for more information, contact me at
sjm1 AT williams.edu**

**Here is a summary sheet of the projects.**

Some papers for potential topics (**REVISING THIS RIGHT NOW!**):

**Irrationality and infinitude of primes:**- Miller-Schiffman-Wieland: Infinitude of primes: there are lots of proofs of the infinitude of the primes. One of my favorite is based on zeta(s) = sum_n 1/n^2 = pi^2/6 is irrational. If you've heard of Euclid's proof, that leads to pi(x) = #{p prime: p <= x} is larger than loglog(x); using some advanced concepts (irrationality measure) we can basically recover this bound by looking at how irrational pi^2 is; unfortunately, the proof of how irrational pi^2 is uses as input pi(x) is about x / log(x). There are ways around this assumption, and we do get something nice.

**Additive and elementary number theory**- Miller-Roman-Sinnott: y^2= n(n+1)(n+2)(n+3): A nice proof when four consecutive integers can and cannot be a square; can you generalize to products of six consecutive integers? This could make a really nice Monthly article.
- More sums than differences: See big papers by
(1)
Nathanson,
(2) Martin and
O'Bryant. Gist: given a finite set of integers A, form A+A and A-A, where
A+A = {a1 + a2: a_i in A} and A-A = {a1 - a2: a_i in A}. As addition is
commutative but subtraction isn't, a generic pair gives two differences and
only one sum. Thus expect A+A to be smaller than A-A. Amazingly, Martin and
O'Bryant showed that a positive percent of the time A+A is larger! If you use
a difference model of choosing sets randomly, however, Hegarty and I showed
almost all sets are difference dominated. In the other direction, I and some
students constructed the world record for densest families of sum-dominated
sets.
- Hegarty-Miller: in a certain model of sets, almost all sets are difference dominated: Some tough open problems at the end.
- Miller-Orosz-Scheinerman: constructing infinite families of sum dominated sets: I have some thoughts about trying to extend this construction.

**Continued Fractions (student reports from Princeton):**Click here for the homepage, with access to computer programs- Digits of continued fractions
- Investigation of lengths of periods of quadratic irrationals
- Average period result: mariusjp.pdf
- Conjecture of Chowla on continued fractions of sqrt(prime): alexajp.pdf

- Closed Form Expansions of Continued Fractions: dfJP2.pdf
- Almost periodic continued fractions: tdJP.pdf

**Differential equations:****Probability topics:**- Miller: derivation of the log5 rule for when one team wins
- Miller: Pythagorean Won-Loss Theorem
- Miller: Die battles: Applications to the game of risk
- Miller: Extending the Pidgeonhole Principle
- Miller: Differentiating identities notes: lots of good math in this, from differentiating identities to matching coefficients of polynomials. This leads to formulas to predict how often you expect to see certain runs / how many runs you expect to see in Heads / Tails.