Research Statement Dominic Klyve

My research concerns explicit inequalities in elementary number theory. Specifically, I am interested in the distribution of twin primes, along with theoretical and computational techniques for putting explicit bounds on classes numbers such as twin primes. One important method I use to examine the density of twins is to look for bounds on Brun’s Constant. 1. Motivation and Background Twin primes are pairs of primes of the form (p, p+ 2). A natural first question to ask is “How many twin primes are there?” This is a very elementary question, in that understanding it requires no mathematics than most students have learned by the eighth grade. It is this simplicity that makes the question so tantalizing, as answering it remains beyond the capabilities of modern mathematics. The Twin Prime Conjecture states that there are infinitely many primes. The Twin Prime Conjecture also has a strong version, which states not only that there are infinitely many twin primes, but also estimates the number of twin primes less than a given bound x. Let π2(x) represent the number of primes p ≤ x − 2 such that p+ 2 is also prime. Then we have the following. Conjecture 1. Let α be the twin prime constant,