# Professor Kevin O'Bryant

I describe my research to other mathematicians as "combinatorial number theory". The truth is that most combinatorialists consider it number theory and most number theorists consider it combinatorics. More recently, I've been working on computational number theory, which is a largely a mix of programming, calculus, and numerical analysis.

Number theory is the study of properties of the natural numbers  0, 1, 2, 3, .... For example, p is a prime number if it is not the product of two smaller numbers. So, 14 is not a prime number (14=2*7), and also 15 and 16 are not prime numbers, but 17 is. So the first question is how many prime numbers are there? The next question is what are they? We know that there are infinitely many, and we even know approximately how many there are less than x, for any number x. To date, we don't even know how many n make 2^n+1 a prime (we believe infinitely many) or make 2^(2^n)+1 a prime (we believe there are only 5).

Combinatorics starts with the science of counting. Sound easy? It starts that way, but as with everything humans touch it has grown in difficulty to the point that progress is more art than science. How many different Su Doku boards are there? Questions about certain objects are more amenable to solution by counting than others, and the study of such objects (graphs, hypergraphs, permutations, et cetera) is also considered to be combinatorics. Here's a favorite game: I put some points on a sheet of paper, and connect some pairs with curves in such a way that you can go from any point to any other point on the curves, but you can't get back to where you started without backtracking. Now you get to label the points with distinct nonnegative integers, and label each curve with the difference of the labels of the points at its two ends. If you have labeled the curves with the numbers 1, 2, ..., each curve getting a different number and no number being skipped, then you win. Otherwise I win. There's evidence that you can always win (for example, you can win if I start with 16 points, no matter how I draw the curves or how many curves I draw), but nobody knows for sure --- yet.

Combinatorial number theory consists of questions about the natural numbers {0, 1, 2, ...} that can be addressed with combinatorics. My favorite example is that of Sidon sets. A Sidon set is a set of naturals with the property that if you secretly pick two and announce their sum, then I can take that sum and figure out which two you picked. For example, the set {2,3,4} is not a Sidon set, if you give the sum 6 then I couldn't deduce if you had picked 2 and 4, or if you had picked 3 twice. The set {0, 1, 4, 6} is a Sidon set. So the question is how many elements can be in a Sidon set if the largest element is n? Even with n=1000 the answer is not known. The problem (and its partial solutions) have been used in cryptography, radio frequency assignments, and in other seemingly unrelated areas.

These problems are appealing for several reasons. A big reason is that they are easy to state and explain. Another reason is that I enjoy programming and some of these problems can be approached experimentally. Another reason is that I enjoy the company of many people who also enjoy these problems.