Dennis Eichhorn




Thursday, May 29, 2008 - 3:00pm


MSTB 254

How many ways can an integer n be expressed as a sum of positive integers? This question is the cornerstone of Partition Theory, and it is surprisingly difficult to answer. For example, if we let p(n) be the number of these expressions of n, even the parity of p(n) remains something of a mystery, despite the fact that it has been studied for over a century. In particular, although empirical evidence (the first several million values) seems to indicate that Po(N) = [the number of odd values of p(n) up to N] is asymptotic to N/2, no one has even been able to show that Po(N) is larger than the square root of N for N sufficiently large. Many advances in discovering the mod 2 behavior of p(n) have been made over the past several years, and most of them have required properties of l-adic Galois representations and the theory of modular forms. However, one lower bound for Po(N) (which was the state-of-the-art for a brief period) was proven using only elementary generating function techniques and results from classical analytic number theory. In this talk, we develop the history of the mystery, and we prove the latter lower bound. The talk will be aimed at the partition theoretically uninitiated, and a great deal of background will be provided.