by Professor Frank Garvan
Abstract: This talk concerns divisibility properties of p(n), the
number of partitions of n; i.e. the number of ways of writing n as a sum
of positive integers disregarding order. In the June 18 issue of Science
News there was an article entitled "Pieces of Numbers". This article
contained some history of Ramanujan's partition congruences and described
recent work of Karl Mahlburg, a graduate student at the University of
Wisconsin. In this talk we will try to explain some of the mathematics
behind this article. We will start with Ramanujan's simplest partition
(*) p(5n + 4) = 0 (mod 5).
We will outline two proofs of this congruence. The first proof is
elementary and the second uses the theory of modular forms. The second
proof leads to more general congruences, which involve calculating
eigenvalues of certain Hecke operators. This idea was used by Ono
to prove the existence of infinitely many partition congruences.
For More Information: Paul Norbury tel.8344 5534 email: email@example.com