# Partition Congruences

*by Professor Frank Garvan*

*Institution:*University of Florida, Gainesville

*Date: Wed 20th July 2005*

*Time: 11:00 AM*

*Location: Russell Love Theatre, Richard Berry Building*

*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

congruence:

(*) 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: pnorbury@ms.unimelb.edu.au