Extension of Ramanujan's Congruences for the  Partition Function Modulo Powers of 5

Abstract

We investigate the optimality of Ramanujan's congruences for the partition function modulo powers of 5.  In particular, are there subprogressions of Ramanujan's arithmetic progressions, other than those found by Ramanujan himself, where the congruence modulo 5^j is actually a congruence modulo a higher power of 5?  We answer this question in the affirmative byeplicitly exhibiting infinitely many such systematic extensions.