Congruences of the Partition Function

Ramanujan also conjectured that congruences (1) exist for the cases A = 5 , 7 , or 11 . This conjecture was proved by Watson [17] for the cases of powers of 5 and 7 and Atkin [3] for the cases of powers of 11. Since then, the problem of finding more examples of such congruences has attracted a great deal of attention. However, Ramanujan-type congruences appear to be very sparse. Prior to the late twentieth century, there are only a handful of such examples [4, 6]. In those examples, the integers A are no longer prime powers. It turns out that if we require the integer A to be a prime, then the congruences proved or conjectured by Ramanujan are the only ones. This was proved recently in a remarkable paper of Ahlgren and Boylan [2]. On the other hand, if A is allowed to be a non-prime power, a surprising result of Ono [12] shows that for each prime m ≥ 5 and each positive integer k, a positive proportion of primes l have the property