New congruences for the partition function

Abstract

Let p(n) denote the number of unrestricted partitions of a non-negative integer n. In 1919, Ramanujan proved that for every non-negative n p(5n + 4) ≡ 0 (mod 5), p(7n + 5) ≡ 0 (mod 7), p(11n + 6) ≡ 0 (mod 11). Recently, Ono proved for every prime m ≥ 5 that there are infinitely many congruences of the form p(An + B) ≡ 0 (mod m). However, his results are theoretical and do not lead to an effective algorithm for finding such congruences. Here we obtain such an algorithm for primes 13 ≤ m ≤ 31 which reveals 76,065 new congruences.