The Ramanujan Journal

, Volume 5, Issue 1, pp 53-63

First online:

New Congruences for the Partition Function

  • Rhiannon L. WeaverAffiliated withCarnegie Mellon University

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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 \(\begin{gathered} p(5 + 4) \equiv 0(\bmod 5), \hfill \\ p(7n + 5) \equiv 0(\bmod 7), \hfill \\ p(11n + 6) \equiv 0(\bmod 11). \hfill \\ \end{gathered} \)

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.

partition function congruences