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New Congruences for the Partition Function


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.

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Weaver, R.L. New Congruences for the Partition Function. The Ramanujan Journal 5, 53–63 (2001).

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  • partition function congruences