Congruences of the Partition Function

Abstract

Let p(n) denote the partition function. In this article, we will show that congruences of the form p(ml(k)n+B)= 0 mod m for all n >= 0 exist for all primes m and l satisfying m >= 13 and l l = 2, 3, m, where B is a suitably chosen integer depending on m and l. Here, the integer k depends on the Hecke eigenvalues of a certain invariant subspace of S(m/2-1)(G(0)(576), chi(12)) and can be explicitly computed. More generally, we will show that for each integer i > 0 there exists an integer k such that with a properly chosen B the congruence p(m(i) l(k) n+B) equivalent to 0 mod m(i) holds for all integers n not divisible by l.