Congruences of the partition function

Abstract

Let $p(n)$ denote the partition function. In this article, we will show that congruences of the form $$p(m^j{\ell }^{k}n+B)\equiv 0\mathrm{mod}m\text{for all}n\ge 0$$ exist for all primes $m$ and $\ell$ satisfying $m\ge 13$ and $\ell \ne 2,3,m$. Here the integer $k$ depends on the Hecke eigenvalues of a certain invariant subspace of $S_{m/2-1}({\Gamma }_0(576),{\chi }_{12})$ and can be explicitly computed. More generally, we will show that for each integer $i>0$ there exists an integer $k$ such that for every non-negative integers $j\ge i$ with a properly chosen $B$ the congruence $$p(m^j{\ell }^{k}n+B)\equiv 0\mathrm{mod}{m}^i$$ holds for all integers $n$ not divisible by $\ell$.

19 pages