Congruences of the partition function

TL;DR

This paper proves the existence of infinite families of congruences for the partition function p(n) modulo primes m, involving powers of primes and explicit conditions related to Hecke eigenvalues.

Contribution

It establishes new infinite congruences for p(n) modulo primes m, with explicit formulas depending on Hecke eigenvalues and extends to higher powers of m.

Findings

Existence of congruences p(m^j ext{l}^kn+B) ≡ 0 mod m for primes m ≥ 13

Explicit computation of the integer k based on Hecke eigenvalues

Generalization to congruences modulo higher powers m^i for all i > 0

Abstract

Let $p(n)$ denote the partition function. In this article, we will show that congruences of the form $$ p(m^j\ell^kn+B)\equiv 0\mod m \text{for all} n\ge 0 $$ exist for all primes $m$ and $\ell$ satisfying $m\ge 13$ and $\ell\neq 2,3,m$. 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 $$ p(m^j\ell^kn+B)\equiv 0\mod m^i $$ holds for all integers $n$ not divisible by $\ell$.