Theta-type congruences for colored partitions

TL;DR

This paper explores specific congruence relations for the number of r-colored partitions, focusing on conditions involving primes and linear forms, to deepen understanding of partition congruences.

Contribution

It introduces new congruence relations for r-colored partitions involving primes and linear forms, expanding the theory of partition congruences.

Findings

Established congruences for r-colored partitions modulo primes

Identified conditions on primes and linear forms for these congruences

Enhanced understanding of partition congruence patterns

Abstract

We investigate congruence relations of the form $p_r(\ell m n + t) \equiv 0 \pmod{\ell}$ for all $n$, where $p_r(n)$ is the number of $r$-colored partitions of $n$ and $m,\ell$ are distinct primes.