Congruence relations for r-colored partitions

TL;DR

This paper establishes infinite congruences for r-colored partition functions modulo primes, extending known results by employing modular Galois representations to prove new congruence relations for partition functions.

Contribution

It proves new infinite congruences for partition functions and their r-colored variants using modular Galois representations, generalizing previous specific cases.

Findings

Proves infinitely many congruences for p(n) modulo primes.

Extends results to r-colored partition functions under certain conditions.

Uses modular Galois representations to establish these congruences.

Abstract

Let $\ell \geq 5$ be prime. For the partition function $p(n)$ and $5 \leq \ell \leq 31$, Atkin found a number of examples of primes $Q \geq 5$ such that there exist congruences of the form $p(\ell Q^{3} n+\beta) \equiv 0 \pmod{\ell}.$ Recently, Ahlgren, Allen, and Tang proved that there are infinitely many such congruences for every $\ell$. In this paper, for a wide range of $c \in \mathbb{F}_{\ell}$, we prove congruences of the form $p(\ell Q^{3} n+\beta_{0}) \equiv c \cdot p(\ell Q n+\beta_{1}) \pmod{\ell}$ for infinitely many primes $Q$. For a positive integer $r$, let $p_{r}(n)$ be the $r$-colored partition function. Our methods yield similar congruences for $p_{r}(n)$. In particular, if $r$ is an odd positive integer for which $\ell > 5r+19$ and $2^{r+2} \not \equiv 2^{\pm 1} \pmod{\ell}$, then we show that there are infinitely many congruences of the form $p_{r}(\ell Q^{3}n+\beta) \equiv 0 \pmod{\ell}$. Our methods involve the theory of modular Galois representations.