3-Divisibility of 9- and 27-Regular Partitions

TL;DR

This paper uses advanced modular form theory to determine precise conditions under which the counts of 9- and 27-regular partitions of an integer are divisible by 3, enhancing understanding of partition divisibility properties.

Contribution

It provides exact criteria for the 3-divisibility of 9- and 27-regular partition counts using Hecke eigenform theory, a novel application in partition analysis.

Findings

Established criteria for 3-divisibility of b_9(n)

Established criteria for 3-divisibility of b_{27}(n)

Applied Serre's theory of Hecke eigenforms

Abstract

A partition of $n$ is $l$-regular if none of its parts is divisible by $l$. Let $b_l(n)$ denote the number of $l$-regular partitions of $n$. In this paper, using the theory of Hecke eigenforms explored by J.-P. Serre, we establish exact criteria for the $3$-divisibility of $b_9(n)$ and $b_{27}(n)$.