Distribution of the k-regular partition function modulo composite integers M

TL;DR

This paper investigates the distribution of the k-regular partition function modulo composite integers coprime to 6, proving the existence of infinitely many Ramanujan-type congruences for these functions.

Contribution

It establishes the existence of infinitely many Ramanujan-type congruences for the k-regular partition function modulo certain composite integers.

Findings

Infinitely many Ramanujan-type congruences exist for b_k(n) modulo M.

Results apply to all k-regular partition functions with M coprime to 6.

Provides new insights into the modular behavior of partition functions.

Abstract

Let $b_k(n)$ denote the $k-$regular partitons of a natural number $n$. In this paper, we study the behavior of $b_k(n)$ modulo composite integers $M$ which are coprime to $6$. Specially, we prove that for arbitrary $k-$regular partiton function $b_k(n)$ and integer $M$ coprime to $6$, there are infinitely many Ramanujan-type congruences of $b_k(n)$ modulo $M$.