Divisibility and distribution of 5-regular partitions

TL;DR

This paper investigates the divisibility and distribution properties of 5-regular partitions using modular forms, establishing infinitely many Ramanujan-type congruences modulo primes greater than or equal to 5.

Contribution

It introduces new theorems on divisibility and distribution of 5-regular partitions, demonstrating the existence of infinitely many Ramanujan-type congruences for primes ≥ 5.

Findings

Proved theorems on divisibility of 5-regular partitions.

Established distribution properties modulo primes.

Proved infinitely many Ramanujan-type congruences.

Abstract

In this paper we study $b_5(n)$, the $5$-regular partitions of $n$. Using the theory of modular forms, we prove several theorems on the divisibility and distribution properties of $b_5(n)$ modulo prime $m\geq5$. In particular, we prove that there are infinitely many Ramanujan-type congruences modulo prime $m\geq5$.