Distribution of 3-regular and 5-regular partitions

TL;DR

This paper investigates the arithmetic properties of 3-regular and 5-regular partition functions using modular forms, revealing new congruences and behaviors modulo primes greater than 3.

Contribution

It introduces new results on the modular form-based arithmetic properties of 3-regular and 5-regular partition functions, expanding understanding of their congruences.

Findings

Proves new congruences for $b_3(n)$ and $b_5(n)$ modulo primes greater than 3.

Establishes arithmetic properties of these partition functions using modular forms.

Provides a framework for analyzing regular partitions through modular form theory.

Abstract

In this paper we study the function $b_3(n)$ and $b_5(n)$, which denote the number of $3$-regular partitions and $5$-regular partitions of $n$ respectively. Using the theory of modular forms, we prove several arithmetic properties of $b_3(n)$ and $b_5(n)$ modulo primes greater than $3$.