Some New Congruences for $l$-Regular Partitions Modulo $l$

TL;DR

This paper derives new infinite families of congruences for the number of $l$-regular partitions modulo $l$ using Ramanujan's theta function identities, expanding understanding of partition congruences for specific primes.

Contribution

It introduces novel congruences for $b_l(n)$ modulo $l$ for primes $l=13,17,23$, utilizing Ramanujan's theta function identities.

Findings

Established new congruences for $b_{13}(n)$, $b_{17}(n)$, and $b_{23}(n)$.

Connected partition congruences with Ramanujan's theta identities.

Extended the theory of partition congruences to additional prime moduli.

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 theta function identities due to Ramanujan, we establish some new infinite families of congruences for $b_l(n)$ modulo $l$, where $l=13,17,23$.