Elementary Proofs of Recent Congruences for Overpartitions Wherein Non-Overlined Parts are Not Divisible by 6

TL;DR

This paper provides elementary proofs for recent congruences related to overpartition functions where non-overlined parts are not divisible by 6, confirming conjectures and simplifying previous modular form-based proofs.

Contribution

It offers elementary, accessible proofs of complex overpartition congruences, including confirming a conjecture about divisibility by 128.

Findings

Confirmed the conjecture that R_6^*(n) is divisible by 128 for infinitely many n.

Provided elementary proofs for two previously modular form-based congruences.

Simplified the understanding of overpartition congruences using elementary methods.

Abstract

We define $\overline{R_l^*}(n)$ as the number of overpartitions of $n$ in which non-overlined parts are not divisible by $l$. In a recent work, Nath, Saikia, and the second author established several families of congruences for $\overline{R_l^*}(n)$, with particular focus on the cases $l=6$ and $l=8$. In the concluding remarks of their paper, they conjectured that $\overline{R_6^*}(n)$ satisfies an infinite family of congruences modulo $128$. In this paper, we confirm their conjectures using elementary methods. Additionally, we provide elementary proofs of two congruences for $\overline{R_6^*}(n)$ previously proven via the machinery of modular forms by Alanazi, Munagi, and Saikia.