New Ramanujan-type congruences for overpartitions modulo $11$ and $13$

TL;DR

This paper discovers new Ramanujan-type congruences for the overpartition function modulo 11 and 13 using modular forms, and proposes conjectures for similar congruences modulo other primes.

Contribution

It introduces two novel Ramanujan-type congruences for overpartitions modulo 11 and 13, expanding the understanding of overpartition congruences.

Findings

Established overpartition congruences modulo 11 and 13

Used modular forms to prove the congruences

Conjectured further congruences for other primes

Abstract

In this paper, we establish two new Ramanujan-type congruences for the overpartition function: $\overline{p}(11\times(8n+5))\equiv 0 \pmod{11}$ and $\overline{p}(13\times 2^6(8n+7))\equiv 0 \pmod{13}$. The proofs rely on the theory of modular forms. We conjecture potential Ramanujan-type congruences for overpartitions modulo 7, 17, 19 and 23.