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

XuanLing Wei

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.

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

Abstract

In this paper, we establish two new Ramanujan-type congruences for the overpartition function: and . The proofs rely on the theory of modular forms. We conjecture potential Ramanujan-type congruences for overpartitions modulo 7, 17, 19 and 23.
Paper Structure (4 sections, 7 theorems, 60 equations)

This paper contains 4 sections, 7 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Closing remarks

Key Result

Theorem 1.1

For all $n \geq 0$, the following congruences hold:

Theorems & Definitions (19)

  • Theorem 1.1: Main Theorem
  • Definition 1.2
  • Definition 1.3: q-Pochhammer symbol
  • Lemma 1.4
  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • ...and 9 more