A short Proof of a conjecture by Hirschhorn and Sellers on Overpartitions

2014/07/21 by Liuquan Wang, Wang, Liuquan
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary 05A17 #Secondary 11P83

paper · pdf · doi:10.48550/arxiv.1407.5430

openalex publication_date 2014/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p(n) be the number of overpartitions of n, we establish and give a short elementary proof of the following congruence p(4^α(40n+35))≡ 0 (\bmod 40), where α,n are nonnegative integers. By letting α=0 we proved a conjecture of Hirschhorn and Sellers. Some new congruences for p(n) modulo 3 and 5 have also been found, including the following two infinite families of Ramanujan-type congruences: for any integers n≥ 0 and α≥ 1, p(5^2α+1(5n+1))≡ p(5^2α+1(5n+4))≡ 0 (\bmod 5).

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• Proof of a Conjecture of Hirschhorn and Sellers on Overpartitions
• Ramanujan-type Congruences for Overpartitions Modulo 5
• Arithmetic Properties of Overpartition Pairs
• Extending Congruences for the number of smallest parts in overpartitions with smallest part even
• Some congruences modulo 5 and 25 for overpartition