Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice

2024/09/18 by James A. Sellers, Sellers, James A. · 1 citation
Mathematics · #05A17 #11P83 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2409.12321

openalex publication_date 2024/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In a recent article on overpartitions, Merca considered the auxiliary function a(n) which counts the number of partitions of n where odd parts are repeated at most twice (and there are no restrictions on the even parts). In the course of his work, Merca proved the following: For all n≥ 0, a(4n+2) amp;≡ 0 \pmod2, \textrm and
a(4n+3) amp;≡ 0 \pmod2. Merca then indicates that a classical proof of these congruences would be very interesting. The goal of this short note is to fulfill Merca's request by providing two truly elementary (classical) proofs of these congruences.

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