Arithmetic density and congruences of ℓ-regular bipartitions II

2024/06/12 by Nabin Kumar Meher, Meher, Nabin Kumar
Computer Science · Mathematics · #05A17 #11P83 #Advanced Algebra and Logic #Analytic Number Theory Research #FOS: Mathematics #Graph theory and applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2406.07905

openalex publication_date 2024/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let B_ℓ(n) denote the number of ℓ-regular bipartitions of n. In 2013, Lin \citeLin2013 proved a density result for B₄(n). He showed that for any positive integer k, B₄(n) is almost always divisible by 2^k. In this article, we improved his result. We prove that B_2^αm(n) and B_3^αm(n) are almost always divisible by arbitrary power of 2 and 3 respectively. Further, we obtain an infinities families of congruences and multiplicative formulae for B₂(n) and B₄(n) by using Hecke eigenform theory. Next, by using a result of Ono and Taguchi on nilpotency of Hecke operator, we also find an infinite families of congruences modulo arbitrary power of 2 satisfied by B_2^α(n).

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