Arithmetic density and congruences of ℓ-regular bipartitions

2024/06/10 by Meher, Nabin Kumar, Jindal, Ankita · 1 citation
#05A17 #11P83 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2406.06224

Abstract

Let B_ℓ(n) denote the number of ℓ-regular bipartitions of n. In this article, we prove that B_ℓ(n) is always almost divisible by p_i^j if p_i^2a_i≥ ℓ, where j is a fixed positive integer and ℓ=p₁^a₁p₂^a₂… p_m^a_m, where p_i are prime numbers ≥ 5. Further, we obtain an infinities families of congruences for B₃(n) and B₅(n) by using Hecke eigen form theory and a result of Newman \citeNewmann1959. Furthermore, by applying Radu and Seller's approach, we obtain an algorithm from which we get several congruences for B_p(n), where p is a prime number.

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