Distribution and congruences of (u,v)-regular bipartitions

2024/07/27 by Nabin Kumar Meher, Meher, Nabin Kumar
Mathematics · #05A17 #11P83 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2407.19230

openalex publication_date 2024/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let B_u,v(n) denote the number of (u,v)-regular bipartitions of n. In this article, we prove that B_p,m(n) is always almost divisible by p, where p≥ 5 is a prime number and m=p₁^α₁ p₂^α₂⋯ p_r^α_r, where α_i ≥ 0 and p_i ≥ 5 be distinct primes with gcd(p,m)=1 . Further, we obtain an infinities families of congruences modulo 3 for B_3,7(n), B_3,5(n) and B_3,2(n) by using Hecke eigenform theory and a result of Newman \citeNewmann1959. Furthermore, we get many infinite families of congruences modulo 7, 11 and 13 respectively for B_2,7(n), B_2,11(n) and B_2,13(n), by employing an identity of Newman \citeNewmann1959. In addition, we prove infinite families of congruences modulo 2 for B_4,3(n), B_8,3(n) and B_4,5(n) by applying another result of Newman \citeNewmann1962.

Related

• Arithmetic density and congruences of ℓ-regular bipartitions
• Some congruences for (s,t)-regular bipartitions modulo t
• Arithmetic density and congruences of ℓ-regular bipartitions II
• Ramanujan type of congruences modulo m for (l, m)-regular bipartitions
• Congruences for the number of partitions and bipartitions with distinct even parts