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

TL;DR

This paper investigates the divisibility and congruence properties of $(u,v)$-regular bipartition functions, establishing infinite families of congruences modulo various primes using modular form theory and identities.

Contribution

It introduces new infinite families of congruences for bipartition functions with specific parameters, extending previous results through advanced number theory techniques.

Findings

Proves $B_{p,m}(n)$ is almost divisible by $p$ for primes $p \\geq 5$.

Establishes infinite congruence families modulo 3, 7, 11, 13, and 2 for various bipartition functions.

Utilizes Hecke eigenform theory and identities of Newman to derive these results.

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 $p\geq 5$ is a prime number and $m=p_1^{\alpha_1} p_2^{\alpha_2}\cdots p_r^{\alpha_r}, $ where $\alpha_i \geq 0$ and $p_i \geq 5$ 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 \cite{Newmann1959}. 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 \cite{Newmann1959}. 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 \cite{Newmann1962}.