Arithmetic density and congruences of $\ell$-regular bipartitions

TL;DR

This paper investigates divisibility properties and congruences of $ ext{ell}$-regular bipartition functions, providing new theoretical results and algorithms for congruences using modular forms and number theory techniques.

Contribution

It establishes almost divisibility conditions for $B_ ext{ell}(n)$ and develops an algorithm to find congruences for $B_p(n)$ using advanced modular form methods.

Findings

Proves almost divisibility of $B_ ext{ell}(n)$ by prime powers under certain conditions.

Derives infinite families of congruences for $B_3(n)$ and $B_5(n)$.

Provides an algorithm to generate congruences for $B_p(n)$ for prime $p$.

Abstract

Let $ B_{\ell}(n)$ denote the number of $\ell$-regular bipartitions of $n.$ In this article, we prove that $ B_{\ell}(n)$ is always almost divisible by $p_i^j$ if $p_i^{2a_i}\geq \ell,$ where $j$ is a fixed positive integer and $\ell=p_1^{a_1}p_2^{a_2}\ldots p_m^{a_m},$ where $p_i$ are prime numbers $\geq 5.$ Further, we obtain an infinities families of congruences for $B_3(n)$ and $B_5(n)$ by using Hecke eigen form theory and a result of Newman \cite{Newmann1959}. 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.