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

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 $p_i^{2a_i}\ge \ell ,$ where $j$ is a fixed positive integer and $\ell =p_1^{a_1}{p}_2^{a_2}\dots p_m^{a_m},$ where $p_i$ are prime numbers $\ge 5.$ 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.