Arithmetic density and congruences of -regular bipartitions

Let B_(n) denote the number of -regular bipartitions of $n.$ In this article, we prove that B_(n) is always almost divisible by pᵢʲ if pᵢ2aᵢ≥ , where j is a fixed positive integer and =p₁a₁p₂a₂… pₘaₘ, where pᵢ 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 {Newmann1959}. Furthermore, by applying Radu and Seller's approach, we obtain an algorithm from which we get several congruences for Bₚ(n), where p is a prime number.