Arithmetic density and congruences of $\ell$-regular bipartitions $II$
Nabin Kumar Meher

TL;DR
This paper extends divisibility and congruence results for $ ext{ell}$-regular bipartitions, providing new infinite families of congruences and multiplicative formulas using Hecke eigenform theory and nilpotency of Hecke operators.
Contribution
It improves previous divisibility results for bipartitions and introduces new infinite families of congruences and formulas leveraging advanced modular form techniques.
Findings
$B_{2^{eta}m}(n)$ and $B_{3^{eta}m}(n)$ are almost always divisible by arbitrary powers of 2 and 3.
Established infinite families of congruences for $B_2(n)$ and $B_4(n)$ using Hecke eigenform theory.
Derived congruences modulo arbitrary powers of 2 for $B_{2^{eta}}(n)$ using nilpotency of Hecke operators.
Abstract
Let denote the number of regular bipartitions of In 2013, Lin \cite{Lin2013} proved a density result for He showed that for any positive integer is almost always divisible by In this article, we improved his result. We prove that and are almost always divisible by arbitrary power of and respectively. Further, we obtain an infinities families of congruences and multiplicative formulae for and by using Hecke eigenform theory. Next, by using a result of Ono and Taguchi on nilpotency of Hecke operator, we also find an infinite families of congruences modulo arbitrary power of satisfied by
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Taxonomy
TopicsAdvanced Algebra and Logic · Analytic Number Theory Research · Graph theory and applications
