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Arithmetic density and congruences of $\ell$-regular bipartitions $II$
- Publication Year :
- 2024
-
Abstract
- Let $ B_{\ell}(n)$ denote the number of $\ell-$regular bipartitions of $n.$ In 2013, Lin \cite{Lin2013} proved a density result for $B_4(n).$ He showed that for any positive integer $k,$ $B_4(n)$ is almost always divisible by $2^k.$ In this article, we improved his result. We prove that $B_{2^{\alpha}m}(n)$ and $B_{3^{\alpha}m}(n)$ are almost always divisible by arbitrary power of $2$ and $3$ respectively. Further, we obtain an infinities families of congruences and multiplicative formulae for $B_2(n)$ and $B_4(n)$ 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 $2$ satisfied by $B_{2^{\alpha}}(n).$<br />Comment: Comments are welcome. arXiv admin note: substantial text overlap with arXiv:2406.06224; text overlap with arXiv:2302.11830; text overlap with arXiv:2405.05274 by other authors
- Subjects :
- Mathematics - Number Theory
05A17, 11P83
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2406.07905
- Document Type :
- Working Paper