Arithmetic density and congruences of $\ell$-regular bipartitions $II$
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| Format: | Preprint |
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2024
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| _version_ | 1866914833033592832 |
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| author | Meher, Nabin Kumar |
| author_facet | Meher, Nabin Kumar |
| contents | 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 $B_{2^αm}(n)$ and $B_{3^αm}(n)$ 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 $B_{2^α}(n).$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_07905 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Arithmetic density and congruences of $\ell$-regular bipartitions $II$ Meher, Nabin Kumar Number Theory 05A17, 11P83 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 $B_{2^αm}(n)$ and $B_{3^αm}(n)$ 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 $B_{2^α}(n).$ |
| title | Arithmetic density and congruences of $\ell$-regular bipartitions $II$ |
| topic | Number Theory 05A17, 11P83 |
| url | https://arxiv.org/abs/2406.07905 |