Arithmetic density and congruences of $\ell$-regular bipartitions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916281714737152 |
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| author | Meher, Nabin Kumar Jindal, Ankita |
| author_facet | Meher, Nabin Kumar Jindal, Ankita |
| contents | 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 $p_i^{2a_i}\geq \ell,$ where $j$ is a fixed positive integer and $\ell=p_1^{a_1}p_2^{a_2}\ldots p_m^{a_m},$ where $p_i$ are prime numbers $\geq 5.$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06224 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Arithmetic density and congruences of $\ell$-regular bipartitions Meher, Nabin Kumar Jindal, Ankita Number Theory 05A17, 11P83 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 $p_i^{2a_i}\geq \ell,$ where $j$ is a fixed positive integer and $\ell=p_1^{a_1}p_2^{a_2}\ldots p_m^{a_m},$ where $p_i$ are prime numbers $\geq 5.$ 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. |
| title | Arithmetic density and congruences of $\ell$-regular bipartitions |
| topic | Number Theory 05A17, 11P83 |
| url | https://arxiv.org/abs/2406.06224 |