On an indivisibility version of Iizuka's conjecture
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916888869601280 |
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| author | R, Muneeswaran Krishnamoorthy, Srilakshmi Bhakta, Subham |
| author_facet | R, Muneeswaran Krishnamoorthy, Srilakshmi Bhakta, Subham |
| contents | Iizuka's conjecture predicts that, given $m \in \mathbb{N}$ and a prime $p$, there exists infinitely many integers $n$ such that the class numbers of \textit{all} of the following quadratic number fields, \[ \mathbb{Q}(\sqrt{n}),\ \mathbb{Q}(\sqrt{n+1}),\ \ldots,\ \mathbb{Q}(\sqrt{n+m}), \] are divisible by $p$. In this article, given $k$ and $m$, we study the proportion of $n$ such that the class numbers of \textit{none} of the successive fields \[ \mathbb{Q}(\sqrt{n}),\ \mathbb{Q}(\sqrt{n+1}),\ \ldots,\ \mathbb{Q}(\sqrt{n+m}), \] are divisible by \( 3^k \). Moreover, we study the proportion of imaginary biquadratic fields whose class numbers are not divisible by $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08772 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On an indivisibility version of Iizuka's conjecture R, Muneeswaran Krishnamoorthy, Srilakshmi Bhakta, Subham Number Theory Iizuka's conjecture predicts that, given $m \in \mathbb{N}$ and a prime $p$, there exists infinitely many integers $n$ such that the class numbers of \textit{all} of the following quadratic number fields, \[ \mathbb{Q}(\sqrt{n}),\ \mathbb{Q}(\sqrt{n+1}),\ \ldots,\ \mathbb{Q}(\sqrt{n+m}), \] are divisible by $p$. In this article, given $k$ and $m$, we study the proportion of $n$ such that the class numbers of \textit{none} of the successive fields \[ \mathbb{Q}(\sqrt{n}),\ \mathbb{Q}(\sqrt{n+1}),\ \ldots,\ \mathbb{Q}(\sqrt{n+m}), \] are divisible by \( 3^k \). Moreover, we study the proportion of imaginary biquadratic fields whose class numbers are not divisible by $3$. |
| title | On an indivisibility version of Iizuka's conjecture |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.08772 |