Divisibility of class numbers of quadratic fields and a conjecture of Iizuka
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910479359672320 |
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| author | Ouyang, Yi Song, Qimin |
| author_facet | Ouyang, Yi Song, Qimin |
| contents | Assume $x,\ y,\ n$ are positive integers and $n$ is odd. In this note, we show that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{x^{2}-y^{n}})$ is divisible by $n$ for fixed $x, n$ if $\gcd(2x,y)=1$ and $y>C$ where $C$ is a constant depending only on $x$ and $n$. Based on this result, for any odd integer $n$ and any positive integer $m$, we construct an infinite family of $m+1$ successive imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1^{2}})$, $\cdots$, $\mathbb{Q}(\sqrt{d+m^{2}})$ $(d\in \mathbb{Z})$ whose class numbers are all divisible by $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05975 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Divisibility of class numbers of quadratic fields and a conjecture of Iizuka Ouyang, Yi Song, Qimin Number Theory Assume $x,\ y,\ n$ are positive integers and $n$ is odd. In this note, we show that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{x^{2}-y^{n}})$ is divisible by $n$ for fixed $x, n$ if $\gcd(2x,y)=1$ and $y>C$ where $C$ is a constant depending only on $x$ and $n$. Based on this result, for any odd integer $n$ and any positive integer $m$, we construct an infinite family of $m+1$ successive imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1^{2}})$, $\cdots$, $\mathbb{Q}(\sqrt{d+m^{2}})$ $(d\in \mathbb{Z})$ whose class numbers are all divisible by $n$. |
| title | Divisibility of class numbers of quadratic fields and a conjecture of Iizuka |
| topic | Number Theory |
| url | https://arxiv.org/abs/2406.05975 |