The divisibility of the class number of the imaginary quadratic fields $\mathbb{Q}(\sqrt{1-2m^k})$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| author | Krishnamoorthy, Srilakshmi Muneeswaran, R. |
| author_facet | Krishnamoorthy, Srilakshmi Muneeswaran, R. |
| contents | Let $h_{(m,k)}$ be the class number of $\mathbb{Q}(\sqrt{1-2m^k}).$
We prove that for any odd natural number $k,$ there exists $m_0$ such that $k \mid h_{(m,k)}$ for all odd $m > m_0.$ We also prove that for any odd $m \geq 3,$ $k \mid h_{(m,k)}$ (when
$k$ and $1-2m^k$ square-free numbers) and $p \mid h_{(m,p)}$ (except finitely many primes $p$). We deduce that for any pair of twin primes $p_1,p_2=p_1+2$,
$p_1 \mid h_{(m,p_1)}$ or $p_2 \mid h_{(m,p_2)}.$
For any odd natural number $k$, we construct an infinite family of pairs of imaginary quadratic fields $\mathbb{Q}(\sqrt{d}), \mathbb{Q}(\sqrt{d+1})$ whose class numbers are divisible by $k$, which settles a generalized version of Iizuka's conjecture (cf : Conjecture 2.2) for the case $n=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_04387 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The divisibility of the class number of the imaginary quadratic fields $\mathbb{Q}(\sqrt{1-2m^k})$ Krishnamoorthy, Srilakshmi Muneeswaran, R. Number Theory 11R29, 11R11 Let $h_{(m,k)}$ be the class number of $\mathbb{Q}(\sqrt{1-2m^k}).$ We prove that for any odd natural number $k,$ there exists $m_0$ such that $k \mid h_{(m,k)}$ for all odd $m > m_0.$ We also prove that for any odd $m \geq 3,$ $k \mid h_{(m,k)}$ (when $k$ and $1-2m^k$ square-free numbers) and $p \mid h_{(m,p)}$ (except finitely many primes $p$). We deduce that for any pair of twin primes $p_1,p_2=p_1+2$, $p_1 \mid h_{(m,p_1)}$ or $p_2 \mid h_{(m,p_2)}.$ For any odd natural number $k$, we construct an infinite family of pairs of imaginary quadratic fields $\mathbb{Q}(\sqrt{d}), \mathbb{Q}(\sqrt{d+1})$ whose class numbers are divisible by $k$, which settles a generalized version of Iizuka's conjecture (cf : Conjecture 2.2) for the case $n=1$. |
| title | The divisibility of the class number of the imaginary quadratic fields $\mathbb{Q}(\sqrt{1-2m^k})$ |
| topic | Number Theory 11R29, 11R11 |
| url | https://arxiv.org/abs/2111.04387 |