论文标题
Lehmer序列方法的划分级数假想二次字段的分数
Lehmer sequence approach to the divisibility of class numbers of imaginary quadratic fields
论文作者
论文摘要
令$ k \ geq 3 $和$ n \ geq 3 $为奇数，让$ m \ geq 0 $为任何整数。对于素数$ \ ell $，我们证明了假想二次字段的类$ \ mathbb {q}（\ sqrt {\ sqrt {\ ell^{2m} -2K^n}）$是$ n $的，或者由$ n $ $ n $的特定divisor划分。应用此结果，我们构建了一个$$ \ left（\ Mathbb {q}}（\ sqrt {d}），\ Mathbb {q}（\ sqrt {\ sqrt {d+1}），\ Mathbb {q Mathbb {Q}}（Q}（Q}（Q}（Q}）， \ Mathbb {q}（\ sqrt {2d+4}），\ Mathbb {q}（\ sqrt {2d+16}），\ cdots，\ cdots，\ mathbb {q}（\ sqrt {\ sqrt {2d+4^t}） 4^t \ leq 2 | d | $，其班级都可以除以$ n $。我们的证明对莱默序列的原始分隔线进行了一些深刻的结果。
Let $k\geq 3$ and $n\geq 3$ be odd integers, and let $m\geq 0$ be any integer. For a prime number $\ell$, we prove that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{\ell^{2m}-2k^n})$ is either divisible by $n$ or by a specific divisor of $n$. Applying this result, we construct an infinite family of certain tuples of imaginary quadratic fields of the form $$\left(\mathbb{Q}(\sqrt{d}), \mathbb{Q}(\sqrt{d+1}), \mathbb{Q}(\sqrt{4d+1}), \mathbb{Q}(\sqrt{2d+4}), \mathbb{Q}(\sqrt{2d+16}), \cdots, \mathbb{Q}(\sqrt{2d+4^t}) \right)$$ with $d\in \mathbb{Z}$ and $1\leq 4^t\leq 2|d|$ whose class numbers are all divisible by $n$. Our proofs use some deep results about primitive divisors of Lehmer sequences.
