Lehmer sequence approach to the divisibility of class numbers of imaginary quadratic fields

Abstract

Let $k\ge 3$ and $n\ge 3$ be odd integers, and let $m\ge 0$ 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:

$$\begin{aligned} \left( {\mathbb {Q}}(\sqrt{d}), {\mathbb {Q}}(\sqrt{d+1}), {\mathbb {Q}}(\sqrt{4d+1}), {\mathbb {Q}}(\sqrt{2d+4}), {\mathbb {Q}}(\sqrt{2d+16}), \cdots , {\mathbb {Q}}(\sqrt{2d+4^t}) \right) \end{aligned}$$

with $d\in {\mathbb {Z}}$ and $1\le 4^t\le 2|d|$ whose class numbers are all divisible by n. Our proofs use some deep results about primitive divisors of Lehmer sequences.