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.