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

Let $k\ge 3$ and $n\ge 3$ be odd integers, and let $m\ge 0$ be any integer. For a prime number $\mathrm{\ell }$, we prove that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{{\mathrm{\ell }}^{2m}-2k^n})$ 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 $$(\mathbb{Q}(\sqrt{d}),\mathbb{Q}(\sqrt{d+1}),\mathbb{Q}(\sqrt{4d+1}),\mathbb{Q}(\sqrt{2d+4}),\mathbb{Q}(\sqrt{2d+16}),\cdots ,\mathbb{Q}(\sqrt{2d+4^t}))$$ with $d\in \mathbb{Z}$ and $1\le 4^t\le 2\mathrm{\mid }d\mathrm{\mid }$ whose class numbers are all divisible by $n$. Our proofs use some deep results about primitive divisors of Lehmer sequences.