On a conjecture of Iizuka (Q2162800)

The author proves that for any given odd prime number \(p\) and any positive integer \(n\), there are infinitely many quadruples of imaginary quadratic fields \[ \mathbb{Q}(\sqrt{d}),\, \mathbb{Q}(\sqrt{d+1}),\,\mathbb{Q}(\sqrt{d+4}),\, \mathbb{Q}(\sqrt{d+4p^2}), \] the class numbers of which are all divisible by \(n\). He also proved that there are infinitely many quintuples of imaginary quadratic fields \[ \mathbb{Q}(\sqrt{d}),\, \mathbb{Q}(\sqrt{d+1}),\,\mathbb{Q}(\sqrt{d+4}),\,\mathbb{Q}(\sqrt{d+36}),\, \mathbb{Q}(\sqrt{d+100}) \] with \(d\in \mathbb{Z}\), the class numbers of which are all divisible by \(n\). The primitive divisors of the Lehmer numbers come into play. Also the divisibility of the class numbers of \(\mathbb{Q}(\sqrt{1-4U^n})\) and \(\mathbb{Q}(\sqrt{1-V^n})\). This nice article shows some progress in the direction of the conjecture of Iizuka.