The divisibility of the class number of the imaginary quadratic fields \({\mathbb {Q}}(\sqrt{1-2m^k})\)

Let \(h_{(m,k)}\) $h_{(m,k)}$ be the class number of \(Q(\sqrt{1-2m^k})\) $Q(\sqrt{1-2m^k})$ . We prove that for any odd natural number k,  there exists \(m_0\) $m_0$ such that \(k \mid h_{(m,k)}\) $k\mid h_{(m,k)}$ for all odd \(m > m_0\) $m>m_0$ . We also prove that for any odd \(m \ge 3,\) $m\ge 3,$ \(k \mid h_{(m,k)}\) $k\mid h_{(m,k)}$ (when k and \(1-2m^k\) $1-2m^k$ square-free numbers) and \(p \mid h_{(m,p)}\) $p\mid h_{(m,p)}$ (except finitely many primes p). We deduce that for any pair of twin primes \(p_1,p_2=p_1+2\) $p_1,p_2=p_1+2$ , \(p_1 \mid h_{(m,p_1)}\) $p_1\mid h_{(m,p_1)}$ or \(p_2 \mid h_{(m,p_2)}\) $p_2\mid h_{(m,p_2)}$ . For any odd natural number k, we construct an infinite family of pairs of imaginary quadratic fields \(Q(\sqrt{d}), {\mathbb {Q}}(\sqrt{d+1})\) $Q(\sqrt{d}),\mathbb{Q}(\sqrt{d+1})$ whose class numbers are divisible by k, which settles a generalized version of Iizuka’s conjecture (cf: Conjecture [2.2]) for the case \(n=1\) $n=1$ .