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

Abstract

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