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We prove that for any odd natural number $y$, there exists $m_0$ such thatthe class number of $\mathbb{Q}(\sqrt{1-2m^y})$ is divisible by $y$, for allodd number $m> m_0$. We also prove that for any odd integer $m\geq3$, for allbut finitely many primes $p$, the class number of $\mathbb{Q}(\sqrt{1-2m^p})$is divisible by $p$. This collection of fields is infinite for a fixed $m$. Weprove that the class number of $\mathbb{Q}(\sqrt{1-2m^p})$ is divisible by $p$whenever $1-2m^p$ is a square-free integer. We deduce analogous results byreplacing $p$ by square-free odd integer $t\geq3$. We also deduce that for anypair of twin primes $p_1,p_2=p_1+2$, at least one of the $p_i$ divides theclass number of $\mathbb{Q}(\sqrt{1-2m^{p_i}})$. For any odd natural number$y$, we construct an infinite family of pairs of imaginary quadratic fields$\mathbb{Q}(\sqrt{d}), \mathbb{Q}(\sqrt{d+1})$ whose class numbers aredivisible by $y$, which settles a generalized version of Iizuka's conjecturefor the case $k=1$.

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