Divisibility of class numbers of quadratic fields and a conjecture of Iizuka

Assume $x,y,n$ are positive integers and $n$ is odd. In this note, we show that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{x^2-y^n})$ is divisible by $n$ for fixed $x,n$ if $\gcd (2x,y)=1$ and $y>C$ where $C$ is a constant depending only on $x$ and $n$. Based on this result, for any odd integer $n$ and any positive integer $m$, we construct an infinite family of $m+1$ successive imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1^2})$, $\cdots$, $\mathbb{Q}(\sqrt{d+m^2})$ $(d\in \mathbb{Z})$ whose class numbers are all divisible by $n$.