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 Q(√x²-yⁿ) is divisible by n for fixed $x, n$ if (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 Q(√d), Q(√d+1²), ⋯, Q(√d+m²) (d∈ Z) whose class numbers are all divisible by n.