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 (x^2-y^{n}) 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+1^2), , Q (d+m^2) (d Z) whose class numbers are all divisible by n.