A note on the divisibility of class numbers of imaginary quadratic fields $\mathbf{Q}(\sqrt{a^{2} - k^{n}})$

Abstract

Let $n > 1$ be an integer, $k > 1$ be an odd integer and $a > 0$ be an even integer. Suppose $a^{2} + b^{2}d = k^{n}$, where $d \neq 1, 3$ is a positive odd square-free integer and $\gcd(a, bd) = 1$. In this paper, we describe imaginary quadratic fields $\mathbf{Q}(\sqrt{a^{2} - k^{n}})$ explicitly whose class numbers are divisible by $n$ if $d \equiv 1, 5, 7\bmod 8$ or $d \equiv 3\bmod 8$ with $(n, 3) = 1$ under certain conditions.