The class numbers of some imaginary quadratic fields and a class of Diophantine equations (Q2742276)

Let \(h(-D)\) be the ideal class number of an imaginary quadratic field \(\mathbb{Z}(\sqrt {-D})\), where \(D\equiv 3\pmod 4\), \(a^2+b^2 D = 4k^n\) for some integers \(a, b \), and \(\max\{D, k\}\geq \exp\exp\exp 1000\), etc. It is proved that \(h(-D)\equiv 0\pmod n\) except in some cases. Under similar conditions, \(a^2+b^2 D = 4p^n\) is proved to have at most one positive integer solution \((n,a,b)\), with some exceptions.