A note on the diophantine equation \(x^ 2+4D=y^ p\) (Q1314866)

For a positive square free integer \(D\), let \(h(-D)\) denote the class number of \(\mathbb{Q}(\sqrt{-D})\). Further, let \(p\) be an odd prime with \(h(- D)\not\equiv 0\pmod p\). The author shows that the diophantine equation \(x^ 2+ 4D= y^ p\) has no solution \((x,y)\in\mathbb{Z}^ 2\), provided \(p\in\{5,7\}\) or \(p> 3\cdot 10^ 6\). This result is a direct consequence of Theorem 5.11 of [\textit{M. Mignotte} and \textit{M. Waldschmidt}, Ann. Fac. Sci. Toulouse, V. Sér., Math. 1989, Spec., Issue 43-75 (1989; Zbl 0702.11044)].