On the generalized Ramanujan-Nagell equation \(x^2+D=p^z\) (Q1125385)

Let \(p\in \{3,23\}\) and \(D\) a positive integer non divisible by \(p\), with \((D,p)\neq (2,3)\). Then the authors prove that the Diophantine equation \(x^2+D=p^z\) has at most one solution \((x,z)\). Moreover, they give an explicit upper bound for \(z\). The proof is quite involved. The first general result on the equation \(x^2+D=p^z\) is Apéry's who proved in 1960 that for squarefree \(D\) the number of solutions is at most two.