A polynomial-exponential equation related to the Ramanujan-Nagell equation (Q1743617)

The author considers the generalized Ramanujan-Nagell equation \(x^2+D^m=p^n\) in positive integers \(x,m,n\), where \(p\) is a fixed odd prime and \(D\) is a fixed positive integer not divisible by \(p\). In the case where \(D\) is the product of \(p-1\) and a square, under some further assumptions, the author provides all solutions of the equation. The main tools in the proof are classical results of Dem'janenko concerning certain quadratic Diophantine equations and a deep theorem of Bilu, Hanrot and Voutier about the existence of primitive divisors of Lucas sequences.