On the Diophantine Equation $$cx^2+p^{2m}=4y^n$$ c x 2 + p 2 m = 4 y n

Let c be a square-free positive integer and p a prime satisfying $$p\not \mid c$$ . Let $$h(-c)$$ denote the class number of the imaginary quadratic field $$\mathbb {Q}(\sqrt{-c})$$ . In this paper, we consider the Diophantine equation $$\begin{aligned}&cx^2+p^{2m}=4y^n,~~x,y\ge 1, m\ge 0, n\ge 3,\\&\quad \gcd (x,y)=1, \gcd (n,2h(-c))=1, \end{aligned}$$ and we describe all its integer solutions. Our main tool here is the prominent result of Bilu, Hanrot and Voutier on existence of primitive divisors in Lehmer sequences.