On the Diophantine equation x2 − 4pm = ±yn

Let m and n be positive integers and p any odd prime. In this paper we consider the Diophantine equation x2 − 4pm = ±yn in positive integers x and y where (x, y) = 1, and we show that under some not very restrictive conditions, this equation has only finitely many solutions (x, y, m, n), and we provide a small explicit upper bound for n which only depends on p.