On the diophantine equation x2 + p2k = yn

Let 2 <= p < 100 be a rational prime and consider equation (3) in the title in integer unknowns x, y, n, k with x > 0, y > 1, n >= 3 prime, k >= 0 and gcd(x, y) = 1. Under the above conditions we give all solutions of the title equation (see the Theorem). We note that if in (3) gcd(x, y) = 1, our Theorem is an extension of several earlier results [15], [27], [2], [3], [5], [23].