ON THE DIOPHANTINE EQUATION x² + C = 2yn

In this paper, we study the Diophantine equation x 2 + C = 2y n in positive integers x,y with gcd (x,y) = 1, where n ≥ 3 and C is a positive integer. If C ≡ 1 (mod 4), we give a very sharp bound for prime values of the exponent n; our main tool here is the result on existence of primitive divisors in Lehmer sequences due to Bilu, Hanrot and Voutier. We illustrate our approach by solving completely the equations x 2 + 17 a 1 = 2y n , x 2 + 5 a 1 13 a 2 = 2y n and x 2 + 3 a 1 11 a 2 = 2y n .