Type: Article

On the Diophantine equation x^2+C=2y^n

Fadwa S. Abu Muriefah; Florian Luca; Samir Siksek; Szabolcs Tengely;

Abstract:

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(a1) = 2y(n), x(2) + 5(a1)13(a2) = 2y(n) and x(2) + 3(a1)11(a2) = 2y(n).