On the Diophantine Equation n^x + 13^y = z^2 where n = 2 (mod 39) and n + 1 is not a Square Number

Abstract

The purpose of the present article is to prove that the Diophantine equation n^x + 13^y = z^2 has exactly one solution (n, x, y, z) = (2, 3, 0, 3) where x, y and z are non-negative integers and n is a positive integer with n ≡ 2 (mod 39) and n + 1 is not a square number. In particular, (3, 0, 3) is a unique solution (x, y, z) in non-negative integers of the Diophantine equation 2^x + 13^y = z^2