On the Diophantine equation 3^x+p^y=z^2 where p ≡ 2 (mod 3)

Abstract

Let p be a prime number where p ≡ 2 (mod 3). In this work, we give a nonnegative integer solution for the Diophantine equation 3x+py = z2. If y = 0, then (p, x, y, z) = (p, 1, 0, 2) is the only solution of the equation for each prime number p. If y is not divisible by 4, then the equation has a unique solution (p, x, y, z) = (2, 0, 3, 3). In case that y is a positive integer that is not divisible by 4, we give a necessary condition for an existence of a solution and give a computational result for p < 1017. We also give a necessary condition for an existence of a solution for qx + py = z2 when p and q are distinct prime numbers.