On the Diophantine Equation aˣ+ (a+5b) ʸ=z²

In this paper, we study the Diophantine equation aˣ+ (a+5b) ʸ=z² when a 1 (5) and b is a positive integer. We establish that the equation has no solutions in positive integers x, y and z. We start with the Diophantine equation pˣ+ (p+5a) ʸ=z² where p and p+5a are both primes and p 1 (5) and a is a positive integer.