**Abstract:**

In this paper, we study the Diophantine equation a^x+(a+5b)^y=z^2 when a\equiv 1\left( \bmod 5 \right) 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 p^x+(p+5a)^y=z^2 where p and p+5a are both primes and p\equiv 1\left( \bmod 5 \right) and a is a positive integer.