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