In this paper, we gave solutions of the Diophantine equations
16^{x}+p^{y}=z^{2}, 64^{x}+p^{y}=z^{2} where p is an odd prime, n is a positive
integer and x,y,z are non-negative integers. Finally we gave a generalization
of the Diophantine equation (4^{n})^{x}+p^{y}=z^{2}.