on the solution of the exponential diophantine equation 2x+m2y=z2, for any positive integer m

It is well known that the exponential diophantine equation 2x+ 1=z2 has the unique solution x=3 and z=3 innon-negative integers, which is closely related to the catlan’s conjecture. in this paper, we show that for m isin;n, m gt;1, the exponential diophantine equation 2x+m2y=z2 admits a solution in positive integers (x, y,z) if and only if m=2 alpha;mn, alpha; ne;0 for some mersenne number mn. when m=2 alpha;mn, alpha; ne;0, the unique solution is (x,y,z)=(2+n+2 alpha;,1, 2 alpha;(2n+1)). finally,we conclude with certain examples and non-examples alike! the novelty of the paper is that we mainly use elementary methods to solve a particular class of exponential diophantine equations.