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.