On the solution of the exponential Diophantine equation ۲x+m۲y=z۲, for any positive integer m

It is well known that the exponential Diophantine equation ۲x+ ۱=z۲ has the unique solution x=۳ and z=۳ in non-negative integers, which is closely related to the Catlan's conjecture. In this paper, we show that for m∈N, m>۱, the exponential Diophantine equation ۲x+m۲y=z۲ admits a solution in positive integers (x, y,z) if and only if m=۲αMn, α≠۰ for some Mersenne number Mn. When m=۲αMn, α≠۰, the unique solution is (x,y,z)=(۲+n+۲α,۱, ۲α(۲n+۱)). 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.

Keywords:

Mersenne numbers , Catalan's Conjecture , Exponential Diophantine equations