How to Cite: Borah, P. B., & Dutta, M. (2022). On the diophantine equation 2x + M 2y = Zn2m for mersenne number Mn. International Journal of Health Sciences, 6(S6), 3214–3219. https://doi.org/10.53730/ijhs.v6nS6.10028 International Journal of Health Sciences ISSN 2550-6978 E-ISSN 2550-696X © 2022. Manuscript submitted: 9 March 2022, Manuscript revised: 27 May 2022, Accepted for publication: 18 June 2022 3214 n n n ∈ On the diophantine equation 2 x + M 2y = Zn 2m for mersenne number M n Padma Bhushan Borah Department of Mathematics, Gauhati University, Guwahati, Assam, India Email: padmabhushanborah@gmail.com Mridul Dutta Department of Mathematics, Dudhnoi College, Goalpara, Assam, India *Corresponding author email: mridulduttamc@gmail.com Abstract---In this paper, we first show that the exponential Diophantine equation 2 x + 1 = z 2 has the unique solution (x, z)n = (3, 3). We then show that for n > 1, the exponential. Diophantine equation 2 x + M 2y = z 2 where Mn := 2 n − 1 is the n th Mersenne number, has exactly two solutions in non-negative integers viz., (3, 0, 3) and (n + 2, 1, 2 n + 1). Also, we prove that the exponential Diophantine equation 2 x + M 2y = w 4 has the unique solution (x, y, w, n) = (5, 1, 3, 3) . Finally, we prove that the exponential Diophantine equation 2 x + M 2y = w 2m , m > 2 has no non-negative integral solutions. We conclude with some examples to illustrate our results. Keywords---mersenne numbers, catalan’s conjecture, diophantine equations, expo- nential equations. Introduction The word ‘Diophantine’ comes from ‘Diophantus’ a mathematician in Alexandria sometimes around 250 AD. The Diophantine equation can be used in various fields. The field of Diophan- tine equations is old and vast, and there is no universal method to determine whether a given Diophantine equation has any solutions, or not. The Diophantine equations usually refer to any equation in one or more unknowns to be solved in Z . The simplest known equation is the linear Diophantine equation in two unknowns, written as ax + by = c, where a, b, c Z. It is believed that the Indian mathematician Brahmagupta first described linear equations. Now, the general solution is very mature. Euclidean Algorithm is one of the methods to solve Diophantine linear equation. If there are linear Diophantine equations, there are also nonlinear Diophantine equations [4]. In most cases, we are reduced to studying