On the diophantine equation 2x + M 2y = Zn2m for mersenne number Mn

Abstract

In this paper, we first show that the exponential Diophantine equation 2x + 1 = z2has the unique solution (x, z) = (3, 3). We then show that for n > 1, the exponential. Diophantine equation 2x + M 2y = z2 where Mn := 2n − 1 is the nth Mersenne number, has exactly two solutions in non-negative integers viz., (3, 0, 3) and (n + 2, 1, 2n + 1). Also, we prove that the exponential Diophantine equation 2x + M 2y = w4 has the unique solution (x, y, w, n) = (5, 1, 3, 3) . Finally, we prove that the exponential Diophantine equation 2x + M 2y = w2m, m > 2 has no non-negative integral solutions. We conclude with some examples to illustrate our results.