The shuffle variant of a Diophantine equation of Miyazaki and Togbé

Abstract

In 2012, T. Miyazaki and A. Togbe gave all of the solutions of the Diophantine equations (2am - 1)(x) + (2m)(y) = (2am + 1)(z) and b(x) + 2(y) = (b + 2)(z) in positive integers x, y, z, a > 1 and b >= 5 odd. In this paper, we propose a similar problem (which we call the shuffle variant of a Diophantine equation of Miyazaki and Togbe). Here we first prove that the Diophantine equation (2am + 1)(x) + (2m)(y) = (2am - 1)(z) has only the solutions (a, m, x, y, z) = (2, 1, 2, 1, 3) and (2, 1, 1, 2, 2) in positive integers a > 1, m, x, y, z. Then using this result, we show that the Diophantine equation b(x) + 2(y) = (b - 2)(z) has only the solutions (b, x, y, z) = (5,2, 1, 3) and (5,1, 2, 2) in positive integers x, y, z and b odd.