On the Exponential Diophantine Equation (m2 + m+1)x + my = (m+1)z

Abstract

Let m >= 1 be a positive integer. We show that the exponential Diophantine equation (m(2) + m + 1)(x) + m(y) = (m + 1)(z) has no positive integer solutions other than (x, y, z) = (1, 1, 2) when m is not an element of {1, 2, 3}.