An application of the Baker method to a new conjecture on exponential Diophantine equations

Let n be a positive integer with n > 1 and let a, b be fixed coprime positive integers with min{a, b} > 2. In this paper, using the Baker method, we proved that, for any n, if a > max{15064b, b(3/2)}, then the equation (an)(x )+ (bn)(y )= ((a + b)n)(z) has no positive integer solutions (x, y, z) with x > z > y. Further, let A, B be coprime positive integers with min{A, B} > 1 and 2|B. Combining the above conclusion with some existing results, we deduced that, for any n, if (a, b) = (A(2), B-2), A > max{123B, B-3/2} and B equivalent to 2 (mod 4), then this equation has only the positive integer solution (x, y, z) = (1, 1, 1). Thus, we proved that the conjecture proposed by Yuan and Han is true for this case.