A note on the exponential diophantine equation A2n)x + (B2n)y = ((A2 + B2)n)z

Abstract

Let A, B be positive integers such that. inin{A, B} > 1, gcd(A, B) = 1 and 2 vertical bar B. In this paper, using an upper bound for solutions of ternary purely exponential Diophantine equations due to R. Scott and R. Styer, we prove that, for any positive integer n, if A > B-3/8, then the equation (A(2)n)(x) + (B(2)n)(y) = ((A(2) + B-2)n)(z) has no positive integer solutions (x, y, z) with x > z > y; if B > A(3)/6, then it has no solutions (x, y, z) with y > z > x. Thus, combining the above conclusion with some existing results, we can deduce that, for any positive integer n, if B 2 (mod 4) and A > B-3/8, then this equation has only the positive integer solution (x, y, z)= (1,1,1).