A note on the exponential diophantine equation (A 2 n) x +(b 2 n) y = ((a 2/sup +b 2 )n) z

Abstract

Let A, B be positive integers such that min{A,B} > 1, gcd(A,B) = 1 and 2|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 > B3/8, then the equation (A2n)x + (B2n)y = ((A2 + B2 )n)z has no positive integer solutions (x,y,z) with x > z > y; if B > A3/6, 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 > B3 /8, then this equation has only the positive integer solution (x,y,z) = (1,1,1).