Summary: Let \(n, a, b, c\) be positive integers with \({\text{gcd}} (a, b, c) = 1\), \(a, b \geq 3\) and the Diophantine equation \({a^x} + {b^y} = {c^z}\) has only the positive integer solution \( (x, y, z) = (1, 1, 1)\). In this paper, the authors prove that if \( (x, y, z)\) is a positive integer solution of the Diophantine equation \( (an)^x + (bn)^y = (cn)^z\) with \( (x, y, z) \neq (1, 1, 1)\), then \(y < z < x\) or \(x < z < y\). The authors also show that when \( (a, b, c) = (3, 5, 8), (5, 8, 13), (8, 13, 21), (13, 21, 34)\), the Diophantine equation \( (an)^x + (bn)^y = (cn)^z\) has only the positive integer solution \( (x, y, z) = (1, 1, 1)\).