ON THE DIOPHANTINE EQUATION axy + byz + czx = 0

Let a, b, c be integers. In this paper, we prove the integer solutions of the equation ax y + by z + cz x = 0 satisfy max {|x|, |y|, |z|} ≤ 2 max {a, b, c} when a, b, c are odd positive integers, and when a = b = 1, c = -1, the positive integer solutions of the equation satisfy max {x, y, z} < exp ( exp ( exp (5))).