Yan Ni LIU, Xiao Yan GUO
The main purpose of this paper is using the elementary method and the divisible properties of the integers to study this problem, and solve it completely. That is, we shall prove that the Diophantine equation xy + yz + zx = 0 has and only has six integer solutions. They are: (x, y, z) = (-2, 1, 1), (1, -2, 1), (1, 1, -2), (1,-1,-2), (-1, -2, 1), (-2,1,-1).