一个丢番图方程及其它的整数解

A Diophantine Equation and Its Integer Solutions

研究丢番图方程x^y+y^z+z^x=0的可解性,并求该方程的所有整数解.本文利用初等方法及整数的整除性质研究这一问题,获得了彻底解决.即就是证明了方程x^y+y^z+z^x=0有且仅有六组整数解(x,y,z)=(-2,1,1),(1,-2,1),(1,1,-2),(1,-1,-2),(-1,-2,1),(-2,1,-1)

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 x^y ＋ y^z ＋ z^x = 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）.