关于Diophantine方程xy-(x±1)~z=1

On the Diophantine Equation x^y-(x±1)~z=1

证明了方程xy-(x+1)z=1仅有正整数解(x,y,z)=(2,2,1);方程xy-(x-1)z=1仅有正整数解(x,y,z)=(1,s,t),(2,1,t),(r,1,1)和(3,2,3),其中r,s,t为任意正整数且r≥3,这一结果推广和改进了文献[4]中的结论.

It has been proven by elementary methods that：（1）the equation xy-（x＋1）z=1 has only a positive integral solution（x,y,z）=（2,2,1）;（2）the equation xy-（x-1）z=1 has only the positive integral solutions（x,y,z）=（1,s,t）,（2,1,t）,（r,1,1） and（3,2,3）,where r,s and t are arbitrary positive integers,with r≥3.The conclusion of has been improved and promoted by this result.