关于Diophantine方程(x^m-1)(x^n-1)=y^2

On the Diophantine Equation (x^m-1)(x^n-1)=y^2

设N是全体正整数的集合证明了:方程(xm-1)(xn-1)=y2,x,y,m,n∈N,x>1,n>m≥1的全部整数解为(x,y,m,n)=(7,120,1,4),(3,22,1,5),(3,44,2,5),(2,21,3,6)(k2-1,k3-2k,1,2),其中k∈Z,k>1.

In this paper, we prove that the euqation (x^m-1)(x^n-1)=y^2,x,y,m,n∈N,x>1,n>m≥1 has only solutions (x,y,m,n)=(7,120,1,4),(3,22,1,5),(3,44,2,5),(2,21,3,6)(k^2-1,k^3-2k,1,2), with k∈Z,k>1.