不定方程x^(2)-k(k+1)y^(2)=1与y^(2)-Dz^(2)=4的公解

On common solutions of Diophantine equations x^(2)-k(k+1)y^(2)=1 and y^(2)-Dz^(2)=4

设p_(1),…,p_(r)是不同的奇素数,x_(1)=2k+1,u,v均为正整数.该文证明了当D=2p_(1)…p_(r)(1≤r≤4)时,除开2(4x_(1)^(2)-3)(4x_(1)^(2)-1)(2x_(1)^(2)-1)=Du^(2)或2(2x_(1)^(2)-1)=Dv^(2)外,不定方程组x2-k(k+1)y^(2)=1与y^(2)-Dz^(2)=4仅有平凡解(x,y,z)=(±(2k+1),±2,0).

Let p_(1),…,p_(s) are diverse odd p_(r)imes,x 1=2k+1 and u,v are positive integers.In this paper,it is p_(r)oved that if D=2p_(1)…p r(1≤r≤4),then the equations x 2-k(k+1)y^(2)=1 and y^(2)-Dz^(2)=4 have only trivial solutions(x,y,z)=(±(2k+1),±2,0),with the exceptions that 2(4x_(1)^(2)-3)(4x_(1)^(2)-1)(2x_(1)^(2)-1)=Du^(2) or 2(2x_(1)^(2)-1)=Dv^(2).