关于丢番图方程x(x+1)=Dy^(3)

ON THE DIOPHANTINE EQUATIONS x(x+1)=Dy^(3)

设p为素数 ,证明了丢番图方程x(x+ 1) =Dy3在D=p 1(mod 3)时仅有解 (p ,x ,y) =(2 ,1,1) ,(2 ,- 2 ,1) ,(17,5 831,12 6 ) ,(17,- 5 832 ,12 6 ) ;在D =2p ,p≡ 2 ,3,5 (mod 9)时仅有解 (x ,y ,p) =(2 ,1,3) ,(- 3,1,3) ;在D =4p ,p=5或p≡ 2 ,3(mod 9)时仅有解 (p ,x ,y) =(3 ,3,1) ,(3,- 4,1) ,(5 ,4,1) ,(5 ,- 5 ,1) ,(5 ,6 85 9,133) ,(5 ,- 6 86 0 ,133)。

This paper proves that Diophntione equations x(x+1)=Dy 3 positive integer solutions merely are ( p,x,y )=(2,1,1),(2,-2,1),(17,5 831,126),(17,-5 832,126) with D=p1 (mod 3);and (x,y,p) =(3,2,1),(3,-3,1) with D=2p,p≡2,3,5 (mod 9);and (p,x,y)= (3,3,1),(3,-4,1),(5,4,1),(5,-5,1),(5,6 859,133),(5,-6 860,133) with D=4p,p=5 or p≡2,3 (mod 9).