关于丢番图方程x(x+1)=Dy^6

ON THE DIOPHANTINE EQUATIONS x(x+1)=Dy^6

设 p为素数 ,本文证明了丢番图方程 x( x+ 1 ) =Dy6在 D=p时仅有正整数解 ( p,x,y) =( 2 ,1 ,1 ) ;在 D=2 p,p ± 1 ,± 1 7,1 9( mod72 )时仅有解( p,x,y) =( 3,2 ,1 ) ;在 D=4p,p 1 ,5 ,37,41 ( mod 72 )时仅有正整数解 ( p,x,y) =( 3,3,1 ) ;在 D=8p时仅有解 ( p,x,y) =( 7,7,1 ) ;在 D=1 6 p,p 1 ,1 7( mod72 )和 D=32 p,p ± 1 ,31 ( mod32 )时均无正整数解 .

We make use elementary theory of number prore diophntione equations x(x+1)=Dy 6 positive integer solutions merely are ( p,x,y)=(2,1,1) with D=p; and (p,x,y) =(7,7,1) with D=8p.