关于丢番图方程x(x+1)=Dy^4

On the Diophantine Equations x(x+1)=Dy^4

设P为素数 ,本文用初等数论方法 ,证明了丢番图方程x(x +1 ) =Dy4 在D =2P ,P≡± 5,7,1 3 (mod1 6)和D =8P ,P≡± 3 (mod8)时均无正整数解 ;在D =P ,P 1 (mod1 6)时仅有正整数解 (D ,x,y) =( 2 ,1 ,1 ) ,( 5,80 ,6) ;在D =4P时仅有正整数解 (D ,x ,y) =( 1 2 ,3 ,1 ) ,( 2 0 ,4 ,1 ) .

With the help of elementary theory of number,this paper shows that the Diophantine equations x(x+1)=2py 4 has no positive integer solutions when p ±5,7,13(mod16).