关于丢番图方程x^3+b=Dy^2(Ⅰ)

On diophantine equation x^3+b=Dy^2(Ⅰ)

设p为奇素数,(x,y)=1,方程x3+p3=y2的全部整数解为:(ⅰ)(x,y)=(3β4+6α2β2-α4,6αβ(α4+3β4)),且α、β满足(α2+3β2)2-12β4=p;(ⅱ)(x,y)=(2α4+2β4-4α3β-4αβ3,3(α+β)(α-β)5+6αβ(α4-β4)),且α、β满足(α+β)4-12α2β2=p;(ⅲ)(x,y)=(α4+6α2β2-3β4,6αβ(α4+3β4)),且α、β满足12β4-(α2-3β2)2=p 其中α,β一奇一偶,(α,β)=1,α>β>0。

Let p is a odd prime number , (x,y)=1,all integral solution of diophantine equation x^3+p^3=y^2 can be expressed as :(ⅰ) (x,y)=(3β~4+6α~2β~2-α~4,6αβ(α~4+3β~4)),where (α,β) is a solution of (α~2+3β~2)~2-12β~4=p which satisfies;(ⅱ) (x,y)=(2α~4+2β~4-4α~3β-4αβ~3,3(α+β)(α-β)~5+6αβ(α~4-β~4)), where (α,β) is a solution of (α+β)~4-12α~2β~2=p which satisfies;(ⅲ) (x,y)=(α~4+6α~2β~2-3β~4,6αβ(α~4+3β~4)),where (α,β) is a solution of 12β~4-(α~2-3β~2)~2=p which satisfies.In this paper (α,β)=1,α>β>0,and if α is odd number then β is even number.If α is even number then β is odd number.