三次Diophantine方程x^(3)+1=pQy^(2)的整数解

The Integer Solutions of the Cubic Diophantine Equation x^(3)+1=pQy^(2)

利用同余式、Legendre符号、递归数列、Pell方程解的性质以及一些初等数论方法得到以下结论:当p、Q分别为6k+1和6k-1型奇素数或p、Q为2个互不相同的6k+1型奇素数时,丢番图方程x^(3)+1=pQy^(2)仅有整数解(x,y)=(-1,0).

Using some elementary methods,such as congruence,Legendre symbol,recursive sequence,properties of the solutions of Pell equation,it is proved that when p,Q are 6k+1 and 6k-1 odd prime numbers respectively or p,Q are two different 6k+1 odd prime numbers,the Diophantine equation x^(3)+1=pQy^(2)has only integer solution(x,y)=(-1,0).