关于三次Diophantine方程x^3+1=3py^2

On the cubic Diophantine equation x^3+1=3py^2

目的研究丢番图方程x3+1=3py2的正整数解问题。方法运用Pell方程的基本性质。结果设p是适合p≡1(mod 6)的奇素数,如果p=3k2-2或者3p=k2+2,其中k是正整数,则方程x3+1=3py2无正整数解。结论部分解决了该方程的可解性问题。即对某些P,该方程无正整数解。

Aim To study the positive integer solution of the Diophantine equation x3 ＋ 1 = 3py2. Methods By using the basic properties of Pell equations. Results Let p be an odd prime with p =- 1 （ mod 6）. For p = 3k2 - 2 or 3p = k2 ＋ 2, where k is a positive integer, the equation x3 ＋ 1 = 3py2 has no positive integer solution. Conclusion It is proved that the Diophantine equation has not integer solution for some special integers p.