关于Diophantine方程x^3+1=3py^2

On the Diophantine Equation x^3+1=3py^2

设p是奇素数,研究了丢番图方程x3+1=3py2正整数解的情况.利用初等数论的方法得到了丢番图方程x3+1=3py2无整数解的一个充分条件,即p为素数且p=3 3k+13k+2+1,其中k是非负整数,则方程x3+1=3py2无正整数解.

Let p be an odd prime. This paper studies the positive integer solutions situation of the Diophantine equation x^3＋1=3py^2. Using elementary theory of numbers methods, a sufficient condition is obtained that Diophantine equation without integer solution, i.e. is a prime, is a nonnegative integers, then the equation has no positive integer solution.