方程x^3+1=3py^2有正整数解的必要条件

The Necessary Conditions for the Equation x^3+1=3py^2 Has Positive Integer Solutions

对于正整数n,设Q(n)是n的无平方因子部分;设p是适合p≡1(mod 6)的奇素数.运用Petr组的性质证明了:如果方程x3+1=3py2有正整数解(x,y),则p≠Q(3s2-2),p≠Q(12s2+1),且3p≠Q(s2+2),其中s是正整数.

For any positive integer n,let Q（n）denote the quadratfrei of n.Let p be an odd prime with p ≡1（mod 6）.In this paper,using the properties of Petr’s triplet,we prove that if the eqation x 3 ＋1=3py 2 has positive integer solutions （x ,y ）,then p ≠Q（3s 2 -2）,p ≠Q（12s 2 ＋1）and 3p ≠Q（s 2 ＋2）,where s is a positive integer.