Diophantine方程x^3-8=py^2解的研究

On the Diophantine equation x^3-8=py^2

研究丢番图方程正整数解的情况.运用初等方法及同余理论,证明了Diophantine方程x3-8=py2,当p是奇素数且p=3(24k+19)(24k+20)+1,其中k是非负整数,则方程x3-8=py2无正整数解.给出了丢番图方程x3-8=py2无正整数解的一个充要条件.

The positive integer solution of the Diophantine equation is studied.Using the elementary and the theory of congruence,Diophantine equation x^3-8=py^2 is proved,when p is an odd prime,p=3（24k＋19）（24k＋20）＋1 and k is non-negative integer,then the equation has no positive integer solutions.A necessary condition of Diophantine equation x^3-8=py^2 has no positive integer solutions is given.