一个三次Diophantine方程的初等解法

A solution of cubic diophantine equations

针对三次Diophantine方程x的立方加减1等于2倍p1,p2,…,直至pi(i≥2)(其中pi(i≥2)与1对模6同余,且pi(i≥2)为互异的奇素数)与y的平方之积的整数解问题至今仍未解决的问题,主要利用同余式、平方剩余、递归序列、Pell方程的解的性质得出了Diophantine方程x的立方加减1等于2倍p,q(其中p与q对模6同余,且p,q为互异的奇素数)与y的平方之积无正整数解的两个充分条件,从而推进了该类三次Diophantine方程的研究.

Let p1≡1（mod 6）（i≥2）be different odd primes. The primary solution of the Diophantine equation x-3±1=2p1p2…ps y-2（s≥2） still remains unresolved. Using congruence, quadratic residue, recursive sequence and some properties of the solutions to Pell equation, two sufficient conditions are obtained that the Diophantine equation x-3±1=2pqy -2 have no integer solutions, where p-=q-l（mod 6）, p and q be different odd primes. These results promote the studies on this kind of cubic Diophantine equation.