关于Diophantine方程x^3±1=3pqry^2的整数解

On integer solution of the Diophantine equation x^3±1=3pqry^2

利用同余式、平方剩余、勒让德符号的性质、Pell方程解的性质、递归序列等理论得到了Diophantine方程x^3±1=3pqry^2仅有平凡解的两个充分条件.其中r≡5(mod6)为奇素数,p≡q≡1(mod6)为奇素数,(p/q)=-1.

By using congruence,quadratic remainder,some properties of Legendre symbol and some properties of the solutions to Pell equation and recursive sequence,two sufficient conditions are given to keep the Diophantine equation x^3±1=3pqry^2 just has trivial solution,where r,p,q be odd primes with r≡5（mod6）,p≡q≡1（mod6）,and （p/q）=-1.