关于不定方程x^3±1=2PDy^2的整数解

Integer solution of the indefinite equation x^3±1=2PDy^2

设D=∏ni=1ri(n≥2),ri≡-1(mod 6)(1≤i≤n)为互异的奇素数,P∏sj=1pj(s≥2),pj≡1(mod 6)(1≤j≤s)为互异的奇素数,利用Pell方程解的性质、同余式、平方剩余、递归序列等,得到当s=2且(p1/p2)=-1时,方程x3±1=2PDy2仅有平凡解的2个充分条件.

LetD=∏ni=1ri（n≥2）,ri≡-1（mod 6）（1≤i≤n）be different odd primes, and P∏sj=1pj（s≥2）,pj≡1（mod 6）（1≤j≤s） be different odd primes. Based on some properties of the solutions to Pell equation, congruence, quadratic residue and i＇ecursive sequence, two sufficient conditions of the indefinite equation x3±1=2PDy2 are given when it only has trivial solutions where s = 2 and （p1/p2）=-1.