关于丢番图方程x^3+p^(3n)=Dy^2

ON THE DIOPHANTINE EQUATION x^3+p^(3n)=Dy^2

设D是大于 2且不含σk +1之形素因数的无平方因子正整数 ,p是适合p D的素数。本文证明了 :当p>3且p ± 1(mod 12 )时 ,如果D有素因数q适合q≡ 1(mod 4) ,则方程x3 +p3n =Dy2 没有适合gcd(x ,y) =1的正整数解 (x,y ,n)。

Let D be a positive integer such that D>2,D is square free and has no prime divisor with the form 6k+1 ,let P be a prime with pD ,if p>3,p±1 (mod 12) and D has a prime divisor q with q≡1 (mod 4),then the equation x+p 3n =Dy 2 has no positive integer solution (x,y,n) with gcd (x,y)=1 .