关于Diophantine方程x^3-p^(3n)=Dy^2

On The Diophantine Equation x^3-p^(3n)=Dy^2

设p是奇素数,D是无平方因子正奇数,本文证明了:当p≡5(mod12),D≡1(mod4)时,如果D不能被p或6k+1之形素数整除,则方程x3-p3n=Dy2没有适合gcd(x,y)=1的正整数解(x,y,n).

Let p be an odd prime,and let D be a positive odd integer woth square free.In this paper we prove that if p≡5(mod 12),D≡1(mod 4)and D is not divisible by p or primes of the form 6k+1,then the equation x^3-p^(3n)=Dy^2 has no positive integer solutions ( x,y,n)with gcd (x,y)=1.