关于Diophantine方程x^3+3^(3n)=Dy^2

On the Diophantine Equation x^3+3^(3n)=Dy^2

设D是无平方因子正整数.证明了:当D不能被形如6k+1之形素数整除时,如果D含有素因数p适合p≡5(mod12),则方程x3+33n=Dy2没有适合gcd(x,y)=1的正整数解(x,y,n).

Let D be a positive integer with square free.This paper proves that D is not divisible by primes of the form 6k+1and D has a prime divisor p with p≡5(mod 12),then the equation x^3+3^(3n)=Dy^2 has no positive integer solutions (x,y,n)with gcd(x,y)=1.