关于Diophantine方程x^3-3^(3m)=Dy^2

On The Diophantine Equation x^3-3~ 3m=Dy^2

设D是无平方因子正奇数.本文证明了:当D不能被6k+1之形素数整除时,如果方程x3-33m=Dy2有适合gcd(x,Y)=1的正整数解(x,y,m),则D≡7(mod 8),D的素因数p都满足了p≡11(mod 12),而且D的素因数个数必为奇数.

Let D be a positive odd integer with square free. In this paper we prove that if D is not divisible by primes of the form 6k+1 and the equation x^3-3~ 3m=Dy^2 has positive integer solutions (x,y,m), then D≡7 (mod 8), the prime divisors P of D satisfy p≡11 (mod 12) and the number of prime divisors of D is odd.