关于Diophantine方程x^3+3^(3m)=2Dy^2

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

证明了当D(无平方因子正奇数)不能被6k+1之形素数整除时,若方程x3+33m=2Dy2有适合gcd(x,y)=1的正整数解(x,y,m),则D≡1(mod4),D的素因数p都满足p≡11(mod12),而且D的素因数个数必为偶数.

Let D be a positive odd integer with square free.In this paper,it is proved that if D is not divisible by primes of the form 6k+1 and the equation x3+33m=2Dy2 has positive integer solutions (x,y,m) with gcd (x,y)=1,then D≡1 (mod 4),the prime divisors p of D satisfy p≡11 (mod 12) and the number of prime divisors of D is even.