关于Diophantine方程x^3±2^(3m)=3Dy^2

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

设m是正整数,D是无平方因子正整数.证明了:当m>1时,如果D不能被3或6k+1之形素数整除,则方程x3±23m=3Dy2没有适合gcd(x,y)=1的正整数解(x,y).

Let m be a positive integer, and D be a positive integer with square free. It is proved that if m 〉 1 and D is not divisible by 3 or primes of the form 6k ＋ 1, then the equation x^3 ＋ 2^3m = 3Dy^2 has no positive integer solutions （ x, y） with gcd（x,y） = 1.