一类Diophantine方程及其他的整数解

One kind Diophantine equation and its integer solutions

目的研究不定方程x3±8=Dy2的可解性问题。方法利用初等及代数方法。结果设D是不含3和6k+1之形素因数的无平方因子正整数。当D>5时,如果D的素因数p都满足p≡1,3(mod 8)或者p≡5,7(mod 8),则方程x3±8=Dy2没有适合gcd(x,y)=1的正整数解(x,y)。结论部分地解决了该方程的可解性问题。即对某些特殊D,该方程无解。

Aim To study the solvability of the Diophantine equation x3 ± 8 = By2. Methods The elementary and algebra methods. Results Let D be a positive integer such that 3 ,D, D is square free and D has no prime divisor with form 6k ＋ 1. If D 〉 5 and every prime divisor p of D satisfies either p= 1,3 （ rood 8） or p = 5,7 （ mod 8 ）, then the equation x3 ± 8 = Dy2 has no positive integer solution （x ,y） with gcd （x,y） = 1. Conclusion It is proved that the Diophantine equation has not integer solutions for some special integers D.