关于不定方程x^3-27=19y^2

On the Diophantine Equation

不定方程x3±27=Dy2(D>0)的研究,曾引了一些学者的兴趣,例如曹玉书确立了当D不含6k+1形状的素数的奇次幂因子时的全部整数解,而当含有6k+1形状的素数因子时,方程的求解比较困难.本文利用递归数列、同余式和平方剩余证明了不定方程x3-27=19y2仅有整数解(x,y)=(3,0).

In this paper the author has proved that the Diophantine equation x^3 -27 = 19y^2 has only one integer solution （x,y） = （3,0） with the methods of recurrent sequence, congruence and quadratic residue.