关于不定方程x^3+27=19y^2  

On the Diophantine Equation x^3+27=19y^2

作　　者：李双娥[1] 

机构地区：[1]重庆师范大学数学与计算机科学学院,重庆400047

出　　处：《重庆师范大学学报（自然科学版）》2007年第4期30-32,共3页Journal of Chongqing Normal University:Natural Science

摘　　要：不定方程x3±27=Dy2(D>0)的研究曾引起了一些学者的兴趣,曹玉书确立了当D不含6k+1形状的素数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、同余式和平方剩余的方法,讨论了不定方程x3+27=19y2在3|x及3x情况下的整数解。其中3x对又分了情形Ⅰx+3=19u2,x2-3x+9=v2,y=uv;情形Ⅱx+3=u2,x2-3x+9=19v2,y=uv这两种情况。最后得到不定方程x3+27=19y2仅有整数解(x,y)=(-3,0),(24,±9),(-2,±1)的结论。The study of the Diophantine equation x^3±27=Dy^2 （ D 〉 0） has caused some authors＇interests, such as Cao Yushu, who determines all the integer solutions when D has no prime odd powers factors in the form of 6k ＋ 1. But it is difficult when D has prime factors in the form of 6k ＋ 1. In this paper it is by using recursive series,and with the I-square residual method,discussed the Diophantine equation x^3＋27=19y^2 in both cases of 3｜x and 3 x the integer solution. The case of 3 x also divided into case Ⅰ x ＋ 3 = 19u^2 , x^2 - 3x ＋ 9 = v^2 ,y = uv ; And case Ⅱx ＋ 3 = u^2 ,x^2 - 3x ＋ 9 = 19v^2 , y = uv. At last, it is proved that the Diophantine equation has only integer solutions （x,y） = （-3,0） ,（24, ±9） ,（ -2, ±1）.

关 键 词：不定方程 整数解 递归数列 平方剩余 

分 类 号：O156.1[理学—数学]