求方程(x+y)/(x^2-xy+y^2)=3/7的整数解1
问题描述：
求方程(x+y)/(x^2-xy+y^2)=3/7的整数解
1
答
(x+y)/(x^2-xy+y^2)=x^3+y^3=3/7 x y 要是整数的话它们各自的三次幂之和怎么可能是分数啊 还真不知道 有答案之后告诉我一下吧 还真得好好请教一下
答
(x+y)/(x^2-xy+y^2)=3/7
(x+y)^2/(x^3+y^3)=3/7
(x^3+y^3)/(x+y)^2=7/3
(x^3+y^3)/(x+y)^2=189/81
(x^3+y^3)=189
(x+y)^2=81
x+y=±9
x=5
y=4
或
x=4
y=5
答
(x+y)/(x^2-xy+y^2)=3/7设x+y=3tx^2-xy+y^2=7tt为整数于是x^2+2xy+y^2=9t^23xy=9t^2-7t(x-y)^2=(28t-9t^2)/3则t是3的倍数,于是,设t=3k则(x-y)^2=28k-27k^2=k(28-27k)≥0又k是整数,于是k=0或1当k=0时,分母为0,舍弃,于...