求出方程组的所有有理数的解,并证明你的答案是正确的4x^2/1+4x^2=y4y^2/1+4y^2=z4z^2/1+4z^2=x

Question

求出方程组的所有有理数的解,并证明你的答案是正确的
4x^2/1+4x^2=y
4y^2/1+4y^2=z
4z^2/1+4z^2=x

Answer 1

4x^2/1+4x^2=y,因为4x^2/1=4x^2,所以4x^2+4x^2=y,8x^2=y；同理8y^2=z；8z^2=x.
三个方程相乘得512x^2y^2z^2=xyz.
解一：x=y=z=0
512xyz=1
xyz=1/512
x y z
1 1 1/512
1 1/2 1/256
1 1/4 1/128
.
1/512 1 1