设有理数x,y满足方程x5+y5=2x2y2,证明1-xy是有理数的平方.

Question

设有理数x,y满足方程x5+y5=2x2y2,证明1-xy是有理数的平方.

Answer 1

(x^5+y^5)^2=x^10+y^10+2x^5y^5=(2x^2y^2)^2=4x^4y^4,
——》x^10+y^10=4x^4y^4-2x^5y^5,
——》(x^5-y^5)^2
=x^10+y^10-2x^5y^5
=4x^4y^4-4x^5y^5
=4x^4y^4(1-xy),
——》1-xy=(x^5-y^5)^2/4x^4y^4=[(x^5-y^5)/2x^2y^2]^2,
x,y为有理数,
——》(x^5-y^5)/2x^2y^2为有理数,
命题得证.