A polynomial P with integer coefficients is square-free if it is not expressible in the form P=Q² R, where Q and R are polynomials with integer coefficients and Q is not constant. For a positive integer n, let P_n be the set of polynomials of the form

1+a₁ x+a₂ x²+⋯+a_n xⁿ

with a₁,a₂,⋯,a_n∈{0,1}. Prove that there exists an integer N so that, for all integers n>N, more than 99% of the polynomials in P_n are square-free.

【译】我们称整系数多项式P是无平方因子的，如果其不能表示为P=Q² R的形式，这里Q,R为整系数多项式且Q不为常数.对于正整数n，记P_n为如下 形式的多项式组成的集合：

1+a₁ x+a₂ x²+⋯+a_n xⁿ

这里a₁,a₂,⋯,a_n∈{0,1}.证明：存在整数N，使得对任意的整数n≥N,P_n中超过99%的多项式都是无平方因子的.