Question 83
A polynomial $P$ with integer coefficients is *square-free* if it is not expressible in the form $P = Q^2R$, 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
$$
1 + a_1x + a_2x^2 + \dots + a_nx^n
$$
with $a_1, a_2, \dots, a_n \in \{0, 1\}$. Prove that there exists an integer $N$ so that, for all integers $n \ge N$, more than 99% of the polynomials in $P_n$ are square-free.
$$
1 + a_1x + a_2x^2 + \dots + a_nx^n
$$
with $a_1, a_2, \dots, a_n \in \{0, 1\}$. Prove that there exists an integer $N$ so that, for all integers $n \ge N$, more than 99% of the polynomials in $P_n$ are square-free.