Question 2
For a polynomial $P$ and a positive integer $n$, let $P_n$ be the number of positive integer pairs $(a,b)$ with $a\lt b\le n$ for which $|P(a)|-|P(b)|$ is divisible by $n$. Determine all polynomials $P$ with integer coefficients satisfying $P_n\le2021$ for every positive integer $n$.