Question 162
For integral $m$, let $p(m)$ be the greatest prime divisor of $m$. By convention, we set $p(\pm 1) = 1$ and $p(0) = \infty$. Find all polynomials $f$ with integer coefficients such that the sequence $\{p(f(n^2)) - 2n\}_{n \ge 0}$ is bounded above. (In particular, this requires $f(n^2) \ne 0$ for $n \ge 0$.)