Question 3
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
$$
\{ p(f(n^2)) - 2n \}_{n \geq 0}
$$
is bounded above. (In particular, this requires $f(n^2) \neq 0$ for $n \ge 0$.)
Proposed by Titu Andreescu, Gabriel Dospinescu (credited in the supplied source).
$$
\{ p(f(n^2)) - 2n \}_{n \geq 0}
$$
is bounded above. (In particular, this requires $f(n^2) \neq 0$ for $n \ge 0$.)
Proposed by Titu Andreescu, Gabriel Dospinescu (credited in the supplied source).