Problem

Source: ELMO 2024/6

Tags: number theory, polynomial, Elmo



For a prime $p$, let $\mathbb{F}_p$ denote the integers modulo $p$, and let $\mathbb{F}_p[x]$ be the set of polynomials with coefficients in $\mathbb{F}_p$. Find all $p$ for which there exists a quartic polynomial $P(x) \in \mathbb{F}_p[x]$ such that for all integers $k$, there exists some integer $\ell$ such that $P(\ell) \equiv k \pmod p$. (Note that there are $p^4(p-1)$ quartic polynomials in $\mathbb{F}_p[x]$ in total.) Aprameya Tripathy