Let $Q\in \mathbb{Z}[x]$ be a polynomial defining an injective function $\mathbb{Z}\to\mathbb{Z}$. Does it define an injective function $\mathbb{Z}/p\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}$ for some prime $p$?



Injective integer polynomial is injective modulo some prime
Consider $Q(x)=x(2x-1)(3x-1)$. This gives an injective map $\mathbb Z\to \mathbb Z$, because $n<m \implies Q(n)<Q(m)$. However, this $Q$ is not injective over $\mathbb Z/p\mathbb Z$ for any $p$ because $Q(x)=0$ has three solutions when $p\geq 5$ and two solutions when $p\in \{2,3\}$.


