Square-free values of f(p), f cubic

Abstract

Let $f\in \mathbb{Z}[x]$, $degf=3$. Assume that f does not have repeated roots. Assume as well that, for every prime q, $f(x)\not\equiv 0$ mod q² has at least one solution in ${(\mathbb{Z}/q^2\mathbb{Z})}^{\ast }$. Then, under these two necessary conditions, there are infinitely many primes p such that f(p) is square-free.