Abstract
Let \({f \in \mathbb{Z}[x]}\), \({\deg f =3}\). Assume that f does not have repeated roots. Assume as well that, for every prime q, \({f(x)\not\equiv 0}\) mod q 2 has at least one solution in \({(\mathbb{Z}/q^2 \mathbb{Z})^*}\). Then, under these two necessary conditions, there are infinitely many primes p such that f(p) is square-free.
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Helfgott, H.A. Square-free values of f(p), f cubic. Acta Math 213, 107–135 (2014). https://doi.org/10.1007/s11511-014-0117-2
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DOI: https://doi.org/10.1007/s11511-014-0117-2