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

TL;DR

This paper proves that under certain conditions, there are infinitely many primes p for which a cubic polynomial f(p) takes square-free values, extending understanding of polynomial values at primes.

Contribution

It establishes the infinitude of primes p with square-free f(p) for cubic polynomials satisfying specific local conditions, a new result in number theory.

Findings

Infinitely many primes p with square-free f(p) exist under given conditions.

Conditions include no repeated roots and local solubility mod q^2.

The result applies to cubic polynomials with particular local properties.

Abstract

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