The largest prime factor of an irreducible cubic polynomial

TL;DR

This paper generalizes Heath-Brown's result on large prime factors of integers of the form n^3+2 to all monic irreducible cubic polynomials, establishing that a positive proportion of such values have large prime factors.

Contribution

It extends Heath-Brown's theorem from the specific polynomial n^3+2 to all monic irreducible cubic polynomials, introducing polynomial-dependent exponents.

Findings

Positive proportion of polynomial values have large prime factors

Generalization from specific to all monic irreducible cubics

Introduction of polynomial-dependent exponent c_p

Abstract

Heath-Brown proved that for a positive proportion of integers $n$, $n^3+2$ has a prime factor larger than $n^{1+c}$ with $c=10^{-303}$. We generalize this result to arbitrary monic irreducible cubic polynomial of $\mathbb{Z}[x]$ with $c$ replaced by an exponent $c_p$ dependent on the polynomial.