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 polynomials of $\mathbb{Z}[x]$ with $c$ replaced by an exponent $c_p$ dependent on the polynomial.

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