The largest prime factor of $X^3+2$

TL;DR

This paper improves a theorem by Heath-Brown, demonstrating that for large X, a positive proportion of numbers of the form n^3+2 have a prime factor exceeding X^{1+10^{-52}}.

Contribution

It advances previous results by establishing a stronger lower bound on the size of prime factors for values of n^3+2.

Findings

A positive proportion of n^3+2 values have prime factors larger than X^{1+10^{-52}}.

The result applies for sufficiently large X.

Improves on Heath-Brown's earlier theorem.

Abstract

Improving on a theorem of Heath-Brown, we show that if $X$ is sufficiently large then a positive proportion of the values $\{n^3+2:n\in (X,2X]\}$ have a prime factor larger than $X^{1+10^{-52}}$.