On the largest prime factor of quadratic polynomials

TL;DR

This paper improves understanding of the largest prime factors of quadratic polynomials, specifically showing that for large integers, the prime factors of n^2 + 1 can be very large, and provides bounds on primitive divisors.

Contribution

It demonstrates that the largest prime factor of n^2 + 1 exceeds x^{1.317} infinitely often, refining previous bounds and applying this to primitive divisor counts.

Findings

Largest prime factor of n^2 + 1 > x^{1.317} infinitely often

New upper bounds for integers n ≤ x with n^2 + 1 having a primitive divisor

Improves previous results on prime factors of quadratic polynomials

Abstract

Let $x$ denote a sufficiently large integer. We show that the recent result of Grimmelt and Merikoski actually yields the largest prime factor of $n^2 +1$ is greater than $x^{1.317}$ infinitely often. As an application, we give a new upper bound for the number of integers $n \leqslant x$ which $n^2 +1$ has a primitive divisor.