On the largest prime factor of $n^2+1$

TL;DR

This paper proves that the largest prime factor of numbers of the form n^2+1 exceeds n^{1.279} infinitely often, improving previous bounds by employing new sieve methods and bounds on Kloosterman sums.

Contribution

It introduces a new Type II estimate and applies Harman's sieve, advancing the lower bound on the largest prime factor of n^2+1.

Findings

Largest prime factor of n^2+1 > n^{1.279} infinitely often

Conditional bound increase to n^{1.312} under Selberg's eigenvalue conjecture

Utilizes bounds on linear forms of Kloosterman sums for the estimate

Abstract

We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bret\`eche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof are a new Type II estimate and using this estimate by applying Harman's sieve method. To prove the Type II estimate we use the bounds of Deshouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selberg's eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$