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

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 Bretèche and Drappeau (2020) who obtained this with $1.2182$ in place of $\mathrm{1.279.}$ 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 $\mathrm{1.312.}$