  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 $1.312.$
