On the largest prime factor of n2 C 1

We show that the largest prime factor of n2 + 1 is infinitely often greater than n1.279. This improves the result of de la Breteche and Drappeau specialIntscript 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.