Largest prime factor of n^2+1

A famous open problem in number theory is to show that there are infinitely many prime numbers of the form n^2+1. To approach this we may consider the largest prime factor of n^2+1. In this talk I will 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 who obtained this with the exponent 1.2182 in place of 1.279. The main new ingredients in the proof are the use Harman's sieve method and a new bilinear estimate which is proved by applying the Deshouillers-Iwaniec bounds for sums of Kloosterman sums. Assuming Selberg's eigenvalue conjecture I show that the exponent 1.279 can be increased to 1.312.