On the largest prime factor of n^2+1

Abstract: It is an open conjecture 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 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 the exponent 1.2182, improving the exponent 1.2024 obtained by Deshouillers and Iwaniec. The main new ingredients in the proof are 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 the exponent 1.279 may be increased to 1.312.

number theory

Audience: researchers in the topic

CRM-CICMA Québec Vermont Seminar Series

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