On the greatest prime factor of \((n^2)+1\). (English) Zbl 0489.10038

There exist infinitely many integers \(n\) such that the greatest prime factor of \(n^2 + 1\) is at least \(n^{6/5}\). The proof is a combination of Hooley’s method – for reducing the problem to the evaluation of Kloosterman sums – and the majorization of Kloosterman sums on average due to the authors.


11N05 Distribution of primes
11L05 Gauss and Kloosterman sums; generalizations
Full Text: DOI Numdam EuDML

Online Encyclopedia of Integer Sequences:

Largest prime factor of n^2 + 1.


[1] J.-M. DESHOUILLERS and H. IWANIEC, Kloosterman sums and Fourier coefficients of cusp forms, Inv. Math. (to appear). · Zbl 0502.10021
[2] C. HOOLEY, On the greatest prime factor of a quadratic polynomial, Acta Math., 117 (1967), 281-299. · Zbl 0146.05704
[3] C. HOOLEY, Applications of sieve methods to the theory of numbers, Cambridge Univ. Press, London, 1976. · Zbl 0327.10044
[4] H. IWANIEC, Rosser’s sieve, Acta Arith., 36 (1980), 171-202. · Zbl 0435.10029
[5] H.J.S. SMITH, Report on the theory of numbers, Collected Mathematical Papers, vol. I, reprinted, Chelsea, 1965.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.