On the greatest prime factor of n 2 +1
Annales de l'Institut Fourier, Volume 32 (1982) no. 4, pp. 1-11.

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.

Il existe une infinité d’entiers n tels que le plus grand facteur premier de n 2 +1 soit au moins n 6/5 . La démonstration de ce résultat combine la méthode de Hooley – pour ramener le problème à l’évaluation de sommes de Kloosterman – et la majoration de sommes de Kloosterman en moyenne obtenue par les auteurs.

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Deshouillers, Jean-Marc; Iwaniec, Henryk. On the greatest prime factor of $n^2+1$. Annales de l'Institut Fourier, Volume 32 (1982) no. 4, pp. 1-11. doi : 10.5802/aif.891. https://aif.centre-mersenne.org/articles/10.5802/aif.891/

[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. | MR: 34 #4225 | 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.

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