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

@article{AIF_1982__32_4_1_0,
     author = {Deshouillers, Jean-Marc and Iwaniec, Henryk},
     title = {On the greatest prime factor of $n^2+1$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {32},
     number = {4},
     year = {1982},
     pages = {1-11},
     doi = {10.5802/aif.891},
     zbl = {0489.10038},
     mrnumber = {84m:10033},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1982__32_4_1_0}
}
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. http://www.numdam.org/item/AIF_1982__32_4_1_0/

[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.