On the greatest prime factor of $n^2+1$

Deshouillers, Jean-Marc; Iwaniec, Henryk

doi:10.5802/aif.891

On the greatest prime factor of

n^{2} + 1

Deshouillers, Jean-Marc ; Iwaniec, Henryk

Annales de l'Institut Fourier, Tome 32 (1982) no. 4, pp. 1-11.

Résumé
Abstract

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.

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.

MR Zbl | 4 citations dans Numdam

DOI : 10.5802/aif.891

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

Bibliographie
Cité par

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[5] H.J.S. Smith, Report on the theory of numbers, Collected Mathematical Papers, vol. I, reprinted, Chelsea, 1965.

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Cité par 23 documents. Sources : Crossref, zbMATH