On the largest prime factor of $n!+{2}^{n}-1$
Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 859-870.

Pour un entier $n\ge 2$, notons $P\left(n\right)$ le plus grand facteur premier de $n$. Nous obtenons des majorations sur le nombre de solutions de congruences de la forme $n!+{2}^{n}-1\equiv 0\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}q\right)$ et nous utilisons ces bornes pour montrer que

 $\underset{n\to \infty }{lim sup}P\left(n!+{2}^{n}-1\right)/n\ge \left(2{\pi }^{2}+3\right)/18.$

For an integer $n\ge 2$ we denote by $P\left(n\right)$ the largest prime factor of $n$. We obtain several upper bounds on the number of solutions of congruences of the form $n!+{2}^{n}-1\equiv 0\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}q\right)$ and use these bounds to show that

 $\underset{n\to \infty }{lim sup}P\left(n!+{2}^{n}-1\right)/n\ge \left(2{\pi }^{2}+3\right)/18.$

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author = {Luca, Florian and Shparlinski, Igor E.},
title = {On the largest prime factor of $n!+ 2^n-1$},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {859--870},
publisher = {Universit\'e Bordeaux 1},
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Luca, Florian; Shparlinski, Igor E. On the largest prime factor of $n!+ 2^n-1$. Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 859-870. doi : 10.5802/jtnb.524. http://www.numdam.org/articles/10.5802/jtnb.524/

[1] R. C. Baker, G. Harman, The Brun-Titchmarsh theorem on average. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995), Progr. Math. 138, Birkhäuser, Boston, MA, 1996, 39–103, | MR 1399332 | Zbl 0853.11078

[2] R. C. Baker, G. Harman, Shifted primes without large prime factors. Acta Arith. 83 (1998), 331–361. | MR 1610553 | Zbl 0994.11033

[3] P. Erdős, R. Murty, On the order of $a\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}p\right)$. Proc. 5th Canadian Number Theory Association Conf., Amer. Math. Soc., Providence, RI, 1999, 87–97. | Zbl 0931.11034

[4] P. Erdős, C. Stewart, On the greatest and least prime factors of $n!+1$. J. London Math. Soc. 13 (1976), 513–519. | MR 409334 | Zbl 0332.10028

[5] É. Fouvry, Théorème de Brun-Titchmarsh: Application au théorème de Fermat. Invent. Math. 79 (1985), 383–407. | MR 778134 | Zbl 0557.10035

[6] H.-K. Indlekofer, N. M. Timofeev, Divisors of shifted primes. Publ. Math. Debrecen 60 (2002), 307–345. | MR 1898566 | Zbl 1017.11042

[7] F. Luca, I. E. Shparlinski, Prime divisors of shifted factorials. Bull. London Math. Soc. 37 (2005), 809–817. | MR 2186713 | Zbl 05009917

[8] M.R. Murty, S. Wong, The $ABC$ conjecture and prime divisors of the Lucas and Lehmer sequences. Number theory for the millennium, III (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, 43–54. | MR 1956267 | Zbl 1030.11012

[9] F. Pappalardi, On the order of finitely generated subgroups of ${ℚ}^{*}\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}p\right)$ and divisors of $p-1$. J. Number Theory 57 (1996), 207–222. | MR 1382747 | Zbl 0847.11049

[10] K. Prachar, Primzahlverteilung. Springer-Verlag, Berlin, 1957. | MR 87685 | Zbl 0080.25901

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