On the largest prime factor of $n!+2^n-1$

Luca, Florian¹ ; Shparlinski, Igor E.² 

¹ Instituto de Matemáticas Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México
² Macquarie University Sydney, NSW 2109, Australia

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 859-870.

Abstract (VO)

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 $n!+2^n-1\equiv 0(modq)$ and use these bounds to show that

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

Résumé

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 $n!+2^n-1\equiv 0(modq)$ et nous utilisons ces bornes pour montrer que

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

Détail

DOI : 10.5802/jtnb.524

Bibliographie

[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, | Zbl | MR

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

[3] P. Erdős, R. Murty, On the order of $a(modp)$. Proc. 5th Canadian Number Theory Association Conf., Amer. Math. Soc., Providence, RI, 1999, 87–97. | Zbl

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

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

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

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

[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. | Zbl | MR

[9] F. Pappalardi, On the order of finitely generated subgroups of ${\mathbb{Q}}^{*}(modp)$ and divisors of $p-1$. J. Number Theory 57 (1996), 207–222. | Zbl | MR

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