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

Florian Luca; Igor E. Shparlinski

doi:10.5802/jtnb.524

On the largest prime factor of

n! + 2^{n} - 1

Florian Luca ¹; Igor E. Shparlinski 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, Volume 17 (2005) no. 3, pp. 859-870.

Abstract
Résumé

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

\underset{n \to \infty}{lim sup} P (n! + 2^{n} - 1) / n \geq (2 π^{2} + 3) / 18 .

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

\underset{n \to \infty}{lim sup} P (n! + 2^{n} - 1) / n \geq (2 π^{2} + 3) / 18 .

Published online: 2009-03-16

MR: 2212129 | Zbl: 1097.11006

DOI: 10.5802/jtnb.524
Author's affiliations:

Florian Luca ¹; Igor E. Shparlinski ²

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

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Florian Luca; Igor E. Shparlinski. On the largest prime factor of $n!+ 2^n-1$. Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 3, pp. 859-870. doi : 10.5802/jtnb.524. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.524/

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