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

Luca, Florian; Shparlinski, Igor E.

doi:10.5802/jtnb.524

On the largest prime factor of

n! + 2^{n} - 1

Luca, Florian ¹ ; 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.

Résumé
Abstract

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 .

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 .

EuDML MR Zbl

DOI : 10.5802/jtnb.524

Affiliations des auteurs :

Luca, Florian ¹ ; 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

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     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},
     volume = {17},
     number = {3},
     year = {2005},
     doi = {10.5802/jtnb.524},
     mrnumber = {2212129},
     zbl = {1097.11006},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/jtnb.524/}
}

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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. https://www.numdam.org/articles/10.5802/jtnb.524/

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