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 .

EuDML: 249430 | 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

@article{JTNB_2005__17_3_859_0,
     author = {Florian Luca and Igor E. Shparlinski},
     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},
     zbl = {1097.11006},
     mrnumber = {2212129},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.524/}
}

TY  - JOUR
AU  - Florian Luca
AU  - Igor E. Shparlinski
TI  - On the largest prime factor of $n!+ 2^n-1$
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2005
SP  - 859
EP  - 870
VL  - 17
IS  - 3
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.524/
UR  - https://zbmath.org/?q=an%3A1097.11006
UR  - https://www.ams.org/mathscinet-getitem?mr=2212129
UR  - https://doi.org/10.5802/jtnb.524
DO  - 10.5802/jtnb.524
LA  - en
ID  - JTNB_2005__17_3_859_0
ER  -

%0 Journal Article
%A Florian Luca
%A Igor E. Shparlinski
%T On the largest prime factor of $n!+ 2^n-1$
%J Journal de théorie des nombres de Bordeaux
%D 2005
%P 859-870
%V 17
%N 3
%I Université Bordeaux 1
%U https://doi.org/10.5802/jtnb.524
%R 10.5802/jtnb.524
%G en
%F JTNB_2005__17_3_859_0

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/

References
Cited by

[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 | Zbl

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

[3] P. Erdős, R. Murty, On the order of $a (mod p)$ . 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. | MR | Zbl

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

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

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

[8] M.R. Murty, S. Wong, The $A B C$ 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 | Zbl

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

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

Cited by Sources: