Distribution of factorials modulo p
Journal de Théorie des Nombres de Bordeaux, Volume 29 (2017) no. 1, pp. 169-177.

We estimate the average number of residue classes missed by the sequence n!(modp) for px.

On s’intéresse à l’estimation du nombre de valeurs prises par la suite n!(modp). Principalement, on obtient en moyenne sur les nombres premiers px, une minoration du nombre de classes modulo p évitées par la suite n!(modp).

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.974
Classification: 11B50,  11B83,  11R09,  11R45
Keywords: Distribution of sequences mod p, polynomials, density results.
Oleksiy Klurman 1; Marc Munsch 2

1 Départment de Mathématiques et de Statistique Université de Montréal, CP 6128 succ. Centre-Ville Montréal QC H3C 3J7, Canada
2 CRM, Université de Montréal 5357 Montréal, Québec, Canada
@article{JTNB_2017__29_1_169_0,
     author = {Oleksiy Klurman and Marc Munsch},
     title = {Distribution of factorials modulo $p$},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {169--177},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {29},
     number = {1},
     year = {2017},
     doi = {10.5802/jtnb.974},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.974/}
}
TY  - JOUR
TI  - Distribution of factorials modulo $p$
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2017
DA  - 2017///
SP  - 169
EP  - 177
VL  - 29
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.974/
UR  - https://doi.org/10.5802/jtnb.974
DO  - 10.5802/jtnb.974
LA  - en
ID  - JTNB_2017__29_1_169_0
ER  - 
%0 Journal Article
%T Distribution of factorials modulo $p$
%J Journal de Théorie des Nombres de Bordeaux
%D 2017
%P 169-177
%V 29
%N 1
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.974
%R 10.5802/jtnb.974
%G en
%F JTNB_2017__29_1_169_0
Oleksiy Klurman; Marc Munsch. Distribution of factorials modulo $p$. Journal de Théorie des Nombres de Bordeaux, Volume 29 (2017) no. 1, pp. 169-177. doi : 10.5802/jtnb.974. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.974/

[1] William D. Banks; Florian Luca; Igor E. Shparlinski; Henning Stichtenoth On the value set of n! modulo a prime, Turkish J. Math., Volume 29 (2005) no. 2, pp. 169-174

[2] Kevin A. Broughan; A. Ross Barnett On the missing values of n!modp, J. Ramanujan Math. Soc., Volume 24 (2009) no. 3, pp. 277-284

[3] Peter J. Cameron; Arjeh M. Cohen On the number of fixed point free elements in a permutation group, Discrete Math., Volume 106/107 (1992), pp. 135-138 (A collection of contributions in honour of Jack van Lint) | Article

[4] Cristian Cobeli; Marian Vâjâitu; Alexandru Zaharescu The sequence n!(modp), J. Ramanujan Math. Soc., Volume 15 (2000) no. 2, pp. 135-154

[5] Moubariz Z. Garaev; J. Hernández A note on n! modulo p (To appear in Monatschefte für Mathematic, http://arxiv.org/abs/1505.05912)

[6] Moubariz Z. Garaev; Florian Luca Character sums and products of factorials modulo p, J. Théor. Nombres Bordeaux, Volume 17 (2005) no. 1, pp. 151-160 | Article

[7] Moubariz Z. Garaev; Florian Luca; Igor E. Shparlinski Character sums and congruences with n!, Trans. Amer. Math. Soc., Volume 356 (2004) no. 12, p. 5089-5102 (electronic) | Article

[8] Moubariz Z. Garaev; Florian Luca; Igor E. Shparlinski Exponential sums and congruences with factorials, J. Reine Angew. Math., Volume 584 (2005), pp. 29-44 | Article

[9] Richard K. Guy Unsolved problems in number theory, Unsolved Problems in Intuitive Mathematics, Volume 1, Springer-Verlag, New York-Berlin, 1981, xviii+161 pages

[10] Henryk Iwaniec; Emmanuel Kowalski Analytic number theory, American Mathematical Society Colloquium Publications, Volume 53, American Mathematical Society, 2004, xi+615 pages

[11] Oleksiy Klurman; Marc Munsch Distribution of factorials modulo p (Preprint, http://arxiv.org/abs/1505.01198)

[12] Jeffrey C. Lagarias; Hugh L. Montgomery; Andrew M. Odlyzko A bound for the least prime ideal in the Chebotarev density theorem, Invent. Math., Volume 54 (1979) no. 3, pp. 271-296 | Article

[13] Jeffrey C. Lagarias; Andrew M. Odlyzko Effective versions of the Chebotarev density theorem, Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) (1977), pp. 409-464

[14] Vsevolod F. Lev Permutations in abelian groups and the sequence n!(modp), European J. Combin., Volume 27 (2006) no. 5, pp. 635-643 | Article

[15] George Pólya; Gábor Szegő Problems and theorems in analysis. Vol. II: Theory of functions, zeros, polynomials, determinants, number theory, geometry, Die Grundlehren der mathematischen Wissenschaften, Volume 216, Springer-Verlag, 1976, xi+391 pages

[16] Barbara Rokowska; Andrzej Schinzel Sur un problème de M. Erdős, Elem. Math., Volume 15 (1960), p. 84-85

[17] Harold M. Stark Some effective cases of the Brauer-Siegel theorem, Invent. Math., Volume 23 (1974), pp. 135-152 | Article

[18] Tim Trudgian There are no socialist primes less than 10 9 , Integers, Volume 14 (2014), Paper No. A63, 4 pages

Cited by Sources: