Automaticity of the sequence of the last nonzero digits of n! in a fixed base
Eryk Lipka
Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 1, p. 283-291

In 2011 Deshouillers and Ruzsa [5] tried to argue that the sequence of the last nonzero digit of n! in base 12 is not automatic. This statement was proven a few years later by Deshoulliers in [4]. In this paper we provide an alternate proof that lets us generalize the problem and give an exact characterization of the bases for which the sequence of the last nonzero digits of n! is automatic.

En 2011, Deshouillers et Ruzsa [5] ont donné des arguments en faveur de la non-automaticité de la suite des derniers chiffres non nuls de n! en base 12. Cette assertion a été prouvée quelques années plus tard par Deshoulliers [4]. Dans cet article, nous donnons une preuve alternative qui nous permet de généraliser le problème et donner une caractérisation complète des bases pour lesquelles la suite des derniers chiffres non nuls de n! est automatique.

Received : 2018-08-27
Revised : 2018-11-06
Accepted : 2018-11-16
Published online : 2019-07-29
DOI : https://doi.org/10.5802/jtnb.1080
Classification:  11B85,  11A63,  68Q45,  68R15
Keywords: automatic sequence, factorial, the last nonzero digit
@article{JTNB_2019__31_1_283_0,
     author = {Eryk Lipka},
     title = {Automaticity of the sequence of the last nonzero digits of $n!$ in a fixed base},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     year = {2019},
     pages = {283-291},
     doi = {10.5802/jtnb.1080},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2019__31_1_283_0}
}
Lipka, Eryk. Automaticity of the sequence of the last nonzero digits of $n!$ in a fixed base. Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 283-291. doi : 10.5802/jtnb.1080. jtnb.centre-mersenne.org/item/JTNB_2019__31_1_283_0/

[1] Jean-Paul Allouche; Jeffrey Shallit Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, 2003 | Zbl 1086.11015

[2] Jakub Byszewski; Jakub Konieczny A density version of Cobham’s theorem (2017) (https://arxiv.org/abs/1710.07261)

[3] Jean-Marc Deshouillers A footnote to The least non zero digit of n! in base 12, Unif. Distrib. Theory, Tome 7 (2012) no. 1, pp. 71-73 | Zbl 1313.11024

[4] Jean-Marc Deshouillers Yet another footnote to The least non zero digit of n! in base 12, Unif. Distrib. Theory, Tome 11 (2016) no. 2, pp. 163-167 | Zbl 06846994

[5] Jean-Marc Deshouillers; Imre Ruzsa The least non zero digit of n! in base 12, Publ. Math., Tome 79 (2011) no. 3-4, pp. 395-400 | Zbl 1249.11044

[6] Adrien-Marie Legendre Théorie des nombres, Firmin Didot frères, 1830 | Zbl 1395.11005

[7] Cameron L. Stewart On the representation of an integer in two different bases, J. Reine Angew. Math., Tome 319 (1980), pp. 63-72 | Zbl 0426.10008