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.

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:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1080
Classification: 11B85,  11A63,  68Q45,  68R15
Keywords: automatic sequence, factorial, the last nonzero digit
Eryk Lipka 1

1 Potok 448 38-404 Krosno, Poland
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Eryk Lipka. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1080/

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[2] Jakub Byszewski; Jakub Konieczny A density version of Cobham’s theorem (2017) (https://arxiv.org/abs/1710.07261) | Zbl: 07178785

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

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

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[7] Cameron L. Stewart On the representation of an integer in two different bases, J. Reine Angew. Math., Volume 319 (1980), pp. 63-72 | MR: 586115 | Zbl: 0426.10008

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