Non-periodicity of the sequence of the last nonzero digits of factorials and its applications to transcendence

2025/09/26 by Kohta Gejima, Gejima, Kohta, Fumichika Takamizo +1
Computer Science · Engineering · #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #graph theory and CDMA systems #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2509.21948

openalex publication_date 2025/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the sequence of the last nonzero digits of factorials in every integer base b>2 is not eventually periodic. We also extend the Adamczewski--Bugeaud criterion, originally formulated for integer base expansions, to Cantor base expansions associated with a periodic Cantor base. As an application, we show that a certain real number expressed through a Cantor base expansion is transcendental when the Cantor base and the digit sequence satisfy suitable conditions.

Citations

• Periodicity and pure periodicity in alternate base systems
• On Periodic Alternate Base Expansions
• Expansions in Cantor real bases
• Automaticity of the sequence of the last nonzero digits of n! in a\n fixed base
• Composition inverses of the variations of the Baum-Sweet sequence
• Representations for real numbers and their ergodic properties

Related

• Periodicity and pure periodicity in alternate base systems
• Periodicity of the last digits of some combinatorial sequences
• An analogue of Solomyak's theorem for periodic Cantor real expansions in alternate bases
• Transcendence of digital expansions and continued fractions generated by a cyclic permutation and k-adic expansion
• On the last digit and the last non-zero digit of nⁿ in base b