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

Kohta Gejima, Fumichika Takamizo

TL;DR

This work advances transcendence theory by extending the Adamczewski–Bugeaud criterion to Cantor-base representations and proving the sequence of last nonzero digits of $n!$ is not eventually periodic for every base $b>2$. It leverages stammering sequences (including all $k$-automatic sequences) and Evertse's Subspace Theorem to derive a Cantor-base transcendence criterion, giving conditions under which a Cantor-base expansion defines a transcendental number. The authors apply this framework to construct explicit transcendental numbers from the nonperiodic last-nonzero-digit sequence, under Pisot Cantor-base settings, providing concrete examples and broadening the class of known transcendental Cantor-base numbers. These results connect automaticity, nonperiodicity, and Pisot-type dynamics to practical transcendence constructions, with potential implications for Diophantine approximation and digital expansion methods.

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.

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

TL;DR

This work advances transcendence theory by extending the Adamczewski–Bugeaud criterion to Cantor-base representations and proving the sequence of last nonzero digits of is not eventually periodic for every base . It leverages stammering sequences (including all -automatic sequences) and Evertse's Subspace Theorem to derive a Cantor-base transcendence criterion, giving conditions under which a Cantor-base expansion defines a transcendental number. The authors apply this framework to construct explicit transcendental numbers from the nonperiodic last-nonzero-digit sequence, under Pisot Cantor-base settings, providing concrete examples and broadening the class of known transcendental Cantor-base numbers. These results connect automaticity, nonperiodicity, and Pisot-type dynamics to practical transcendence constructions, with potential implications for Diophantine approximation and digital expansion methods.

Abstract

We prove that the sequence of the last nonzero digits of factorials in every integer base 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.

Paper Structure

This paper contains 12 sections, 18 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cantor base representations
  4. Automata
  5. Evertse's Subspace Theorem
  6. The Adamczewski--Bugeaud criterion
  7. Stammering sequences
  8. The Adamczewski--Bugeaud criterion for Cantor base representations
  9. Non-periodicity of the sequence of the last nonzero digits of factorials
  10. Preliminaries
  11. The proof of the non-periodicity
  12. An application

Key Result

Theorem 1

Let $b>2$ be an integer with the prime factorization $b=p_1^{a_1}p_2^{a_2}\cdots$ such that $a_1(p_1-1) \geq a_2(p_2-1) \geq \cdots$. Let ${\bm{\mathcal{B}}}=(\beta_k)_{k=1}^\infty$ be a real Cantor base of period $p$ consisting of real numbers $\beta_k>1$ such that $\delta=\prod_{k=1}^p\beta_k$ is is transcendental.

Theorems & Definitions (31)

  • Conjecture 1
  • Theorem 1
  • Proposition 2.1
  • proof
  • Theorem 2.2: CCK, MP
  • Theorem 2.3: Evertse Evertse
  • Proposition 3.1
  • proof
  • Claim 3.2
  • proof : Proof of Claim \ref{['claim:B_m']}
  • ...and 21 more