arXiv subjects

Fumichika Takamizo

Publications and source records attributed to Fumichika Takamizo.

2 recordsLinked to original sources

arXiv2025

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

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.

math.NT↗

arXiv2023

Finite beta-expansions of natural numbers

Let $β>1$. For $x \in [0,\infty)$, we have so-called the $β$-expansion of $x$ in base $β$ as follows: $$x= \sum_{j \leq k} x_{j}β^{j} = x_{k}β^{k}+ \cdots + x_{1}β+x_{0}+x_{-1}β^{-1} + x_{-2}β^{-2} + \cdots$$ where $k \in \mathbb{Z}$, $β^{k} \leq x < β^{k+1}$, $x_{j} \in \mathbb{Z} \cap [0,β)$ for all $j \leq k$ and $\sum_{j \leq n}x_{j}β^{j}<β^{n+1}$ for all $n \leq k$. In this paper, we give a sufficient condition (for $β$) such that each element of $\mathbb{N}$ has the finite beta-expansion in base $β$. Moreover we also find a $β$ with this finiteness property which does not have positive finiteness property.

math.NT↗