On polyadic Liouville numbers

V. G. Chirskii

Čebyševskij sbornik, Tome 22 (2021) no. 5, pp. 243-251

Résumé

We study here polyadic Liouville numbers, which are involved in a series of recent papers. The canonic expansion of a polyadic number $\lambda$ is of the form $$ \lambda= \sum_{n=0}^\infty a_{n} n!, a_{n}\in\mathbb{\mathrm{Z}}, 0\leq a_{n}\leq n.$$ This series converges in any field of $p-$ adic numbers $ \mathbb{\mathrm{Q}}_p $. We call a polyadic number $\lambda$ a polyadic Liouville number, if for any $n$ and $P$ there exists a positive integer $A$ such that for all primes $p$, satisfying $p\leq P$ the inequality $$\left|\lambda -A \right|_{p}^{-n}$$ holds. Let $k\geq 2$ be a positive integer. We denote for a positive integer $m$ $$\Phi(k,m)=k^{k^{\ldots^{k}}}$$ Let $$n_{m}=\Phi(k,m)$$ and let$$\alpha=\sum_{m=0}^{\infty}(n_{m})!.$$ Theorem 1. For any positive integer $k\geq 2$ and any prime number $p$ the series $\alpha$ converges to a transcendental element of the ring $\mathbf{Z}_p.$ In other words, the polyadic number $\alpha$ is globally transcendental.

Détail

Keywords: polyadic number, polyadic Liouville number.