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This article is cited in 1 scientific paper (total in 1 paper)
On polyadic Liouville numbers
V. G. Chirskiiab a Lomonosov Moscow State University (Moscow)
b RANEPA (Moscow)
Abstract:
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 $$\left|\lambda -A \right|_{p}<A^{-n}$$ 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.
Keywords:
polyadic number, polyadic Liouville number.
Received: 23.08.2021 Accepted: 21.12.2021
Citation:
V. G. Chirskii, “On polyadic Liouville numbers”, Chebyshevskii Sb., 22:5 (2021), 243–251; Doklady Mathematics, 106:Suppl. 2 (2022), S137–S141
Linking options:
https://www.mathnet.ru/eng/cheb1130 https://www.mathnet.ru/eng/cheb/v22/i5/p243
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