Journal of Integer Sequences, Vol. 26 (2023), Article 23.3.3

Transcendence of Values of the Iterated Exponential Function at Algebraic Points

Hirotaka Kobayashi
Graduate School of Mathematics
Nagoya University
Furo-cho, Chikusa-ku
Nagoya 464-8602

Kota Saito
Faculty of Pure and Applied Sciences
University of Tsukuba
1-1-1 Tenodai, Tsukuba
Ibaraki 305-8571

Wataru Takeda
Department of Applied Mathematics
Tokyo University of Science
1-3 Kagurazaka, Shinjuku-ku
Tokyo 162-8601


We say that the limit of a sequence of functions

$\displaystyle x,\quad x^x, \quad x^{x^x},\ldots

is the iterated exponential function, denoted by $h(x)$. By a result of Barrow, this limit is convergent for every $x\in[e^{-e}, e^{1/e}]$. In this paper, we prove that, for each fixed integer $k\ge2$, the limit $h(A)$ is transcendental for all but finitely many algebraic numbers $A\in[e^{-e}, e^{1/e}]$ with $k=\min\{n\in\mathbb{N} \mid A^n\in\mathbb{\mathbb{Q}}\}$. Furthermore, let $Q(k)$ be the cardinality of exceptional points $A$. We prove that the ratio $Q(k)/\varphi(k)$ approaches $e-1/e$ as $k\rightarrow \infty$, where $\varphi(k)$ denotes Euler's totient function.

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Received December 24 2022; revised versions received March 8 2023; March 11 2023; March 12 2023. Published in Journal of Integer Sequences, March 13 2023.

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