Transcendence of Values of the Iterated Exponential Function at Algebraic Points

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.

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.