Transcendence of values of the iterated exponential function at algebraic points

Hirotaka Kobayashi; Kota Saito; Wataru Takeda

Transcendence of values of the iterated exponential function at algebraic points

Number Theory 2020-01-08 v2

Authors: Hirotaka Kobayashi , Kota Saito , Wataru Takeda

Abstract

We say that the order of an algebraic number $A$ is the minimum of positive integers $k$ such that $A^k$ is rational. In this paper, we show that the number of algebraic numbers $A$ with order $k$ such that $A,\ A^A,\ A^{A^A},\ \ldots$ converges to an algebraic number is approximated by $(e-1/e)\varphi (k)$ . Here $\varphi(k)$ denotes Euler's totient function.

Keywords

iterated exponential algebraic number transcendence order of algebraic number Euler's totient function convergence

Cite

@article{arxiv.1912.09125,
  title  = {Transcendence of values of the iterated exponential function at algebraic points},
  author = {Hirotaka Kobayashi and Kota Saito and Wataru Takeda},
  journal= {arXiv preprint arXiv:1912.09125},
  year   = {2020}
}

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15 pages

Transcendence of values of the iterated exponential function at algebraic points

Abstract

Keywords

Cite

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