TY - JOUR
T1 - Transcendence of Values of the Iterated Exponential Function at Algebraic Points
AU - Kobayashi, Hirotaka
AU - Saito, Kota
AU - Takeda, Wataru
N1 - Publisher Copyright:
© 2023, University of Waterloo. All rights reserved.
PY - 2023
Y1 - 2023
N2 - We say that the limit of a sequence of functions x, xx, xxx, … is the iterated exponential function, denoted by h(x). By a result of Barrow, this limit is convergent for every (formula presented). In this paper, we prove that, for each fixed integer k ≥ 2, the limit h(A) is transcendental for all but finitely many algebraic numbers (formula presented). Furthermore, let Q(k) be the cardinality of exceptional points A. We prove that the ratio Q(k)/ϕ(k) approaches e − 1/e as k → ∞, where ϕ(k) denotes Euler’s totient function.
AB - We say that the limit of a sequence of functions x, xx, xxx, … is the iterated exponential function, denoted by h(x). By a result of Barrow, this limit is convergent for every (formula presented). In this paper, we prove that, for each fixed integer k ≥ 2, the limit h(A) is transcendental for all but finitely many algebraic numbers (formula presented). Furthermore, let Q(k) be the cardinality of exceptional points A. We prove that the ratio Q(k)/ϕ(k) approaches e − 1/e as k → ∞, where ϕ(k) denotes Euler’s totient function.
KW - Gelfond-Schneider theorem
KW - iterated exponential
KW - Lindemann theorem
KW - transcendental number
UR - https://www.scopus.com/pages/publications/85150596421
M3 - Article
AN - SCOPUS:85150596421
SN - 1530-7638
VL - 26
JO - Journal of Integer Sequences
JF - Journal of Integer Sequences
IS - 3
M1 - 23.3.3
ER -