Transcendence of values of the iterated exponential function at algebraic points

2019/12/19 by Kobayashi, Hirotaka, Saito, Kota, Takeda, Wataru
#11D45 #11J81 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1912.09125

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^AA, … converges to an algebraic number is approximated by (e-1/e)φ(k). Here φ(k) denotes Euler's totient function.

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