Transcendence of values of the iterated exponential function at algebraic points

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 $(e-1/e)φ(k)$. Here $φ(k)$ denotes Euler's totient function.

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