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By the Gelfond-Schneider theorem, if $x^x=2$, $x$ must be transcendental. What can be said of $x$ if $x^{x^x}=2$, $x^{x^{x^x}}=2$ etc.? Must it be transcendental? Of course, $2$ can be replaced by any algebraic number.
Henry Harry Lande M Sc

Daniele Tampieri
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Harry Lande
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  • This is open as far as I know, it's been asked before here: https://math.stackexchange.com/questions/373881/is-the-positive-root-of-the-equation-xxx-2-x-1-47668433-a-transcende?rq=1 I think everyone expects the answer to be yes but it's out of reach. – Qiaochu Yuan Sep 27 '23 at 17:55
  • What is the distinction between x and X ? – Dan Asimov Sep 27 '23 at 19:03

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