Is the positive root of the equation $x^{x^x}=2$, $x=1.47668433...$ a transcendental number?

transcendental-numbers irrational-numbers transcendence-theory tetration transcendental-equations

Solution 1:

I believe, it is a known open problem. Ditto for ${^3 x}=3$, ${^3 x}=4$, ${^3 x}=5$ (left superscript denotes tetration).

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