Is there some relation between the trascendence of $e^\pi$ and that of $\pi^e$? I mean: the transcendence of one implies the other or the proofs are independent? Thanks.
2026-03-25 23:19:51.1774480791
Trascendence of $e^\pi$ and $\pi^e$
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As answered in the comments there is no immediate relation between them. $e^\pi$ is transcendental but it is not known if $\pi^e$ is transcendental or not.
If you wish to create a trivial relation you could say that if $e^{\pi}$ is algebraic then $\pi^e$ is algebraic which is something like saying if $0 = 1$ then I am the King of England.