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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.

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    $\begingroup$ I don't think it's known if $\pi^e$ is transcendental or not. $\endgroup$ – David Mitra Jul 1 '14 at 14:39
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    $\begingroup$ @DavidMitra: Yes, but, if we prove $\pi^e$ is trascendental, this automatically, means $e^\pi$ is trascendental? $\endgroup$ – Riccardo.Alestra Jul 1 '14 at 14:42
  • $\begingroup$ I don't know... However, $e^\pi$ is transcendental; Google "Gelfond's constant". $\endgroup$ – David Mitra Jul 1 '14 at 14:44
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    $\begingroup$ @Shaun Arf... There are has been a slight mix-up between your two links : both are the same. Maybe you can still edit ? $\endgroup$ – jibe Jul 1 '14 at 14:48
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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.

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