# Trascendence of $e^\pi$ and $\pi^e$

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.

• I don't think it's known if $\pi^e$ is transcendental or not. Jul 1 '14 at 14:39
• @DavidMitra: Yes, but, if we prove $\pi^e$ is trascendental, this automatically, means $e^\pi$ is trascendental? Jul 1 '14 at 14:42
• I don't know... However, $e^\pi$ is transcendental; Google "Gelfond's constant". Jul 1 '14 at 14:44
• Jul 1 '14 at 14:45
• @Shaun Arf... There are has been a slight mix-up between your two links : both are the same. Maybe you can still edit ?
– jibe
Jul 1 '14 at 14:48

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.