Any proof to $\pi^{e}$'s irrationality?

Question

I've searched for this for a while but get nothing...

There are plenty of proofs to irrationality of $e$,$\pi$,$e^{\pi}$. However, I can't find a proof for $\pi^e$. More, when searching for this I got a answer Unknown on this site(in Chinese), which says

目前为止， $\pi^e$ 是否有理还是一个谜。

Whether $\pi^e$ is irrational or not is unknown up to now(2010).

Thus, I just wanna know whether there's progressing these years...

How do you know that's irrational. You do not know.(Until its proved). — Inceptio, May 08 '13 at 17:15
Sorry for that but I misunderstood a wrong proof... which is from a collection of that kind of stuff... — Vury Leo, May 08 '13 at 17:17
Somewhat related: http://math.stackexchange.com/questions/159350/why-is-it-hard-to-prove-whether-pie-is-an-irrational-number — Jonas Meyer, May 08 '13 at 17:27

score 4 · Answer 1 · answered Oct 18 '13 at 16:20

In the sixth edition of 'An Introduction to the Theory of Numbers' by Hardy and Wright (revised by Heath-Brown and Silverman) published in 2008, in section 4.2 they state that for $\pi^e$ this is still open. Other numbers whose irrationality has not been proved include $2^e, \pi^{\sqrt{2}}, e+\pi$.

Thus, as of 2008, this was not yet resoled.

Any proof to $\pi^{e}$'s irrationality?

1 Answers1