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

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.
