Any proof to $\pi^{e}$'s irrationality?
Solution 1:
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.