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...

  • $\begingroup$ How do you know that's irrational. You do not know.(Until its proved). $\endgroup$ – Inceptio May 8 '13 at 17:15
  • $\begingroup$ Sorry for that but I misunderstood a wrong proof... which is from a collection of that kind of stuff... $\endgroup$ – Vury Leo May 8 '13 at 17:17
  • 1
    $\begingroup$ Somewhat related: math.stackexchange.com/questions/159350/… $\endgroup$ – Jonas Meyer May 8 '13 at 17:27

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.


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