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A little reading suggests:

It is known that either $\pi + e$ or $\pi \times e$ is transcendental (or possibly both), but no proof is known that one of those two numbers in particular is transcendental.

If we just want irrationality rather than transcendence, is a proof known?

Can we prove $\pi+e$ is irrational? Can we prove $\pi \times e$ is irrational?

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2  
See this: mathoverflow.net/questions/40145/… – PEV Mar 21 '11 at 4:43
1  
And this: mathforum.org/library/drmath/view/51617.html – PEV Mar 21 '11 at 4:45
up vote 17 down vote accepted

It is not known whether $\pi + e$ is irrational, nor whether $\pi \times e$ is irrational. See $\# 22$ here.

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I typeset your answer to change * to $\times$ and to add the webpage as a link. Hope you don't mind. – user17762 Mar 21 '11 at 4:51
    
@Sivaram: Not at all, thanks for cleaning it up. – Alex Becker Mar 21 '11 at 4:53
    
The link does not work anymore. Here is the version from Google cache. – Martin Sleziak Jun 17 '12 at 6:56

protected by T. Bongers Jun 29 '14 at 20:23

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