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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?

idmercer
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1 Answers1

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

Alex Becker
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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. –  Mar 21 '11 at 04:51
  • @Sivaram: Not at all, thanks for cleaning it up. – Alex Becker Mar 21 '11 at 04:53
  • The link does not work anymore. [Here](https://docs.google.com/open?id=1h1YKOImdD9cB3_r26W2h6JMcoubjt1v15HhMTVapZkrBDhUUH_nqO6gpaNsY) is the version from Google cache. – Martin Sleziak Jun 17 '12 at 06:56