Is there a proof that $\pi \times e$ is irrational?

Question

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?

See this: http://mathoverflow.net/questions/40145/irrationality-of-pie-pipi-and-epi2 — NebulousReveal, Mar 21 '11 at 04:43
And this: http://mathforum.org/library/drmath/view/51617.html — NebulousReveal, Mar 21 '11 at 04:45
See here it’s been recently proven that it’s irrational. @NebulousReveal — homosapien, Aug 10 '22 at 01:37

score 27 · Accepted Answer · edited Mar 21 '11 at 04:50

27

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

edited Mar 21 '11 at 04:50

answered Mar 21 '11 at 04:42

Alex Becker

60,569

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 is the version from Google cache. – Martin Sleziak Jun 17 '12 at 06:56

Is there a proof that $\pi \times e$ is irrational?

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