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

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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See this: mathoverflow.net/questions/40145/… –  PEV Mar 21 '11 at 4:43
–  PEV Mar 21 '11 at 4:45

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