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I remembered that a Russian mathematician proved this, but now I cannot find it again.

Could you please recommend a book on these kinds of problems?

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  • $\begingroup$ I like Niven, Irrational Numbers $\endgroup$ – Will Jagy May 29 '17 at 2:59
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    $\begingroup$ well, I don't see anything that would imply your idea, i think you have remembered it incorrectly. $\endgroup$ – Will Jagy May 29 '17 at 3:09
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    $\begingroup$ @WillJagy It's an open question if $2^e$ is irrational... $\endgroup$ – Dair May 29 '17 at 3:18
  • $\begingroup$ A Russian mathematician proved that $2^\sqrt2$ is transcendental, but that's another story. $\endgroup$ – Ivan Neretin May 29 '17 at 7:52
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It is unknown if $2^e$ is irrational. See here (under Open questions).

It is not known whether πe, π/e, $2^e$, ee, $e^{e^e}$, πe, $π\sqrt{2}$, ln π, Catalan's constant, or the Euler–Mascheroni gamma constant γ are irrational.

(Remark: All real numbers that are transcendental are irrational.)

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