Is $2^\sqrt{2}$ irrational? Is it transcendental?
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5See http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem – Julian Rosen Jul 22 '12 at 08:44
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2See Gelfond–Schneider constant at Wikipedia. Somewhat related are also this question Real Numbers to Irrational Powers and this MO question About the proof of the proposition “there exists irrational numbers a, b such that a^b is rational” – Martin Sleziak Jul 22 '12 at 08:48
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@Pink Elephants : Perhaps this should be an answer. – Patrick Da Silva Jul 22 '12 at 08:50
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@Pink Elephants : Thanks, very interesting, but is there an easy way to prove irrationality without such a theorem? – Marco Disce Jul 22 '12 at 08:52
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6If it were so easy, it wouldn't have been on the list of Hilbert's problems, would it? – J. M. ain't a mathematician Jul 22 '12 at 08:58
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4@J.M.: As far as I understand it the Hilber's problem is to decide wheter it is trascendental, not to decide whether it is irrational. – Marco Disce Jul 22 '12 at 11:57
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According to Gel'fond's theorem, if $\alpha$ and $\beta$ are algebraic numbers (which $2$ and $\sqrt 2$ are) and $\beta$ is irrational, then $\alpha^\beta$ is transcendental, except in the trivial cases when $\alpha$ is 0 or 1.
Wikipedia's article about the constant $2^{\sqrt 2}$ says that it was first proved to be transcendental in 1930, by Kuzmin.
MJD
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