I wanna show that if $n^x$ are rational for all positive integers n, then we may conclude $x$ is an integer. First of all, WLOG, we may assume $x$ is a real number between 0,1. I think it suffices to show that for R={$log_p q$, where p,q are coprime positive integers bigger than 1}, then the elements of R+R contained in R must be two pairs with same p. But I stuck at proving my conjectural statement...
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1relevant http://math.stackexchange.com/questions/378130/what-is-the-set-x-in-bbb-r-mid-forall-q-in-bbb-q-qx-in-bbb-q/378143#comment810074_378143 – clark Nov 23 '13 at 03:52
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OK, I think I know how to prove this, by using Baker – lee Nov 23 '13 at 04:17
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I mean, by using Baker's theorem. – lee Nov 23 '13 at 04:18
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http://en.wikipedia.org/wiki/Baker%27s_theorem by Baker's thm I mean this. – lee Nov 23 '13 at 04:22
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Wait, I'm confused by myself... Ignore me... – lee Nov 23 '13 at 04:29
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@lee If you find your comment incomplete, click "edit" to modify it, instead of adding one more. – Shuchang Nov 23 '13 at 05:09
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Baker's Theorem not the way to go. If you want to use a sledgehammer, the Six Exponentials Theorem will work. But this was a problem on the 1971 Putnam Exam, and can be solved without the big tools. – Gerry Myerson Nov 23 '13 at 05:52
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(Almost) duplicate of http://math.stackexchange.com/q/570218/1508. – TonyK Nov 23 '13 at 08:20