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Supposing that a real number $c$ is given, is the following true?

"If $n^c$ is a natural number for every natural number $n$, then $c$ is a non-negative integer."

Though this seems true, I can't prove that. Can anyone help?

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Hint: What does it tell you if this holds for some specified natural number, for example a prime? – Tobias Kildetoft Nov 17 '13 at 9:51
@901301: In my opinion, it does not seem to help. – mathlove Nov 17 '13 at 10:24
@TobiasKildetoft Any proof should use that $n^c$ is an integer, for several integers n, since otherwise the result fails. – Did Nov 17 '13 at 11:27
This is an excellent question! It feels like it should have a simple proof, but I can't see any. – TonyK Nov 17 '13 at 12:54
Perhaps even the following is true: If $2^c$ and $3^c$ are both integers, then $c$ is an integer. – TonyK Nov 17 '13 at 19:39
up vote 13 down vote accepted

A variant of this question was asked on Mathoverflow here by Alon Amit. As Gerry Myerson answers, in particular, it's apparently sufficient to know that only $2^c$ and $3^c$ and $5^c$ are all integers. It's apparently unknown whether it's sufficient to know that $2^c$ and $3^c$ are integers.

He also mentions that the original question (using $n$ instead of $2,3,5$) was actually a 1971 Putnam problem and Chris Phan provides a link to the solution. (It's problem A6).

(Community wiki because I've done nothing.)

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Thank you very much for great information! – mathlove Nov 19 '13 at 17:04
@mathlove: You're welcome! Incidentally, since this is CW, I don't think you can award me the bounty (or maybe you can??) In any case, I don't need it. So if there is something you or I can do to minimize how much rep this costs you, let me know! (For example, I have no qualms with deleting this answer and letting you write your own answer.) – Jason DeVito Nov 20 '13 at 3:23
Well, ok, so if I'm going to do nothing, then nothing happens, right? Is it OK to you or not? – mathlove Nov 20 '13 at 15:01
@mathlove: Well, I was just trying to look out for your bounty. I am ok with anything you decide to do (or not do). – Jason DeVito Nov 20 '13 at 16:11
Thanks. What I said depends on my understanding that a bounty never comes back. – mathlove Nov 20 '13 at 16:16

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