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?
2026-04-13 12:34:02.1776083642
If $n^c\in\mathbb N$ for every $n\in\mathbb N$, then $c$ is a non-negative integer?
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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.)