Is it possible to find an $\alpha\in\mathbb R$, $\alpha>0$ so that for all $n\in\mathbb N$, $n^\alpha\in\mathbb N$?
A stronger(restricted) problem: Can $2^\alpha$ and $3^\alpha$ be simultaneously integers?? Here $\alpha >0$.
For the second one, I can only reduce to $\cfrac{\log 3}{\log 2} = \cfrac{\log s}{\log q}$ for integer solution $(s,q)$, and then no idea..... For the first one, we have excluded all $\alpha\in\mathbb Q$(which is easy). I tried to use large enough $n^\alpha\in\mathbb N$ and to show $(n+1)^\alpha$ fails to be an integer but I failed.
I think there is more advanced number theoretic technique to be used.
Let $$\alpha = log _3 7$$ Then
$$ 3^\alpha ,9^\alpha, 27^\alpha, 81^\alpha,.......$$
are all integers.