TITLE: If $2^x $and $3^x$ are integers, must $x$ be as well?
QUESTION [137 upvotes]: I'm fascinated by this open problem (if it is indeed still that) and every few years I try to check up on its status. Some background: Let $x$ be a positive real number.

If $n^x$ is an integer for every $n \in \mathbb{N}$ then $x$ must be an integer. This is a fun little puzzle.
If $2^x$, $3^x$ and $5^x$ are integers then $x$ must be an integer. This requires fairly sophisticated tools, and can be derived from the results in e.g. Lang, Algebraic values of meromorphic functions. II., Topology 5, 1966.
Finally, if all you know is that $2^x$ and $3^x$ are integers, then as far as I know it is not known if $x$ is forced to be an integer (unbelievable, isn't it?). Although of course one can never be certain, I am quite sure this was still the case as recently as 2003. 

So the question is, is that still an open problem, and is there any sort of relevant progress that may provide some hope?

REPLY [50 votes]: This question came up recently on the NMBRTHRY mailing list and I can't resist paraphrasing a comment I made there. So, perhaps surprisingly, this question has links to automorphic forms! For if $x$ is a complex number and $||.||^x$ is the associated Grossencharacter of the ideles of $\mathbf{Q}$, that is, the map $\mathbf{A}_\mathbf{Q}^\times/\mathbf{Q}^\times\to\mathbf{C}^\times$ sending an idele to the $x$'th power of its norm, then the assumption that $p^x$ is an integer for all primes $p$ (which is clearly equivalent to the assumption that $n^x$ is an integer for all $n$) implies that the grossencharacter is arithmetic. Now a standard conjecture in the theory of automorphic representations is that an automorphic representation is arithmetic iff it's algebraic, and this conjecture is a theorem for tori, so the theorem in this case says that $||.||^x$ is algebraic which is precisely the statement that $x$ is an integer!
So for tori over general number fields it's a theorem of Waldschmidt that arithmetic implies algebraic for automorphic forms. So in practice we get a vast generalisation of the first question above, where integers can be replaced by algebraic integers and where we can add finite order characters and so on.
As an example, one sees that if $x$ is complex and if there's a number field $E$ in the complexes such that $n^x$ is an integer in $E$, for all $n$, then again the grossencharacter is arithmetic, so algebraic, and hence $x$ must be an integer. I don't know if there's any low-level proof of this (but it follows from standard transcendence theorems). As other examples $n$ can be replaced by the algebraic integers in a number field and so on.