TITLE: If $3^x$ and $5^x$ are both integers, is $x$ an integer?
QUESTION [11 upvotes]: Does the following statement hold? $$x\in \mathbb{R}^+ \text{and} \  3^x, 5^x \in \mathbb{Z} \implies x \in \mathbb{Z}$$
In words:
If $x>0$ is a real number, and $3^x$ and $5^x$ are both integers, does
that mean that $x$ is an integer?

This is a slightly modified form of another problem I was working on. A friend of mine claims this is a very hard problem. What do you think?
If one claims it is an open problem, can one show that this problem is equivalent to some other known open problem?

REPLY [4 votes]: This is probably an open question, as the related problem with $2^x$ and $3^x$ is open. Today, it is known that if $2^x$, $3^x$ and $5^x$ are integers, then $x$ is integer as well--it follows from the six exponentials theorem in transcendental number theory. 
I cannot confirm whether the $3^x$, $5^x$ case follows from the four exponentials conjecture, as I do not know the field; so I would be glad if someone could.