If $3^x$ and $5^x$ are both integers, is $x$ an integer?
Solution 1:
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.