Is the sequence of fractional parts of $(3/2)^n$ dense in $[0,1]$?
This problem has a completeley elementary statement, yet noone has any idea how to prove it.
It is known that for almost every $t$, $t (3/2)^n \bmod 1$ is equidistributed in $[0,1]$ (this follows from a much more general result of Weyl), and also that for almost every $\beta>1$, the sequence $\beta^n\bmod 1$ is equidistributed in $[0,1]$ (This was proved by Koksma in the 30s).
It also follows from results of Pisot that $(3/2)^n \bmod 1$ has infinitely many accumulation points.
This question is somewhat related to the question (already mentioned in an answer) of whether famous irrational numbers are normal, but in a sense more elementary and frustrating because it is only about multiplication/division by $2$ and $3$!