$\alpha \in \mathbb{R}$ and $2^\alpha$, $3^\alpha \in \mathbb{N}$, implies $\alpha \in \mathbb{N}$?

Question

Let $\alpha \in \mathbb{R}$. Suppose $2^\alpha$, $3^\alpha \in \mathbb{N}$. Does it implies that $\alpha \in \mathbb{N}$?

Answer 1

This is an open problem, as documented on MathOverflow: https://mathoverflow.net/q/17560

Answer 2

Actually, this is an open problem in mathematics; unfortunately I can't localize a source. (I've read it once on Math Overflow, but I have a hard time tracking it down).