If any integer to the power of $x$ is integer, must $x$ be integer?

Question

My apologies if this has been asked already, I've searched but couldn't find it...

Let $x$ such that for every $y \in N$, $y^x$ is an integer. Does that necessarily mean that $x$ is an integer?

Answer 1

This question was question A6 in the 1971 Putnam competition. A solution using finite differences and the Mean Value Theorem can be found here. You may also be interested in this MO question which discusses a vast generalisation which shows that if $2^x$, $3^x$, and $5^x$ are integral, so is $x$. As established in the answers to the MO question, it is still an open problem as to whether knowing $2^x$ and $3^x$ are integral is enough to deduce that $x$ is integral.

Answer 2

Yes! There is a famous problem which says that if $2^\alpha$, $3^\alpha$ and $5^\alpha$ are integer numbers, then $\alpha$ is a natural number.

Unfortunately I don't remember proof and the reference I know is not in English.