TITLE: If any integer to the power of $x$ is integer, must $x$ be integer?
QUESTION [11 upvotes]: 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?

REPLY [7 votes]: 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.