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

Solution 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.

Solution 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.