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?
Asked
Active
Viewed 787 times
11
-
If $y$ is a perfect square then let $x=0.5$. – Jul 07 '14 at 11:46
-
@M.Knight Um, huh? – anon Jul 07 '14 at 11:54
-
1See http://mathoverflow.net/questions/17560/if-2x-and-3x-are-integers-must-x-be-as-well and also http://math.stackexchange.com/questions/33109/contest-problems-with-connections-to-deeper-mathematics – Gerry Myerson Jul 07 '14 at 12:42
2 Answers
7
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.
Michael Albanese
- 99,526
1
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.
Mohammad Khosravi
- 2,333
-
4That's a special case of the Six Exponentials Theorem, and the proof is long & hard. But the question we have here is not that hard. It was on the Putnam exam in 1971, and solutions to Putnam questions are available in various places. – Gerry Myerson Jul 07 '14 at 12:40