This was Problem 3 (first day) of the 1990 IMO. A full solution can be found here.
How many rationals of the form $\large \frac{2^n+1}{n^2},$ $(n \in \mathbb{N} )$ are integers?
The possible values of $n$ that i am able to find is $n=1$ and $n=3$, so there are two solutions and this seems to be the answer to this problem.
But now we have to prove that no more of such $n$ exists, and thus the proof reduces to: Proving that $n^2$ does not divides $2^n+1$ for any $n \gt 3$.
Does anybody know how to prove this?
Best Answer
This was Problem 3 (first day) of the 1990 IMO. A full solution can be found here.