When is $\frac{2^n+1}{n^2}$ an integer?

Question

Can anyone see how to solve this number puzzle?

Find all integers $n>1$ such that

$$\frac{2^n+1}{n^2}$$ is an integer.

Answer 1

Hint: Since this problem comes from number theory, then I suggest you to look at equivalent congruence problem

$$ 2^n+1=k n^2 \implies 2^n \equiv -1\pmod{n^2} .$$

Check this, page 33.

Answer 2

This was asked here: How many rationals of the form $\large \frac{2^n+1}{n^2}$ are integers?

Where a link to the answer was given: http://www.cs.cornell.edu/~asdas/imo/imo/isoln/isoln903.html