Question

Determine all integers n > 1 such that
\[\frac {2^n + 1}{n^2}\]
is an integer.

Answer 1

For the fraction to be an integer, 2^n+1 should be a multiple of n^2, so:

\[2^n+1 = kn^2, k\]

Answer 2

\[k \in \mathbb{R} \]

Answer 3

ie
\[n^2|2^n+1 \implies n\in\mathbb{N}: 2^n\equiv-1\pmod{n^2} \]

Answer 4

I think you can write:

\[2^n + 1 = kn^2\]
Differentiate for n:
\[2^n log2 = 2kn\]
Differentiate again:
\[2^n (log2)^2 = 2k\]

It could be possible to find n now. But you should wait from another response, because I am not sure about the answer.