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Find all integers n having the property that $\frac{2^n+1}{n^2}$ is an integer.

Just by playing around with the problem and trying possibilities, I was able to find 1 and 3 as solutions, but I'm not sure how to prove that these are the only cases that work. Thank you so much for the help!

ilovemath567
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  • Use Fermat's little theorem to show that if $n > 1$ then the only prime factor of $n$ is $3$ (if $p \mid n$ and $p$ is prime then $p = 3$). So $n = 3^k$ for some $k \geq 1$. Figure out a pattern for the highest power of $3$ dividing $2^{3^k}+1$. – KCd Jun 19 '21 at 17:45

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