Determine all integers (bigger than 1) such that $\frac{2^n+1}{n^2}$ is an integer

Asked Jan 02 '18 at 15:13

Active Jan 02 '18 at 15:22

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Determine all integers $k>1$ such that $$ \dfrac{2^k+1}{k^2}\in\mathbb{N}$$

I have tried to solve for even numbers $(k=2n)$ and for odd numbers $(k=2n+1)$, but i couldn't find any answer to this. I think i have to begin by saying that $$2^k+1\equiv 0 \pmod{k^2} $$ but i don't know how to continue.

asked Jan 02 '18 at 15:13

Varazda

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Just for reference, this is problem 3 from IMO 1990. – projectilemotion Jan 02 '18 at 15:24
Hint: Here you can find an answer https://artofproblemsolving.com/community/c6h60740p366466 – nonuser Jan 02 '18 at 15:22

Determine all integers (bigger than 1) such that $\frac{2^n+1}{n^2}$ is an integer

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