This is a problem I recently found which I could not solve. This was there in a book prescribed for RMO and INMO. Here it is:

Determine with proof the set of all positive integers \(n\) such that \(2^n +1\) is divisible by \(n^2\).

Any hint, or solutions are welcome

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TopNewestThis is IMO 1990 P3. See here. – Ameya Daigavane · 1 week ago

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Since, \(2^{n} +1\) is divisible by \(n^2\).

Let,

\(2^{n}+1= kn^2\)

\(\implies 2^n = (\sqrt{k}n+1)(\sqrt{k}n-1)\)

Let, \(\sqrt{k}n-1 = a\)

Then,

\(2^n = (a)(a+2)\)

Now,

We have \(2\) powers of \(2\) with a difference of \(2\);that add up to an integer value;

Which, must be \(2\) and \(4\)

Hence, \(n=3\) and \(k=1\) is one answer.

Moreover, as @Kushagra Sahni has pointed out; \(n=1\) also works. I cant find a rigorous proof approach to \(n=1\) so, for now I will be satisfied by the fact that since 1 divides every natural number it divides \(2^n +1\).

P,S.: I am not very good at these kind of proofs so, there may be a trick or two I have missed and maybe even another value for \(n\) (though, I somehow feel quite sure about this answer.) – Yatin Khanna · 2 weeks ago

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– Ameya Daigavane · 1 week ago

How do you know \( a \) is an integer?Log in to reply

– Kushagra Sahni · 2 weeks ago

n=1 is also an answer. Put k=9. U believe these are the only 2 solutions.Log in to reply

\(k=3\) and \(n=1\) also works. – Yatin Khanna · 2 weeks ago

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