This one is probably a bit interesting:

Suppose $ K$ be the number of integers $ n $ such that $ \large \frac{2^n+1}{n^2}$ is also an integer.Find $K$.

PS:This was posted (by me) earlier in Mathematics group, I am cross-posting it here as this is more meta material.

Question

This one is probably a bit interesting:

Suppose $ K$  be the number of integers $ n $ such that $ \large \frac{2^n+1}{n^2}$ is also an integer.Find $K$.

PS:This was posted (by me) earlier in Mathematics group, I am cross-posting it here as this is more meta material.

binary3i · Answer

not sure but is K=2

anonymous · Answer

Yes, but how about proving it ? ;)

binary3i · Answer

the graph(2^n+1=y) and the graph (An^2=y) intersect at only two points.

anonymous · Answer

So you have plotted through out infinity ? and graphical proof isn't admissible in number theory.

binary3i · Answer

ok then let me try

binary3i · Answer

2^n=(An^2-1)
take log.
nlog2=log(An^2-1)
take its derivative. 
log2=1/(An^2-1) (A2n)
gives a qudratic which has two values of n.

binary3i · Answer

what do you think is it correct?

anonymous · Answer

can you please explain question properly i didnt get what is relation between  n ,k

anonymous · Answer

IMO-1990-problem 3