Find all the natural numbers such that, the regular $n , n+1 , n+2 $ gons are constructible.
Well this problem can be restated in the following way. Since the construction of the regular n-gon is equivalent to the construction of the primitive root $e^{\frac{2\pi i}{n}}$ , and we know that the degree of this extension over $\Bbb Q$ is $\phi(n)$ , and also knowing that a number is constructible iff it's degree over $\Bbb Q$ is $2^k$ we want all the natural numbers $n$ such that $ \phi(n) , \phi(n+1) , \phi(n+2)$ are powers of 2. I have no idea how can I do this :S, I only know properties of $\phi(n)$ involving products of coprimes. But here ...