Fix an $n$th primitive root of unity $\xi$. I need to understand if we can characterize in an easy way all the solutions $k \in \mathbb{Z}$ of the equation $\left|1-\left(-\frac{\xi^k - 1}{\xi-1}\right)^n\right| = 1$ (note the complex modulus). Actually, I think that the only solutions are the trivial $k=an$, with $a \in \mathbb{Z}$. I know that similar problems can be really difficults, however I am not sure if this is the case.


Thank you very much for any suggestion. :-)



Diophantine equation with primitive nth root of unity
The expression $(- (\xi^k - 1)/(\xi - 1))^n$ is real: it equals the $n$'th power of $\sin(k \pi/n)/\sin(\pi/n)$ times $(-1)^{n+k-1}$, if I calculated correctly. So the thing you're taking absolute value of must be $\pm 1$. If it's $1$, you get the trivial solution. If it's $-1$, that tells you an $n$'th root of $\pm 2$ must be in the cyclotomic field $\mathbb{Q}(\xi\_n).$ But the degree of the field obtained by adjoining a root of the polynomial $x^n \pm 2$ is $n$ (since the poly is irreducible by Eisenstein), and so this leads to $n \leq \phi(n)$, which is a contradiction for any $n > 1$.


