Let $\displaystyle G_n(1)=\{q\in Z_n|(q,n)=1\}$ $\displaystyle S\subset G_n (1) $ (strict subset). Prove that:
$\displaystyle
\sum_{q \in S}\omega^q_n\notin Z
$
where $\displaystyle \omega_n=e^{i\frac{2\pi}n}.$
Let $\displaystyle G_n(1)=\{q\in Z_n|(q,n)=1\}$ $\displaystyle S\subset G_n (1) $ (strict subset). Prove that:
$\displaystyle
\sum_{q \in S}\omega^q_n\notin Z
$
where $\displaystyle \omega_n=e^{i\frac{2\pi}n}.$