TITLE: Is there always a primitive m-th root of unity with imaginary part bigger than 1/2
QUESTION [6 upvotes]: Let $m$ be a positive integer.
I need the existence of a primitive $m$-th root of unity $\zeta_m$ such that its imaginary part is strictly greater than $1/2$.
We can write $\zeta_m = \exp(2\pi i a/m)$ for some $a$ coprime to $m$.
The condition above boils down to $\sin(2 \pi a /m ) > 1/2$. This just means that $$ \frac{m}{12} <  a < \frac{5m}{12}.$$ So I'm looking for the existence of an integer $a$ coprime to $m$ such that $$ \frac{m}{12} < a < \frac{5m}{12}. $$
Is this always possible?
Probably I need that $m> 12$. For $m=12$, there is no such $a$. It's ok if it doesn't work for a finite number of $m$.

REPLY [4 votes]: By Bertrand's postulate, there is always a prime number between $\lceil m/12\rceil$ and $2\lceil m/12\rceil$ and one between $2\lceil m/12\rceil$ and $4\lceil m/12\rceil$. For sufficiently large $m$ we have $4\lceil m/12\rceil\lt5m/12$. For sufficiently large $m$, these primes cannot both divide $m$, so at least one of them is coprime to $m$.