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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 $m$.

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    @joriki: I was giving a start.2011-10-13

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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 $m$.

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    @Rayleigh: I did$a$computer search; the only values of $m$ less than $72$ for which the desired $a$ doesn't exist are $1$ through $4$ and $12$.2011-10-13