Are there any size constraints on the reduced solutions $x$, $(0<x<q)$, to $x^p\equiv 1\mod q$, for $p$ and $q$ specific primes? (Considering the primitive $p$-th roots of unity modulo $q$).
I can see that $x>q^{1/p}$ as $x^p>q$. Also for $p=2$, the solutions are simply $\pm 1$. Besides these, can anything be said so far about the lower or upper bounds?
