Let $r \ge 3$ be a fixed integer. I'm interested in primes p such that no integer in the interval $(-\sqrt{p}, \sqrt{p})$, except $1$ (and $-1$ if $r$ is even), is an r-th root of unity modulo p.


The naive heuristic that $r$-th roots of unity should be "randomly distributed" suggests that there should be infinitely many such primes, and indeed they should have density 1 among all primes. Can this be made rigorous?



Roots of unity near 1 in Z / p Z
OK, thinking a little more clearly about this... (hopefully)
Say $p\le N$ fails to have the property you want. Then $p | n^r-1$ for some 
$|n| < \sqrt{p} \le \sqrt{N}$. There are only $O(\sqrt{N})$ integers of the form $n^r-1$ with $n$ in this range, and each has only $O(\log{N})$ prime factors. 
So there are only $O(\sqrt{N} \log{N})$ exceptional primes $p \leq N$, which is tiny compared to $\pi(N)$.


