Numbers having 𝑚 small 𝑚th roots mod 𝑝

Abstract

Here are two typical results about the numbers mentioned in the title: If p is a prime such that p ≡ 1 ( mod 6 ) p \equiv 1 \pmod 6 and p > 67 p > 67 , then there are exactly six numbers mod p \bmod \;p , each of which has six sixth roots less than 2 3 p 2\sqrt {3p} in absolute value. If p is a prime such that p ≡ 1 ( mod 8 ) p \equiv 1 \pmod 8 , then there is at least one number mod p \bmod \;p which has eight eighth roots less than p 3 / 4 {p^{3/4}} in absolute value.