Simple arguments on consecutive power residues

Abstract

By some extremely simple arguments, we point out the following: (i) If n is the least positive k-th power non-residue modulo a positive integer m, then the greatest number of consecutive k-th power residues mod m is smaller than m/n. (ii) Let O_K be the ring of algebraic integers in a quadratic field $K=Q(\sqrt d)$ with d in {-1,-2,-3,-7,-11}. Then, for any irreducible $\pi\in O_K$ and positive integer k not relatively prime to $\pi\bar\pi-1$, there exists a k-th power non-residue $\omega\in O_K$ modulo $\pi$ such that $|\omega|<\sqrt{|\pi|}+0.65$.