Simple arguments on consecutive power residues

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