Using A. Well's estimates the authors have given bounds for the largest prime P 0 such that all primes P_0 $"> p > P 0 have sequences of quadratic residues of length m . For m ≤ 8 the smallest prime having m consecutive quadratic residues is ≡ 3 ( mod 4 ) and P 0 ≡ 1 ( mod 4 ) . The reason for this phenomenon is investigated in this paper and the theory developed is used to successfully uncover analogous phenomena for r t h power residues, r ≥ 2 , r even.