Abstract

Using A. Well's estimates the authors have given bounds for the largest prime P0 such that all primes p>P0 have sequences of quadratic residues of length m. For m≤8 the smallest prime having m consecutive quadratic residues is ≡3(mod4) and P0≡1(mod4). The reason for this phenomenon is investigated in this paper and the theory developed is used to successfully uncover analogous phenomena for rth power residues, r≥2, r even.