Sequences in power residue classes (Q1076054)

Summary: Using A. Weil's estimates the authors have given bounds for the largest prime \(P_0\) such that all primes \(p>P_0\) have sequences of quadratic residues of length \(m\). For \(m\le 8\) the smallest prime having \(m\) consecutive quadratic residues is \(\equiv 3\pmod 4\) and \(P_0\equiv 1\pmod 4\). The reason for this phenomenon is investigated in this paper and the theory developed is used to successfully uncover analogous phenomena for \(r\)-th power residues, \(r\ge 2\), \(r\) even.