Vinogradov's Conjecture and Beyond

Abstract

In this paper, if prime $p\equiv 3\pmod 4$ is sufficiently large then we prove an upper bound on the number of occurences of any arbitrary pattern of quadratic residues and nonresidues of length $k$ as $k$ tends to $\lceil \log_2 p\rceil$. As an immediate consequence, it proves that, there exist a constant $c$ such that, the least nonresidue for such primes is at most $c\lceil \log_2 p\rceil$.