Least Consecutive Pair of Quadratic Nonresidues

Abstract

Let $p>1$ be a large prime number and let $x=O((\log p{)}^2(\log \log p{)}^3$ be a real number. It is proved that the least consecutive pair of quadratic nonresidues $u\ne \pm 1,v^2$ and $u+1$ satisfies the upper bound $u\ll x$ in the prime field ${\mathbb{F}}_p$.

Eighteen Pages. Keywords: Quadratic nonresidue mod $p$; Least prime quadratic nonresidue; Complexity theory; Finite field