Consecutive Quadratic Residues And Quadratic Nonresidue Modulo $p$

TL;DR

This paper provides a new proof for the existence of specific patterns of consecutive quadratic residues and nonresidues modulo a large prime, and applies this to bound the least quadratic nonresidue.

Contribution

It introduces a novel proof technique for the existence of patterns of quadratic residues and nonresidues, and derives a new bound on the least quadratic nonresidue.

Findings

Existence of any pattern of $k$ consecutive quadratic residues and nonresidues proven.

Bound on the least quadratic nonresidue: $n_p \\ll (\\log p)(\\log \\log p)$.

New proof method simplifies understanding of quadratic residue patterns.

Abstract

Let $p$ be a large prime, and let $k\ll \log p$. A new proof of the existence of any pattern of $k$ consecutive quadratic residues and quadratic nonresidues is introduced in this note. Further, an application to the least quadratic nonresidues $n_p$ modulo $p$ shows that $n_p\ll (\log p)(\log \log p)$.