Upper Bound of the Least Quadratic Nonresidues

TL;DR

This paper improves the known upper bounds on the smallest quadratic nonresidue modulo a large prime, moving from a polynomial bound to a conjectured logarithmic bound, thus breaking a longstanding exponential barrier.

Contribution

It unconditionally establishes a sharper upper bound on the least quadratic nonresidue, approaching the conjectured logarithmic growth.

Findings

Unconditional proof of a near-logarithmic upper bound

Breaks the exponential upper bound barrier

Advances understanding of quadratic nonresidues

Abstract

Let $p\geq3$ be a large prime and let $n(p)\geq2$ denotes the least quadratic nonresidue modulo $p$. This note sharpens the standard upper bound of the least quadratic nonresidue from the unconditional upper bound $n(p)\ll p^{1/4\sqrt{e}+\varepsilon}$ to the conjectured upper bound $n(p)\ll (\log p)^{1+\varepsilon}$, where $\varepsilon>0$ is a small number, unconditionally. This improvement breaks the exponential upper bound barrier.