Upper Bound of the Least Quadratic Nonresidues

Abstract

Let $p\ge 3$ be a large prime and let $n(p)\ge 2$ denotes the least quadratic nonresidue modulo $p$. This note sharpens the standard upper bound of the least quadratic nonresidue from the unconditional upper bound $n(p)\ll p^{1/4\sqrt{e}+\epsilon }$ to the conjectured upper bound $n(p)\ll (\log p{)}^{1+\epsilon }$, where $\epsilon >0$ is a small number, unconditionally. This improvement breaks the exponential upper bound barrier.

Twenty One Pages. Refined Analysis of the Error Term. Keywords: Least quadratic nonresidue; Square root mod $p$; Deterministic algorithm