A new theorem on quadratic residues modulo primes

Abstract

Let $p>3$ be a prime, and let $(\frac{\cdot }{p})$ be the Legendre symbol. Let $b\in \mathbb{Z}$ and $\varepsilon \in \lbrace \pm 1\rbrace $. We mainly prove that \[ \left|\left\lbrace N_p(a,b):\ 1\lbrace ax^2+b\rbrace _p$, and $\lbrace m\rbrace _p$ with $m\in \mathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$.