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 $\epsilon \in \{\pm 1\}$. We mainly prove that $$\left|\left\{N_p(a,b):1<a<p\text{and}\left(\frac{a}{p}\right)=\epsilon \right\}\right|=\frac{3-(\frac{-1}{p})}{2},$$ where $N_p(a,b)$ is the number of positive integers $x<p/2$ with $\{x^2+b{\}}_p>\{a{x}^2+b{\}}_p$, and $\{m{\}}_p$ with $m\in \mathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$.

6 pages, final version