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