A new theorem on quadratic residues modulo primes

TL;DR

This paper establishes a new theorem characterizing the distribution of quadratic residues modulo primes, specifically counting certain inequalities involving residues and Legendre symbols, enhancing understanding of quadratic residue behavior.

Contribution

The paper introduces a novel theorem that precisely counts the number of specific inequalities involving quadratic residues modulo primes, expanding theoretical knowledge in number theory.

Findings

Exact count of certain quadratic residue inequalities for primes greater than 3

Relation between Legendre symbols and the distribution of quadratic residues

Provides a new formula linking residues and quadratic residue counts

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 $p$.