Distribution of residues modulo $p$ using the Dirichlet's class number formula

TL;DR

This paper investigates the distribution of quadratic residues and non-residues modulo an odd prime p, focusing on multiples of 2, 3, or 4 within [1, p-1], using Dirichlet's class number formula.

Contribution

It applies Dirichlet's class number formula to analyze the distribution of specific quadratic residues and non-residues modulo p.

Findings

Distribution patterns of residues and non-residues identified

Quantitative results on residues that are multiples of 2, 3, or 4

Insights into the structure of quadratic residues in relation to class numbers

Abstract

Let $p$ be an odd prime number. In this article, we study the number of quadratic residues and non-residues modulo $p$ which are multiples of $2$ or $3$ or $4$ and lying in the interval $[1, p-1]$, by applying the Dirichlet's class number formula for the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$.