Reciprocity laws for Legendre symbols of the type $(a+b\sqrt{m}/p)$ - Long Version

TL;DR

This paper presents a comprehensive general theorem on reciprocity laws involving Legendre and quartic residue symbols, unifying many classical results and suggesting all such laws may derive from this framework.

Contribution

It introduces a broad, unified reciprocity law encompassing various older results, likely covering all existing laws of this type.

Findings

Unifies multiple classical reciprocity laws under a single general theorem

Demonstrates how to derive known laws like Scholz's, Lehmer's, and Burde's from the main result

Provides a corrected and improved version of previous work

Abstract

We announce a very general statement involving the rational quartic residue symbol $(m/p)_4$ and, more generally, Legendre symbols of the type ${a+b\sqrt{m}/p$. We show how our main theorem can be used to produce many older results such as Scholz's, Lehmer's or Burde's reciprocity laws and many others. It is very likely that all existing reciprocity laws of this type can be obtained from our result. This is a corrected and improved version of version 2.