A century problem related to the Legendre symbol modulo p

TL;DR

This paper presents an elementary, constructive proof of a longstanding number theory problem related to the Legendre symbol modulo p, resolving an issue that has persisted since 1927.

Contribution

It introduces a simple, elementary method to completely solve an old problem concerning the Legendre symbol, fulfilling a long-standing mathematical desire.

Findings

The problem related to the Legendre symbol modulo p has been fully solved.

The proof is elementary and constructive, accessible to a broad mathematical audience.

The solution addresses a question that has remained open since 1927.

Abstract

The main purpose of this paper is using a very simple constructive method to study an old number theory problem related to the Legendre symbol modulo p, and completely solved it. The proving method of the result is purely elementary and has been desired in the literature at least since 1927.