A new proof of Legendre's theorem on the Diophantine equation $ax^2+by^2+cz^2=0$

TL;DR

This paper presents a new proof of Legendre's theorem on the existence of rational solutions to the quadratic Diophantine equation, utilizing Hasse invariants and Jacobi symbols from quadratic form theory.

Contribution

It introduces a novel proof of Legendre's theorem based on advanced concepts like Hasse invariants and Jacobi symbols, offering a different perspective from traditional proofs.

Findings

Provides necessary and sufficient conditions for rational solutions

Uses Hasse invariant and Jacobi symbol in proof

Enhances understanding of quadratic forms and Diophantine equations

Abstract

One of Legendre's theorems on the Diophantine equation $ax^2+by^2+cz^2=0$ provides necessary and sufficient conditions on the existence of nonzero rational solutions of this equation, which helps determine the existence of rational points on a conic. In this paper, we provide a new proof of this famous theorem using Hasse invariant and Jacobi symbol from the theory of quadratic forms.