A new proof of Legendre's theorem on the Diophantine equation $a{x}^2+b{y}^2+c{z}^2=0$

Abstract

One of Legendre's theorems on the Diophantine equation $a{x}^2+b{y}^2+c{z}^2=0$ 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.