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

Jingbo Liu; Bruce McOsker

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arXiv · 1912.11750

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

Jingbo Liu·Bruce McOsker

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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.

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Jingbo Liu, Bruce McOsker. 2019-12-26. A new proof of Legendre's theorem on the Diophantine equation $ax^2+by^2+cz^2=0$. https://arxiv.org/abs/1912.11750

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