Quadratic diophantine equations with applications to quartic equations

TL;DR

This paper develops methods to solve quadratic diophantine equations in four variables, relates them to elliptic curves and quartic equations, and provides formulas for generating rational solutions from known solutions.

Contribution

It introduces a new approach to express solutions of quadratic diophantine equations using bilinear forms and connects these solutions to quartic equations for rational solution generation.

Findings

Solutions can be expressed via bilinear forms in four parameters.

A necessary condition for the solvability of two quadratic equations is established.

New formulas are derived for rational solutions of quartic equations from known solutions.

Abstract

In this paper we first show that, under certain conditions, the solution of a single quadratic diophantine equation in four variables $Q(x_1,\,x_2,\,x_3,\,x_4)=0$ can be expressed in terms of bilinear forms in four parameters. We use this result to establish a necessary, though not sufficient, condition for the solvability of the simultaneous quadratic diophantine equations $Q_j(x_1,\,x_2,\,x_3,\,x_4)=0,\;j=1,\,2,$ and give a method of obtaining their complete solution. In general, when these two equations have a rational solution, they represent an elliptic curve but we show that there are several cases in which their complete solution may be expressed by a finite number of parametric solutions and/ or a finite number of primitive integer solutions. Finally we relate the solutions of the quartic equation $y^2=t^4+a_1t^3+a_2t^2+a_3t+a_4$ to the solutions of a pair of quadratic diophantine equations, and thereby obtain new formulae for deriving rational solutions of the aforementioned quartic equation starting from one or two known solutions.