In this paper we first show that, under certain conditions, the solution of asingle quadratic diophantine equation in four variables$Q(x_1,,x_2,,x_3,,x_4)=0$ can be expressed in terms of bilinear forms infour parameters. We use this result to establish a necessary, though notsufficient, condition for the solvability of the simultaneous quadraticdiophantine equations $Q_j(x_1,,x_2,,x_3,,x_4)=0,;j=1,,2,$ and give amethod of obtaining their complete solution. In general, when these twoequations have a rational solution, they represent an elliptic curve but weshow that there are several cases in which their complete solution may beexpressed by a finite number of parametric solutions and/ or a finite number ofprimitive integer solutions. Finally we relate the solutions of the quarticequation $y^2=t^4+a_1t^3+a_2t^2+a_3t+a_4$ to the solutions of a pair ofquadratic diophantine equations, and thereby obtain new formulae for derivingrational solutions of the aforementioned quartic equation starting from one ortwo known solutions.
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