2014/09/19 by Choudhry, Ajai · 1 citation
#11D09 #11D25 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1409.5527
In this paper we first show that, under certain conditions, the solution of a single quadratic diophantine equation in four variables Q(x1, x2, x3, x4)=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 Qj(x1, x2, x3, x4)=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 y2=t4+a1t3+a2t2+a3t+a4 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.