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2016 Quadratic diophantine equations with applications to quartic equations
Ajai Choudhry
Rocky Mountain J. Math. 46(3): 769-799 (2016). DOI: 10.1216/RMJ-2016-46-3-769

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 \[ Q_j(x_1,\ x_2,\ x_3,\ x_4)=0,\quad j=1,\ 2, \] 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.

Citation

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Ajai Choudhry. "Quadratic diophantine equations with applications to quartic equations." Rocky Mountain J. Math. 46 (3) 769 - 799, 2016. https://doi.org/10.1216/RMJ-2016-46-3-769

Information

Published: 2016
First available in Project Euclid: 7 September 2016

zbMATH: 06628755
MathSciNet: MR3544835
Digital Object Identifier: 10.1216/RMJ-2016-46-3-769

Subjects:
Primary: 11D09, 11D25

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

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Vol.46 • No. 3 • 2016
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