Open Access
Translator Disclaimer
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


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.


Download Citation

Ajai Choudhry. "Quadratic diophantine equations with applications to quartic equations." Rocky Mountain J. Math. 46 (3) 769 - 799, 2016.


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

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

Primary: 11D09, 11D25

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium


Vol.46 • No. 3 • 2016
Back to Top