Rocky Mountain Journal of Mathematics

Quadratic diophantine equations with applications to quartic equations

Ajai Choudhry

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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.

Article information

Source
Rocky Mountain J. Math., Volume 46, Number 3 (2016), 769-799.

Dates
First available in Project Euclid: 7 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1473275762

Digital Object Identifier
doi:10.1216/RMJ-2016-46-3-769

Mathematical Reviews number (MathSciNet)
MR3544835

Zentralblatt MATH identifier
06628755

Subjects
Primary: 11D09: Quadratic and bilinear equations 11D25: Cubic and quartic equations

Keywords
Bilinear solutions of quadratic diophantine equations quartic diophantine equation quartic model of elliptic curve quartic function made a perfect square

Citation

Choudhry, Ajai. Quadratic diophantine equations with applications to quartic equations. Rocky Mountain J. Math. 46 (2016), no. 3, 769--799. doi:10.1216/RMJ-2016-46-3-769. https://projecteuclid.org/euclid.rmjm/1473275762


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References

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  • L.J. Mordell, Diophantine equations, Academic Press, London, 1969.
  • W. Sierpinski, Elementary theory of numbers, PWN-Polish Scientific Publishers, Warszawa, 1987.