## Rocky Mountain Journal of Mathematics

### Quadratic diophantine equations with applications to quartic equations

Ajai Choudhry

#### 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

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

#### 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

#### 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.