## Rocky Mountain Journal of Mathematics

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

### Quadratic diophantine equations with applications to quartic equations

#### 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