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**Quadratic Diophantine equations with applications to quartic equations.**
*(English)*
Zbl 1419.11053

Summary: 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.

### MSC:

11D09 | Quadratic and bilinear Diophantine equations |

11D25 | Cubic and quartic Diophantine equations |

### Keywords:

bilinear solutions of quadratic Diophantine equations; quartic Diophantine equation; quartic model of elliptic curve; quartic function made a perfect square
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\textit{A. Choudhry}, Rocky Mt. J. Math. 46, No. 3, 769--799 (2016; Zbl 1419.11053)

### References:

[1] | L.E. Dickson, History of theory of numbers , Volume 2, Chelsea Publishing Company, New York, 1992. |

[2] | L.J. Mordell, Diophantine equations , Academic Press, London, 1969. |

[3] | W. Sierpinski, Elementary theory of numbers , PWN-Polish Scientific Publishers, Warszawa, 1987. |

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