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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
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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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