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.


11D09 Quadratic and bilinear Diophantine equations
11D25 Cubic and quartic Diophantine equations
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[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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