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Complete solution of a family of simultaneous Pellian equations. (English) Zbl 1024.11014

A set of positive integers \(\{a_1,a_2,\ldots ,a_m\}\) is called a Diophantine \(m\)-tuple with the property \(D(n)\) if \(a_ia_j+n\) is a perfect square for all \(1\leq i < j \leq m\). The author obtains a new result concerning Diophantine quadruples. He proves that the pair \(\{1,2\}\) cannot be extended to a Diophantine quadruple with the property \(D(-1)\).
This result follows directly from the main theorem of the paper, which asserts that for all positive integers \(k\) all solutions of the system of the simultaneous Pellian equations \[ z^2-c_kx^2=c_k-1, \quad 2z^2-c_ky^2=c_k-2 \] (where \(c_k=P_{2k}^2+1\) and \(P_{2k}\) denotes the \(k\)th Pell number) are given by \((x,y,z)=(0, \pm 1, \pm P_{2k})\).

MSC:

11D09 Quadratic and bilinear Diophantine equations
11D25 Cubic and quartic Diophantine equations
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References:

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