## 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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