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arXiv · 1301.5029

Markoff-Rosenberger triples in arithmetic progression

Enrique González-Jiménez·José M. Tornero
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Abstract

We study the solutions of the Rosenberg--Markoff equation ax^2+by^2+cz^2 = dxyz (a generalization of the well--known Markoff equation). We specifically focus on looking for solutions in arithmetic progression that lie in the ring of integers of a number field. With the help of previous work by Alvanos and Poulakis, we give a complete decision algorithm, which allows us to prove finiteness results concerning these particular solutions. Finally, some extensive computations are presented regarding two particular cases: the generalized Markoff equation x^2+y^2+z^2 = dxyz over quadratic fields and the classic Markoff equation x^2+y^2+z^2 = 3xyz over an arbitrary number field.

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Enrique González-Jiménez, José M. Tornero. 2013-01-21. Markoff-Rosenberger triples in arithmetic progression. https://doi.org/10.1016/j.jsc.2012.11.003

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