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arXiv · 1711.04668

Diophantine triples in linear recurrence sequences of Pisot type

Clemens Fuchs·Christoph Hutle·Florian Luca
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Abstract

The study of Diophantine triples taking values in linear recurrence sequences is a variant of a problem going back to Diophantus of Alexandria which has been studied quite a lot in the past. The main questions are, as usual, about existence or finiteness of Diophantine triples in such sequences. Whilst the case of binary recurrence sequences is almost completely solved, not much was known about recurrence sequences of larger order, except for very specialized generalizations of the Fibonacci sequence. Now, we will prove that any linear recurrence sequence with the Pisot property contains only finitely many Diophantine triples, whenever the order is large and a few more not very restrictive conditions are met.

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Clemens Fuchs, Christoph Hutle, Florian Luca. 2017-11-10. Diophantine triples in linear recurrence sequences of Pisot type. https://doi.org/10.1007/s40993-018-0121-2

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