Diophantine triples in linear recurrence sequences of Pisot type
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.
Keywords
Cite
@article{arxiv.1711.04668,
title = {Diophantine triples in linear recurrence sequences of Pisot type},
author = {Clemens Fuchs and Christoph Hutle and Florian Luca},
journal= {arXiv preprint arXiv:1711.04668},
year = {2018}
}
Comments
25 pages. arXiv admin note: text overlap with arXiv:1602.08236