Diophantine triples in linear recurrence sequences of Pisot type

TL;DR

This paper proves that linear recurrence sequences of Pisot type of sufficiently large order contain only finitely many Diophantine triples, extending known results beyond binary sequences.

Contribution

It establishes finiteness of Diophantine triples in linear recurrence sequences of Pisot type under certain conditions, generalizing previous results from binary sequences.

Findings

Finiteness of Diophantine triples in Pisot-type sequences of large order

Extension of known results beyond binary recurrence sequences

Conditions under which finiteness holds

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.