Diophantine triples in linear recurrences 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 specialised generalisations 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.

Only finitely many Tribonacci Diophantine triples exist

“…The problem of finding bounds on the size m for Diophantine m-tuples with values in linear recurrences is one such variation. The first general result is due to Fuchs, Luca and Szalay [2] and states that if {u n } n≥1 is a binary recurrent sequence satisfying certain conditions, then there are at most finitely many triples of positive integers a < b < c such that ab + 1, ac + 1 and bc + 1 are all members of {u n } n≥1 . The sequences of Fibonacci and Lucas numbers satisfy the conditions of the above theorem, and all Diophantine triples with values in the Fibonacci sequence or in the Lucas sequence were computed in [11] and [12], respectively.…”

Diophantine Triples and k-Generalized Fibonacci Sequences

“…A variation of this classical problem is obtained if one changes the set of squares by some different subset of positive integers like k-powers for some fixed k ≥ 3, or perfect powers, or primes, or members of some linearly recurrent sequence, etc. (see [7], [13], [14], [18], [12]). In this paper, we study this problem with the set of values of k-generalized Fibonacci numbers for some integer k ≥ 2.…”

Diophantine triples with values in the sequences of Fibonacci and Lucas numbers

“…Another topic which has received interest is when A is the set of members of some binary recurrent sequence. Some necessary conditions on the binary recurrence for the existence of infinitely many examples with m = 3 appear in [4]. For example, one such condition is that the roots of the characteristic equation of the binary recurrence are integers and the smallest one in absolute value is 1.…”