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.
Highlights
The problem of Diophantus of Alexandria about tuples of integers {a1, a2, a3, . . . , am} such that the product of each distinct two of them plus 1 always results in an integer square has already quite a long history
One of the main questions was, how many such Diophantine m-tuples exist for a fixed m ≥ 3
Much later Arkin, Hoggatt and Strauss [5] proved that every Diophantine triple can be extended to a Diophantine quadruple
Summary
The problem of Diophantus of Alexandria about tuples of integers {a1, a2, a3, . . . , am} such that the product of each distinct two of them plus 1 always results in an integer square has already quite a long history (see [8]). The problem of Diophantus of Alexandria about tuples of integers {a1, a2, a3, . Am} such that the product of each distinct two of them plus 1 always results in an integer square has already quite a long history (see [8]). One of the main questions was, how many such Diophantine m-tuples exist for a fixed m ≥ 3. Already Euler proved that there are infinitely many Diophantine quadruples, demonstrating it with the family. Much later Arkin, Hoggatt and Strauss [5] proved that every Diophantine triple can be extended to a Diophantine quadruple. Let {a, b, c} be a Diophantine triple and ab + 1 = r2, ac + 1 = s2, bc + 1 = t2, where r, s, t are positive integers. {a, b, c, d+} is a Diophantine quadruple. Dujella proved in [9], that there are no Diophantine sextuples and that there are only finitely many Diophantine quintuples
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