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

Luca, Florian; Munagi, Augustine O.

doi:10.3336/gm.52.1.03

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2018

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Cited by 3 publications

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“…The Fibonacci sequence and the Lucas sequence both satisfy these conditions and all Diophantine triples with values in these sequences were computed in [23] and [24]. Further results in this direction can be found in [2], [21] and [22]. Moreover, in [1] it is shown that there are no balancing Diophantine triples; see also [3] for a related result.…”

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confidence: 84%

Diophantine triples in linear recurrences of Pisot type

2018

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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.

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