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Only finitely many Tribonacci Diophantine triples exist

Clemens Fuchs, Christoph Hutle, Nurettin Irmak, Florian Luca and László Szalay
From the journal Mathematica Slovaca


Diophantine triples taking values in recurrence sequences have recently been studied quite a lot. In particular the question was raised whether or not there are finitely many Diophantine triples in the Tribonacci sequence. We answer this question here in the affirmative. We prove that there are only finitely many triples of integers 1 ≤ u < v < w such that uv + 1, uw + 1, vw + 1 are Tribonacci numbers. The proof depends on the Subspace theorem.

(Communicated by Federico Pellarin)

C. Fuchs and C. Hutle were supported by FWF (Austrian Science Fund) grant No. P24574 and by the Sparkling Science project EMMA grant No. SPA 05/172.


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Received: 2015-8-31
Accepted: 2015-11-25
Published Online: 2017-7-14
Published in Print: 2017-8-28

© 2017 Mathematical Institute Slovak Academy of Sciences