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Mathematica Slovaca

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Volume 67, Issue 4


Only finitely many Tribonacci Diophantine triples exist

Clemens Fuchs / Christoph Hutle / Nurettin Irmak / Florian Luca / László Szalay
Published Online: 2017-07-14 | DOI: https://doi.org/10.1515/ms-2017-0015


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.

MSC 2010: Primary 11D72; 11B39; Secondary 11J87

Keywords: Diophantine triples; Tribonacci numbers; Diophantine equations; application of the Subspace theorem


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About the article


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.

Received: 2015-08-31

Accepted: 2015-11-25

Published Online: 2017-07-14

Published in Print: 2017-08-28

Citation Information: Mathematica Slovaca, Volume 67, Issue 4, Pages 853–862, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0015.

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