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Get Access to Full TextDiophantine 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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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.
© 2017 Mathematical Institute Slovak Academy of Sciences.
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