Only finitely many Tribonacci Diophantine triples exist

  • 1 University of Salzburg, Hellbrunner Str. 34/I 5020, Salzburg, Austria
  • 2 Ömer Halisdemir University, Nigde, Turkey
  • 3 University of Witwatersrand, Braamfontein 2000, Johannesburg, South Africa
  • 4 University of Sopron, Bajcsy-Zsilinszky u. 4. 9400, Sopron, Hungary
Clemens Fuchs, Christoph Hutle, Nurettin Irmak, Florian Luca and László Szalay

Abstract

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.

  • [1]

    Bravo, J. J.—Luca, F.: On a conjecture about repdigits in k-generalized Fibonacci sequences, Publ. Math. Debrecen 82 (2013), 623–639.

  • [2]

    Dujella, A.: There are only finitely many Diophantine quintuples. J. Reine Angew. Math. 566 (2004), 183–214.

  • [3]

    Dujella, A.: Diophantine m-tuples, https://web.math.pmf.unizg.hr/duje/dtuples.html.

  • [4]

    Evertse, J.-H.—Schmidt, W. M.—Schlickewei, H.-P.: Linear equations in variables which lie in a multipilicative group, Ann. of Math. 155 (2002), 807–836.

  • [5]

    Evertse, J.-H.: An improvement of the quantitative Subspace theorem, Compos. Math. 101 (1996), 225–311.

  • [6]

    Fuchs, C.: Polynomial-exponential equations and linear recurrences, Glas. Mat. 38 (2003), 233–252.

  • [7]

    Fuchs, C.: Diophantine problems with linear recurrences via the Subspace theorem, Integers 5 (2005), A8.

  • [8]

    Fuchs, C.: Polynomial-exponential equations involving multi-recurrences, Studia Sci. Math. Hungar. 46 (2009), 377–398.

  • [9]

    Fuchs, C.—Luca, F.—Szalay, L.: Diophantine triples with values in binary recurrences, Ann. Sc. Norm. Super. Pisa Cl. Sc. 7 (2008), 579–608.

  • [10]

    Fuchs, C.—Tichy, R. F.: Perfect powers in linear recurrence sequences, Acta Arith. 107 (2003), 9–25.

  • [11]

    Gomez Ruiz, C. A.—Luca, F.: Tribonacci Diophantine quadruples, Glas. Mat. 50 (2015), 17–24.

  • [12]

    Gomez Ruiz, C. A.—Luca, F.: Diophantine quadruples in the sequence of shifted Tribonacci numbers, Publ. Math. Debrecen 86 (2015), 473–491.

  • [13]

    Irmak, N.—Szalay, L.: Diophantine triples and reduced quadruples with the Lucas sequence of recurrence un = Aun−1 − un−2, Glas. Mat.49 (2014), 303–312.

  • [14]

    Luca, F.—Szalay, L.: Fibonacci Diophantine Triples, Glas. Mat. 43 (2008), 253–264.

  • [15]

    Luca, F.—Szalay, L.: Lucas Diophantine Triples, Integers 9 (2009), 441–457.

  • [16]

    Spickerman, W. R.: Binet’s formula for the Tribonacci numbers, Fibonacci Quart. 20 (1982), 118–120.

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