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.
References
[1] Bravo, J. J.—Luca, F.: On a conjecture about repdigits in k-generalized Fibonacci sequences, Publ. Math. Debrecen 82 (2013), 623–639.10.5486/PMD.2013.5390Search in Google Scholar
[2] Dujella, A.: There are only finitely many Diophantine quintuples. J. Reine Angew. Math. 566 (2004), 183–214.10.1515/crll.2004.003Search in Google Scholar
[3] Dujella, A.: Diophantine m-tuples, https://web.math.pmf.unizg.hr/duje/dtuples.html.Search in Google Scholar
[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), 807–836.10.2307/3062133Search in Google Scholar
[5] Evertse, J.-H.: An improvement of the quantitative Subspace theorem, Compos. Math. 101 (1996), 225–311.Search in Google Scholar
[6] Fuchs, C.: Polynomial-exponential equations and linear recurrences, Glas. Mat. 38 (2003), 233–252.10.3336/gm.38.2.03Search in Google Scholar
[7] Fuchs, C.: Diophantine problems with linear recurrences via the Subspace theorem, Integers 5 (2005), A8.Search in Google Scholar
[8] Fuchs, C.: Polynomial-exponential equations involving multi-recurrences, Studia Sci. Math. Hungar. 46 (2009), 377–398.10.1556/sscmath.2009.1098Search in Google Scholar
[9] Fuchs, C.—Luca, F.—Szalay, L.: Diophantine triples with values in binary recurrences, Ann. Sc. Norm. Super. Pisa Cl. Sc. 7 (2008), 579–608.10.2422/2036-2145.2008.4.01Search in Google Scholar
[10] Fuchs, C.—Tichy, R. F.: Perfect powers in linear recurrence sequences, Acta Arith. 107 (2003), 9–25.10.4064/aa107-1-2Search in Google Scholar
[11] Gomez Ruiz, C. A.—Luca, F.: Tribonacci Diophantine quadruples, Glas. Mat. 50 (2015), 17–24.10.3336/gm.50.1.02Search in Google Scholar
[12] Gomez Ruiz, C. A.—Luca, F.: Diophantine quadruples in the sequence of shifted Tribonacci numbers, Publ. Math. Debrecen 86 (2015), 473–491.10.5486/PMD.2015.7118Search in Google Scholar
[13] Irmak, N.—Szalay, L.: Diophantine triples and reduced quadruples with the Lucas sequence of recurrenceun = Aun−1 − un−2, Glas. Mat.49 (2014), 303–312.10.3336/gm.49.2.05Search in Google Scholar
[14] Luca, F.—Szalay, L.: Fibonacci Diophantine Triples, Glas. Mat. 43 (2008), 253–264.10.3336/gm.43.2.03Search in Google Scholar
[15] Luca, F.—Szalay, L.: Lucas Diophantine Triples, Integers 9 (2009), 441–457.10.1515/INTEG.2009.037Search in Google Scholar
[16] Spickerman, W. R.: Binet’s formula for the Tribonacci numbers, Fibonacci Quart. 20 (1982), 118–120.Search in Google Scholar
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