Glasnik Matematicki, Vol. 49, No. 2 (2014), 303-312.

DIOPHANTINE TRIPLES AND REDUCED QUADRUPLES WITH THE LUCAS SEQUENCE OF RECURRENCE un=Aun-1-un-2

Nurettin Irmak and László Szalay

Mathematics Department, Art and Science Faculty, University of Niğde, 51240 Niğde, Turkey
e-mail: nirmak@nigde.edu.tr, irmaknurettin@gmail.com

Institute of Mathematics, University of West Hungary, H-9400 Sopron, Hungary
e-mail: laszlo.szalay@emk.nyme.hu

Abstract.   In this study, we show that there is no positive integer triple {a, b, c} such that all of ab+1, ac+1 and bc+1 are in the sequence {un}n≥ 0 satisfies the recurrence un=Aun-1-un-2 with the initial values u0=0, u1=1. Further, we investigate the analogous question for the quadruples {a,b,c,d} with abc+1=ux, bcd+1=uy, cda+1=uz and dab+1=ut, and deduce the non-existence of such quadruples.

2010 Mathematics Subject Classification.   11D72, 11B39.

Key words and phrases.   Diophantine triples, Diophantine quadruples, binary recurrence.

Full text (PDF) (free access)

DOI: 10.3336/gm.49.2.05

References:

  1. M. Alp, N. Irmak and L. Szalay, Balancing Diophantine Triples, Acta Univ. Sapientiae 4 (2012), 11-19.
    MathSciNet    

  2. R. D. Carmichael, On the numerical factors of the arithmetic forms α n± β n, Ann. of Math. (2) 15 (1913/14), 30-48.

  3. A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183-214.
    MathSciNet     CrossRef

  4. C. Fuchs, F. Luca and L. Szalay, Diophantine triples with values in binary recurrences, Ann. Scuola Norm. Sup. Pisa. Cl. Sci. III 5 (2008), 579-608.
    MathSciNet    

  5. V. E. Hoggat and G. E. Bergum, A problem of Fermat and Fibonacci sequence, Fibonacci Quart. 15 (1977), 323-330.
    MathSciNet    

  6. F. Luca and L. Szalay, Fibonacci Diophantine triples, Glas. Mat. Ser. III 43(63) (2008), 253-264.
    MathSciNet     CrossRef

  7. F. Luca and L. Szalay, Lucas Diophantine triples, INTEGERS 9 (2009), 441-457.
    MathSciNet     CrossRef

  8. G. K. Panda and S. S. Rout, A Class of recurrent sequences exhibiting some exciting properties of balancing numbers, World Acad. of Sci., Eng. and Tech. 61 (2012), 164-166.

  9. L. Szalay, Diophantine equations with binary recurrences associated to Brocard-Ramanujan problem, Portugal. Math. 69 (2012), 213-220.
    MathSciNet     CrossRef
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