Glasnik Matematicki, Vol. 47, No. 1 (2012), 31-51.

ON A FAMILY OF TWO-PARAMETRIC D(4)-TRIPLES

Alan Filipin, Bo He and Alain Togbé

Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, 10000 Zagreb, Croatia
e-mail: filipin@grad.hr

Department of Mathematics, ABa Teacher's College, Wenchuan, Sichuan, 623000, P. R. China
e-mail: bhe@live.cn

Mathematics Department, Purdue University North Central, 1401 S, U.S. 421, Westville IN 46391, USA
e-mail: atogbe@pnc.edu


Abstract.   Let k be a positive integer. In this paper, we study a parametric family of the sets of integers {k,A2k+4A,(A+1)2k+4(A+1),d}. We prove that if d is a positive integer such that the product of any two distinct elements of that set increased by 4 is a perfect square, then

d= (A4 + 2A3 + A2)k3 + (8A3 + 12A2 + 4A)k2 + (20A2 + 20A + 4) k + (16A + 8)
for 1≤ A ≤22 and A ≥ 51767.

2010 Mathematics Subject Classification.   11D09, 11D25, 11J86.

Key words and phrases.   Diophantine m-tuples, Pell equations, Baker's method.


Full text (PDF) (free access)

DOI: 10.3336/gm.47.1.03


References:

  1. A. Baker and H. Davenport, The equations 3x2-2=y2 and 8x2-7=z2, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129-137.
    MathSciNet     CrossRef

  2. M. A. Bennett, On the number of solutions of simultaneous Pell equations, J. Reine Angew. Math. 498 (1998), 173-199.
    MathSciNet     CrossRef

  3. Y. Bugeaud, A. Dujella and M. Mignotte, On the family of Diophantine triples {k - 1, k + 1, 16k3 - 4k}, Glasg. Math. J., 49 (2007), 333-344.
    MathSciNet     CrossRef

  4. A. Dujella, The problem of the extension of a parametric family of Diophantine triples, Publ. Math. Debrecen 51 (1997), 311-322.
    MathSciNet    

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

  6. A. Dujella and A. Pethő, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2) 49 (1998), 291-306.
    MathSciNet     CrossRef

  7. A. Dujella and A. M. S. Ramasamy, Fibonacci numbers and sets with the property D(4), Bull. Belg. Math. Soc. Simon Stevin 12 (2005), 401-412.
    MathSciNet     CrossRef

  8. A. Filipin, An irregular D(4)-quadruple cannot be extended to a quintuple, Acta Arith. 136 (2009), 167-176.
    MathSciNet     CrossRef

  9. A. Filipin, On the size of sets in which xy + 4 is always a square, Rocky Mount. J. Math. 39 (2009), 1195-1224.
    MathSciNet     CrossRef

  10. A. Filipin, There are only finitely many D(4)-quintuples, Rocky Mount. J. Math. 41 (2011), 1847-1859.
    MathSciNet     CrossRef

  11. A. Filipin, There does not exist a D(4)-sextuple, J. Number Theory, 128 (2008), 1555-1565.
    MathSciNet     CrossRef

  12. Y. Fujita, Any Diophantine quintuple contains a regular Diophantine quadruple, J. Number Theory 129 (2009), 1678-1697.
    MathSciNet     CrossRef

  13. Y. Fujita, The extensibility of Diophantine pairs {k-1, k+1}, J. Number Theory, 128 (2008), 322-353.
    MathSciNet     CrossRef

  14. Y. Fujita, The unique representation d=4k(k2-1) in D(4)-quadruples {k-2, k+2, 4k, d}, Math. Commun. 11 (2006), 69-81.
    MathSciNet    

  15. Y. Fujita, The number of Diophantine quintuples, Glas. Mat. Ser. III 45 (2010), 15-29.
    MathSciNet     CrossRef

  16. B. He and A. Togbé, On a family of Diophantine triples {k, A2k+2A ,(A+1)2k+2(A+1)} with two parameters, Acta Math. Hungar. 124(1-2) (2009), 99-113.
    MathSciNet     CrossRef

  17. B. He and A. Togbé, On a family of Diophantine triples {k, A2k+2A ,(A+1)2k+2(A+1)} with two parameters II., Periodica Math. Hungar. 64 (2012), 1-10.

  18. B. He and A. Togbé, On the D(-1)-triple {1, k2+1, k2+2k+2} and its unique D(1)-extension, J. Number Theory 131 (2011), 120-137.
    MathSciNet     CrossRef

  19. K. S. Kedlaya Solving constrained Pell equations, Math. Comp. 67 (1998), 833-842.
    MathSciNet     CrossRef

  20. M. Mignotte, A corollary to a theorem of Laurent-Mignotte-Nesterenko, Acta Arith. 86 (1998), 101-111.
    MathSciNet    

  21. S. P. Mohanty and A. M. S. Ramasamy, The simultaneous Diophantine equations 5y2-20 = x2 and 2y2+1 = z2, J. Number Theory 18 (1984), 356-359.
    MathSciNet     CrossRef

  22. J. H. Rickert, Simultaneous rational approximation and related Diophantine equations, Math. Proc. Cambridge Philos. Soc. 11 (1993), 461-472.
    MathSciNet     CrossRef

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