Glasnik Matematicki, Vol. 45, No.1 (2010), 15-29.

THE NUMBER OF DIOPHANTINE QUINTUPLES

Yasutsugu Fujita

Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan
e-mail: fujita.yasutsugu@nihon-u.ac.jp

Abstract.   A set {a1, ... ,am} of m distinct positive integers is called a Diophantine m-tuple if aiaj+1 is a perfect square for all i, j with 1 ≤ i < j ≤ m. It is known that there does not exist a Diophantine sextuple and that there exist only finitely many Diophantine quintuples. In this paper, we first show that for a fixed Diophantine triple {a,b,c} with a < b < c, the number of Diophantine quintuples {a,b,c,d,e} with c < d < e is at most four. Using this result, we further show that the number of Diophantine quintuples is less than 10276, which improves the bound 101930 due to Dujella.

2000 Mathematics Subject Classification.   11D09, 11J68, 11J86.

Key words and phrases.   Simultaneous Diophantine equations, Diophantine tuples.

Full text (PDF) (free access)

DOI: 10.3336/gm.45.1.02

References:

  1. J. Arkin, V. E. Hoggatt and E. G. Strauss, On Euler's solution of a problem of Diophantus, Fibonacci Quart. 17 (1979), 333-339.
    MathSciNet

  2. 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

  3. M. A. Bennett, M. Cipu, M. Mignotte and R. Okazaki, On the number of solutions of simultaneous Pell equations II, Acta Arith. 122 (2006), 407-417.
    MathSciNet     CrossRef

  4. 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

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

  6. A. Dujella, A proof of the Hoggatt-Bergum conjecture, Proc. Amer. Math. Soc. 127 (1999), 1999-2005.
    MathSciNet     CrossRef

  7. A. Dujella, An absolute bound for the size of Diophantine m-tuples, J. Number Theory 89 (2001), 126-150.
    MathSciNet     CrossRef

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

  9. A. Dujella, On the number of Diophantine m-tuples, Ramanujan J. 15 (2008), 37-46.
    MathSciNet     CrossRef

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

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

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

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

  14. E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II, Izv. Math. 64 (2000), 1217-1269.
    MathSciNet     CrossRef

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