Glasnik Matematicki, Vol. 50, No. 1 (2015), 25-34.

BOUNDS FOR DIOPHANTINE QUINTUPLES

Mihai Cipu and Yasutsugu Fujita

Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit nr. 5, P.O. Box 1-764, RO-014700 Bucharest, Romania
e-mail: Mihai.Cipu@imar.ro

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 of m positive integers {a1,...,am} is called a Diophantine m-tuple if the product of any two elements in the set increased by one is a perfect square. The conjecture according to which there does not exist a Diophantine quintuple is still open. In this paper, we show that if {a,b,c,d,e} is a Diophantine quintuple with a < b < c < d < e , then b >3a; moreover, b > max{21 a, 2 a3/2} in case c>a+b+2(ab+1)1/2.

2010 Mathematics Subject Classification.   11D09, 11B37, 11J68.

Key words and phrases.   Diophantine m-tuples, Pell equations, hypergeometric method.

Full text (PDF) (free access)

DOI: 10.3336/gm.50.1.03

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, Simultaneous approximation to pairs of algebraic numbers, in: CMS Conf. Proc. 15 (1995), 55-65.
    MathSciNet    

  4. M. A. Bennett, Simultaneous rational approximation to binomial functions, Trans. Amer. Math. Soc. 348 (1996), 1717-1738.
    MathSciNet     CrossRef

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

  6. M. Cipu, Further remarks on Diophantine quintuples, Acta Arith. 168 (2015), 201-219.

  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. Math. 15 (2008), 37-46.
    MathSciNet     CrossRef

  10. A. Dujella and A. Pethő, A generalization of a theorem of Baker and Davenport, Q. J. Math. 49 (1998), 291-306.
    MathSciNet     CrossRef

  11. C. Elsholtz, A. Filipin and Y. Fujita, On Diophantine quintuples and D(-1)-quadruples, Monatsh. Math. 175 (2014) 227-239.
    MathSciNet     CrossRef

  12. A. Filipin and Y. Fujita, The number of Diophantine quintuples II, Publ. Math. Debrecen 82 (2013), 293-308.
    MathSciNet     CrossRef

  13. A. Filipin, Y. Fujita, and A. Togbé, The extendibility of Diophantine pairs I: the general case, Glas. Mat. Ser. III 49 (2014), 25-36.
    MathSciNet     CrossRef

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

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

  16. Y. Fujita, The number of Diophantine quintuples, Glas. Mat. Ser. III 45(65) (2010), 15-29.
    MathSciNet     CrossRef

  17. P. E. Gibbs, Computer Bulletin 17 (1978), 16.

  18. P. E. Gibbs, Some rational Diophantine sextuples, Glas. Mat. Ser. III 41 (2006), 195-203.
    MathSciNet     CrossRef

  19. B. W. Jones, A second variation on a problem of Diophantus and Davenport, Fibonacci Quart. 16 (1978), 155-165.
    MathSciNet    

  20. The PARI Group, PARI/GP, version 2.3.5, Bordeaux, 2010, available from http://pari.math.u-bordeaux.fr/.

  21. J. H. Rickert, Simultaneous rational approximations and related Diophantine equations, Math. Proc. Cambridge Philos. Soc. 113 (1993), 461-472.
    MathSciNet     CrossRef

  22. L. Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x) II, Math. Comp. 30 (1976), 337-360.
    MathSciNet     CrossRef

  23. W. Wu and B. He, On Diophantine quintuple conjecture, Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), 84-86.
    MathSciNet     CrossRef
Glasnik Matematicki Home Page