Glasnik Matematicki, Vol. 41, No.2 (2006), 205-216.

THE NON-EXTENSIBILITY OF D(4k)-TRIPLES {1, 4k(k-1), 4k2+1} WITH |k| PRIME

Yasutsugu Fujita

Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
e-mail: fyasut@yahoo.co.jp

Abstract.   For a nonzero integer n, a set of m distinct positive integers {a1, ... , am} is called a D(n)-m-tuple if aiaj+n is a perfect square for each i, j with 1 ≤ i < j ≤ m. Let k be an integer with |k| prime. Then we show that the D(4k)-triple {1, 4k(k-1), 4k2+1} cannot be extended to a D(4k)-quadruple.

2000 Mathematics Subject Classification.   11D09, 11D45.

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

Full text (PDF) (free access)

DOI: 10.3336/gm.41.2.03

References:

  1. F. S. Abu Muriefah and A. Al-Rashed, On the extendibility of the Diophantine triple {1,5,c}, Int. J. Math. Math. Sci. 33 (2004), 1737-1746.
    MathSciNet     CrossRef

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

  3. 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

  4. E. Brown, Sets in which xy+k is always a square, Math. Comp. 45 (1985), 613-620.
    MathSciNet     CrossRef

  5. A. Dujella, Generalization of a problem of Diophantus, Acta Arith. 65 (1993), 15-27.
    MathSciNet

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

  7. A. Dujella, On the exceptional set in the problem of Diophantus and Davenport, in: Applications of Fibonacci numbers, Vol. 7, Kluwer Acad. Publ., Dordrecht, 1998, 69-76.
    MathSciNet

  8. A. Dujella, Diophantine quadruples and quintuples modulo 4, Notes Number Theory Discrete Math. 4 (1998), 160-164.
    MathSciNet

  9. A. Dujella, Complete solution of a family of simultaneous Pellian equations, Acta Math. Inform. Univ. Ostraviensis 6 (1998), 59-67.
    MathSciNet

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

  11. A. Dujella, On the size of Diophantine m-tuples, Math. Proc. Cambridge Philos. Soc. 132 (2002), 23-33.
    MathSciNet

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

  13. A. Dujella, Bounds for the size of sets with the property D(n), Glas. Mat. Ser. III 39(59) (2004), 199-205.
    MathSciNet

  14. A. Dujella, A. Filipin, and C. Fuchs, Effective solution of the D(-1)-quadruple conjecture, preprint.

  15. A. Dujella and C. Fuchs, Complete solution of a problem of Diophantus and Euler, J. London Math. Soc. (2) 71 (2005), 33-52.
    MathSciNet     CrossRef

  16. A. Dujella and F. Luca, Diophantine m-tuples for primes, Int. Math. Res. Not. 47 (2005), 2913-2940.
    MathSciNet     CrossRef

  17. 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

  18. 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

  19. A. Filipin, Nonextendibility of D(-1)-triples of the form {1,10,c}, Int. J. Math. Math. Sci. 14 (2005), 2217-2226.
    MathSciNet     CrossRef

  20. Y. Fujita, The extensibility of D(-1)-triples {1,b,c}, Publ. Math. Debrecen, to appear.
    MathSciNet

  21. Y. Fujita, The unique representation d = 4k(k2-1) in D(4)-quadruples {k-2,k+2,4k,d}, Math. Commun., to appear.
    MathSciNet

  22. Y. Fujita, The extensibility of Diophantine pairs {k - 1, k + 1}, preprint.

  23. P. Gibbs, Some rational Diophantine sextuples, Glas. Mat. Ser. III 41(61) (2006), 195-203.
    MathSciNet     CrossRef

  24. H. Gupta and K. Singh, On k-triad sequences, Internat. J. Math. Math. Sci. 8 (1985), 799-804.
    MathSciNet     CrossRef

  25. C. Long and G. E. Bergum, On a problem of Diophantus, in: Applications of Fibonacci numbers, Vol. 2, Kluwer Acad. Publ., Dordrecht, 1988, 183-191.
    MathSciNet

  26. 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

  27. V. K. Mootha and G. Berzsenyi, Characterizations and extendibility of Pt-sets, Fibonacci Quart. 27 (1989), 287-288.
    MathSciNet

  28. S. P. Mohanty and A. M. S. Ramasamy, On Pr,k sequences, Fibonacci Quart. 23 (1985), 36-44.
    MathSciNet

  29. R. A. Mollin, Quadratics, CRC Press, Boca Raton, 1996.
    MathSciNet

  30. T. Nagell, Introduction to Number Theory, John Wiley & Sons, Inc., New York; Almqvist & Wiksell, Stockholm, 1951.
    MathSciNet

  31. J. E. Shockley, Introduction to Number Theory, Holt, Rinehart and Winston, Inc., New York-Toronto, Ont.-London, 1967.
    MathSciNet
Glasnik Matematicki Home Page