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Periodica Mathematica Hungarica

, Volume 65, Issue 1, pp 75–81 | Cite as

The extension of the D(−k 2)-pair {k 2,k 2 + 1}

  • Yasutsugu Fujita
  • Alain Togbé
Article
  • 79 Downloads

Abstract

Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a positive integer. In this paper, we show that if {k 2, k 2+1, c, d} is a D(−k 2)-quadruple with c < d, then c = 1 and d = 4k 2+1. This extends the work of the first author [20] and that of Dujella [4].

Key words and phrases

Diophantine tuples simultaneous Diophantine equations 

Mathematics subject classification numbers

11D09 11J68 

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References

  1. [1]
    A. Baker and H. Davenport, The equations 3x 2 − 2 = y 2 and 8x 2 − 7 = z 2, Quart. J. Math. Oxford Ser. (2), 20 (1969), 129–137.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    A. Dujella, Generalization of a problem of Diophantus, Acta Arith., 65 (1993), 15–27.MathSciNetMATHGoogle Scholar
  3. [3]
    A. Dujella, The problem of the extension of a parametric family of Diophantine triples, Publ. Math. Debrecen, 51 (1997), 311–322.MathSciNetMATHGoogle Scholar
  4. [4]
    A. Dujella, Complete solution of a family of simultaneous Pellian equations, Acta Math. Inform. Univ. Ostraviensis, 6 (1998), 59–67.MathSciNetMATHGoogle Scholar
  5. [5]
    A. Dujella, On the exceptional set in the problem of Diophantus and Davenport, Applications of Fibonacci Numbers, 7 (1998), 69–76.MathSciNetCrossRefGoogle Scholar
  6. [6]
    A. Dujella, An extension of an old problem of Diophantus and Euler, Fibonacci Quart., 37 (1999), 312–314.MathSciNetMATHGoogle Scholar
  7. [7]
    A. Dujella, An absolute bound for the size of Diophantine m-tuples, J. Number Theory, 89 (2001), 126–150.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    A. Dujella, An extension of an old problem of Diophantus and Euler. II, Fibonacci Quart., 40 (2002), 118–123.MathSciNetMATHGoogle Scholar
  9. [9]
    A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math., 566 (2004), 183–214.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    A. Dujella, A. Filipin and C. Fuchs, Effective solution of the D(−1)-quadruple conjecture, Acta Arith., 128 (2007), 319–338.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    A. Dujella and C. Fuchs, Complete solution of a problem of Diophantus and Euler, J. London Math. Soc., 71 (2005), 33–52.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    A. Dujella and A. Pethő, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford (Ser. 2), 49 (1998), 291–306.MathSciNetMATHGoogle Scholar
  13. [13]
    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.MathSciNetMATHGoogle Scholar
  14. [14]
    A. Filipin, There does not exist a D(4)-sextuple, J. Number Theory, 128 (2008), 1555–1565.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    A. Filipin and Y. Fujita, The D(-k 2)-triple {1, k 2+1, k 2+4} with k prime, Glas. Mat. Ser. III, 46 (2011), 311–323.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    A. Filipin, B. He and A. Togbé, On the D(4)-triple {F 2k, F 2k+6, 4F 2k+4}, Fibonacci Quart., 48 (2010), 219–227.MathSciNetMATHGoogle Scholar
  17. [17]
    A. Filipin, B. He and A. Togbé, On a family of two-parametric D(4)-triples, Glas. Mat. Ser. III, to appear.Google Scholar
  18. [18]
    Y. Fujita, The non-extensibility of D(4k)-triples {1, 4k(k − 1), 4k 2 + 1} with |k| prime, Glas. Mat. Ser. III, 41 (2006), 205–216.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Y. Fujita, The extensibility of Diophantine pairs {k−1, k+1}, J. Number Theory, 128 (2008), 322–353.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Y. Fujita, Extensions of the D(∓k 2)-triples {k 2, k 2 ± 1, 4k 2 ± 1}, Period. Math. Hungar., 59 (2009), 81–98.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    Y. Fujita, The number of Diophantine quintuples, Glas. Mat. Ser. III, 45 (2010), 15–29.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    B. He and A. Togbé, On a family of Diophantine triples {k,A 2 k +2A, (A+1)2 k + 2(A + 1)} with two parameters, Acta Math. Hungar., 124 (2009), 99–113.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    B. He and A. Togbé, On a family of Diophantine triples {k,A 2 k +2A, (A+1)2 k + 2(A + 1)} with two parameters II, Period. Math. Hungar., 64 (2012), 1–10.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    S. P. Mohanty and A. M. S. Ramasamy, The simultaneous Diophantine equations 5y 2 − 20 = x 2 and 2y 2 + 1 = z 2, J. Number Theory, 18 (1984), 356–359.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2012

Authors and Affiliations

  1. 1.Department of Mathematics, College of Industrial TechnologyNihon UniversityNarashino, ChibaJapan
  2. 2.Mathematics DepartmentPurdue University North CentralWestvilleUSA

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