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Science China Mathematics

, Volume 61, Issue 3, pp 421–438 | Cite as

Two-parameter families of uniquely extendable Diophantine triples

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Abstract

Let A and K be positive integers and є ∈ {−2;−1, 1, 2}. The main contribution of the paper is a proof that each of the D(є2)-triples {K,A2K + 2єA, (A + 1)2K + 2є(A + 1)} has unique extension to a D(є2)- quadruple. This is used to slightly strengthen the conditions required for the existence of a D(1)-quintuple whose smallest three elements form a regular triple.

Keywords

Diophantine m-tuples Pell equations hypergeometric method linear forms in logarithms 

MSC(2010)

16S34 16U60 11J68 11J86 

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Notes

Acknowledgements

This work was supported by Grants-in-Aid for Scientific Research (JSPS KAKENHI) (Grant No. 16K05079).

References

  1. 1.
    Arkin J, Hoggatt V E, Strauss E G. On Euler’s solution of a problem of Diophantus. Fibonacci Quart, 1979, 17: 333–339MathSciNetMATHGoogle Scholar
  2. 2.
    Baker A, Davenport H. The equations 3x 2−2 = y 2 and 8x 2−7 = z 2. Quart J Math Oxford Ser (2), 1969, 20: 129–137MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bennett M A. On the number of solutions of simultaneous Pell equations. J Reine Angew Math, 1998, 498: 173–199MathSciNetMATHGoogle Scholar
  4. 4.
    Cipu M. Further remarks on Diophantine quintuples. Acta Arith, 2015, 168: 201–219MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cipu M, Filipin A, Fujita Y. Bounds for Diophantine quintuples II. Publ Math Debrecen, 2016, 88: 59–78MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cipu M, Fujita Y. Bounds for Diophantine quintuples. Glas Mat Ser III, 2015, 50: 25–34MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dujella A. There are only nitely many Diophantine quintuples. J Reine Angew Math, 2004, 566: 183–224MathSciNetMATHGoogle Scholar
  8. 8.
    Dujella A, Petho A. A generalization of a theorem of Baker and Davenport. Quart J Math Oxford Ser (2), 1998, 49: 291–306MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dujella A, Ramasamy A M S. Fibonacci numbers and sets with the property D(4). Bull Belg Math Soc Simon Stevin, 2005, 12: 401–412MathSciNetMATHGoogle Scholar
  10. 10.
    Elsholtz C, Filipin A, Fujita Y. On Diophantine quintuples and D(-1)-quadruples. Monatsh Math, 2014, 175: 227–239MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Filipin A. There does not exist a D(4)-sextuple. J Number Theory, 2008, 128: 1555–1565MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Filipin A. An irregular D(4)-quadruple cannot be extended to a quintuple. Acta Arith, 2009, 136: 167–176MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Filipin A. On the size of sets in which xy + 4 is always a square. Rocky Mountain J Math, 2009, 39: 1195–1224MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Filipin A. There are only finitely many D(4)-quintuples. Rocky Mountain J Math, 2011, 41: 1847–1860MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Filipin A, He B, Togbé A. On a family of two-parametric D(4)-triples. Glas Mat Ser III, 2012, 47: 31–51MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fujita Y. The unique representation d = 4k(k 2-1) in D(4)-quadruples–k-2; k +2; 4k; d. Math Commun, 2006, 11: 69–81MathSciNetMATHGoogle Scholar
  17. 17.
    Fujita Y. Any Diophantine quintuple contains a regular Diophantine quadruple. J Number Theory, 2009, 129: 1678–1697MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    He B, Togbé A. On a family of Diophantine triples–k;A 2 k + 2A; (A + 1)2k + 2(A + 1) with two parameters. Acta Math Hungar, 2009, 124: 99–113MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    He B, Togbé A. On a family of Diophantine triples–k;A 2 k+2A; (A+1)2k+2(A+1) with two parameters II. Period Math Hungar, 2012, 64: 1–10MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Laurent M. Linear forms in two logarithms and interpolation determinants II. Acta Arith, 2008, 133: 325–348MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Matveev E M. An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II. Izv Ross Akad Nauk Ser Mat, 2000, 64: 125–180; English translation in Izv Math, 2000, 64: 1217–1269MathSciNetMATHGoogle Scholar
  22. 22.
    Rickert J H. Simultaneous rational approximation and related Diophantine equations. Math Proc Cambridge Philos Soc, 1993, 113: 461–472MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    The PARI Group. PARI/GP, version 2.6.0. Bordeaux, 2013, http://pari.math.u-bordeaux.fr/Google Scholar
  24. 24.
    Trudgian T S. Bounds on the number of Diophantine quintuples. J Number Theory, 2015, 157: 233–249MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Mihai Cipu
    • 1
  • Yasutsugu Fujita
    • 2
  • Maurice Mignotte
    • 3
  1. 1.Simion Stoilow Institute of Mathematics of the Romanian AcademyBucharestRomania
  2. 2.Department of Mathematics, College of Industrial TechnologyNihon UniversityChibaJapan
  3. 3.Département de MathématiqueUniversité de StrasbourgStrasbourgFrance

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