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Zeckendorf representations with at most two terms to x-coordinates of Pell equations

  • Carlos A. GómezEmail author
  • Florian Luca
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  • 3 Downloads

Abstract

In this paper, we find all positive squarefree integers d satisfying that the Pell equation X2dY2 = ±1 has at least two positive integer solutions (X, Y) and (X′, Y′) such that both X and X′ have Zeckendorf representations with at most two terms.

Keywords

Pell equation Fibonacci numbers lower bounds for linear forms in logarithms reduction method 

MSC(2010)

11B39 11J86 

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Notes

Acknowledgements

The first author was supported by the Project from Universidad del Valle (Grant No. 71079). The second author was supported by NRF of South Africa (Grant No. CPRR160325161141), an A-Rated Scientist Award from the NRF of South Africa and by Czech Granting Agency (Grant No. 17-02804S).

References

  1. 1.
    Cohen H. Number Theory. Volume I: Tools and Diophantine Equations. New York: Springer, 2007zbMATHGoogle Scholar
  2. 2.
    Cohn J H E. The Diophantine equation x 4Dy 2 = 1, II. Acta Arith, 1997, 78: 401–403MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dossavi-Yovo A, Luca F, Togbé A. On the x-coordinates of Pell equations which are rep-digits. Publ Math Debrecen, 2016, 88: 381–399MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dujella A, Pethő A. A Generalization of a theorem of Baker and Davenport. Quart J Math Oxford, 1998, 49: 291–306MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Faye B, Luca F. On x-coordinates of Pell equations which are repdigits. ArXiv:1605.03756, 2016zbMATHGoogle Scholar
  6. 6.
    Laurent M, Mignotte M, Nesterenko Y. Formes linéaires en deux logarithmes et déterminants d’interpolation. J Number Theory, 1995, 55: 285–321MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ljunggren W. Zur Theorie der Gleichung X 2 + 1 = DY 4. Avh Norske Vid Akad Oslo I Mat-Naturv, 1942, 5: 27Google Scholar
  8. 8.
    Luca F, Montejano A, Szalay L, Togbé A. On the x-coordinates of Pell equations which are Tribonacci numbers. Acta Arith, 2017, 179: 25–35MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Luca F, Togbé A. On the x-coordinates of Pell equations which are Fibonacci numbers. Math Scand, 2018, 122: 18–30MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Matveev E M. An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. Izv Math, 2000, 64: 1217–1269MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Spickerman W R. Binet’s formula for the Tribonacci numbers. The Fibonacci Quarterly, 1982, 20: 118–120MathSciNetzbMATHGoogle Scholar
  12. 12.
    Sun Q, Yuan P Z. On the Diophantine equation x 4Dy 2 = 1. Adv Math (China), 1996, 25: 84Google Scholar
  13. 13.
    Sun Q, Yuan P Z. A note on the Diophantine equation x 4Dy 2 = 1. Sichuan Daxue Xuebao, 1997, 34: 265–268MathSciNetGoogle Scholar
  14. 14.
    Zeckendorf E. Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas. Bulletin de la Société Royale des Sciences de Liège, 1972, 41: 179–182MathSciNetzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad del ValleCaliColombia
  2. 2.School of MathematicsUniversity of the WitwatersrandWitsSouth Africa
  3. 3.Max Planck Institute for MathematicsBonnGermany
  4. 4.Department of Mathematics, Faculty of SciencesUniversity of OstravaOstrava 1Czech Republic

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