Zeckendorf representations with at most two terms to x-coordinates of Pell equations
Articles
First Online:
- 3 Downloads
Abstract
In this paper, we find all positive squarefree integers d satisfying that the Pell equation X2–dY2 = ±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 methodMSC(2010)
11B39 11J86Preview
Unable to display preview. Download preview PDF.
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.Cohen H. Number Theory. Volume I: Tools and Diophantine Equations. New York: Springer, 2007zbMATHGoogle Scholar
- 2.Cohn J H E. The Diophantine equation x 4–Dy 2 = 1, II. Acta Arith, 1997, 78: 401–403MathSciNetCrossRefGoogle Scholar
- 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.Dujella A, Pethő A. A Generalization of a theorem of Baker and Davenport. Quart J Math Oxford, 1998, 49: 291–306MathSciNetCrossRefzbMATHGoogle Scholar
- 5.Faye B, Luca F. On x-coordinates of Pell equations which are repdigits. ArXiv:1605.03756, 2016zbMATHGoogle Scholar
- 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.Ljunggren W. Zur Theorie der Gleichung X 2 + 1 = DY 4. Avh Norske Vid Akad Oslo I Mat-Naturv, 1942, 5: 27Google Scholar
- 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.Luca F, Togbé A. On the x-coordinates of Pell equations which are Fibonacci numbers. Math Scand, 2018, 122: 18–30MathSciNetCrossRefzbMATHGoogle Scholar
- 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.Spickerman W R. Binet’s formula for the Tribonacci numbers. The Fibonacci Quarterly, 1982, 20: 118–120MathSciNetzbMATHGoogle Scholar
- 12.Sun Q, Yuan P Z. On the Diophantine equation x 4–Dy 2 = 1. Adv Math (China), 1996, 25: 84Google Scholar
- 13.Sun Q, Yuan P Z. A note on the Diophantine equation x 4–Dy 2 = 1. Sichuan Daxue Xuebao, 1997, 34: 265–268MathSciNetGoogle Scholar
- 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