Revista Integración, temas de matemáticas.

Revista Integración, temas de matemáticas

Vol. 42 No. 1 (2024): Revista Integración, temas de matemáticas
Research and Innovation Articles

Fibonacci and Lucas numbers of the form -2^a-3^b-5^c+7^d

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Sofía Ibarra
Universidad Nacional Autónoma de México
Luis Manuel Rivera
Universidad Autónoma de Zacatecas
DOI: https://doi.org/10.18273/revint.v42n1-2024004

Published 2024-05-16

Keywords

  • Fibonacci and Lucas sequences,
  • Linear form in logarithms,
  • reduction method

How to Cite

Ibarra, S., & Rivera Martínez, L. M. (2024). Fibonacci and Lucas numbers of the form -2^a-3^b-5^c+7^d. Revista Integración, Temas De matemáticas, 42(1), 43–50. https://doi.org/10.18273/revint.v42n1-2024004

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This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

In this note we find all Fibonacci and Lucas numbers of the form -2^a-3^b-5^c+7^d where a, b, c, d are non-negative integers, with 0 ≤ max{a, b, c} ≤ d. This result gives an answer to a question posed by Qu, Zeng and Cao.

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References

  1. Baker A. and Davenport H., “The equations 3X^2 −2 = Y^2 and 8X^2 −7=Z^2”, Q. J. Math. Oxf., 20 (1969), no. 2, 129–137.
  2. Bravo J. J., Gómez C.A. and Luca F. “Powers of two as sums of two k-Fibonacci numbers”, Miskolc Math. Notes, 17 (2016), no. 1, 85–100.
  3. Bugeaud Y., Mignotte M. and Siksek S., “Classical and modular approaches to exponential
  4. Diophantine equations. I. Fibonacci and Lucas perfect powers”, Ann. Math., 163 (2006), 969–1018.
  5. Dujella A. and Petho A., “A generalization of a theorem of Baker and Davenport”, Quart. J. Math. Oxford, 49 (1998), no. 3, 291–306.
  6. García-Lomelí A. C. and Hernández S., “Powers of two as sums of two Padovan numbers”, Integers 18 (2018), Paper No. A84, 11 pp. .
  7. Hernández S. H., “The Fibonacci numbers of the form 2^a ± 2^b + 1”, Fibonacci Quart., 56 (2018), no. 4, 354–359.
  8. Hernández S. H., Luca F. and Rivera L. M., “On Pillai’s problem with the Fibonacci and Pell sequences”, Bol. Soc. Mat. Mex., 25 (2019), no. 3, 495–507.
  9. Qu Y., Zeng J. and Cao Y., “Fibonacci and Lucas Numbers of the Form 2^a + 3^b + 5^c + 7^d”, Symmetry, 10 (2018), no. 509, 7 pp. [9] Luca F., “Fibonacci numbers of the form k^2 + k + 2”, In Proceedings of the Applications of Fibonacci Numbers, Rochester, NY, USA, 22–26 June 1998; Kluwer Academic Publishers: Dordrecht, The Netherlands, Vol. 8, pp. 241–249 (1999).
  10. Luca F. and Szalay L., “Fibonacci numbers of the form pa ± pb + 1”, Fibonacci Quart., 45 (2007), 98–103.
  11. Luo M., “On triangular Fibonacci numbers”, Fibonacci Quart., 27 (1989), 98–108.
  12. Luo M. , “On triangular Lucas numbers”, In Applications of Fibonacci Numbers; Springer: Dordrecht, The Netherlands, 231–240 (1991).
  13. Marques D. and Togbé A., “Fibonacci and Lucas numbers of the form 2^a + 3^b + 5^c”, Proc. Japan Acad. Ser. A Math. Sci., 89 (2013) no. 3, 45–50.
  14. Matveev E. M., “An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II”, Izv. Math., 64 (2000), no. 6, 1217–1269.
  15. Sánchez S. G. and Luca F., “Linear combinations of factorials and S-units in a binary recurrence sequence”, Ann. Math. Québec, 38 (2014), 169–188.