Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

REPDIGITS AS SUM OF FIBONACCI AND LUCAS NUMBERS

  • SALIFOU NIKIEMA (Department of Mathematics and Informatic, University Ledea Bernard OUEDRAOGO) ;
  • CHARLES WEND-WAOGA TOUGMA (Department of Mathematics, UFR Sciences et Technologies, University Thomas SANKARA)
  • Received : 2025.06.20
  • Accepted : 2026.03.03
  • Published : 2026.05.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we explore the Diophantine equation $$d(\frac{10^m-1}{9})=F_n+L_p$$ where (Fn)n∈ℕ and (Ln)n∈ℕ denote the Fibonacci and Lucas sequences, respectively, d ∈ {1, . . . , 9} and m ≥ 1 is an integer. We present all solutions of this equation.

Keywords

Acknowledgement

The authors would like to thank the anonymous reviewers for their valuable comments and interest in this work.

References

  1. R.D. Carmichael, On the numerical factors of the arithmetic forms αn ± βn, Ann. Math. 15 (1913), 30-70. https://doi.org/10.2307/1967797
  2. Z. Siary, R. Keskinz, Repdigits As Sums Of Two Lucas Numbers, Applied Mathematics E-Notes 20 (2020), 33-38.
  3. F. Luca, Fibonacci and Lucas numbers with only one distinct digit, Port. Math. 57 (2000), 243-254.
  4. F. Luca, Distinct digits in base b expansions of linear recurrence sequences, Quaest. Math. 23 (2000), 389-404. https://doi.org/10.2989/16073600009485986
  5. F. Luca, Repdigits which are sums of at most three Fibonacci number, Math. Comm. 17 (2012), 1-11.
  6. J.J. Bravo and F. Luca, On a conjecture about repdigits in k-generalized Fibonacci sequences, Publ. Math. Debrecen 82 (2013), 623-639. https://doi.org/10.5486/PMD
  7. B. Faye and F Luca, Pell and Pell-Lucas numbers with only one distinct digit, Ann. Math. Inform. 45 (2015), 55-60.
  8. S.S. Guzman and F. Luca, Linear combinations of factorials and s-units in a binary recurrence sequence, Ann. Math. 38 (2014), 169-188.
  9. F. Erduvan, R. Keskin and F. Luca, Fibonacci and Lucas numbers as difference of two repdigits, Rend. Circ. Mat. Palermo, II. Ser. 71 (2022), 575-589. https://doi.org/10.1007/s12215-021-00645-3
  10. E.M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers, II, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 125-180. English translation in Izv. Math. 64 (2000), 1217-1269.
  11. Y. Bugeaud, M. Mignotte, On integers with identical digits, Mathematika 46 (1999), 411-417. https://doi.org/10.1112/mtk.v46.2
  12. C.W.-W. Tougma, S. Nikiéma, Repdigits As Absolute Value of Difference Between Lucas and Fibonacci Numbers. GJOM 19 (2025), 1-19.
  13. K. Bhoi, P.K. Ray, Repdigits as difference of two Fibonacci or Lucas numbers, Mat. Stud. 56 (2021), 124-132. https://doi.org/10.30970/ms.56.2
  14. Y. Bugeaud, M. Mignotte, and S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. of Math. 163 (2006), 969-1018. https://doi.org/10.4007/annals
  15. S. D. Alvarado, F. Luca, Fibonacci numbers which are sums of two repdigits, Proceedings of the XIVth International Conference on Fibonacci numbers and their applications, Sociedad Matematica Mexicana, Aportaciones Matematicas, Investigacion, 20 (2011), 97-108.
  16. B. V. Normenyo, B. Kafle and A. Togbe, repdigit as sum of two Fibonacci numbers and two Lucas numbers, Integers 19 (2019), 1-16.
  17. D. Marques, A. Togbe, On repdigits as product of consecutioe Fibonacci numbers, Rend. Istit. Mat. Univ. Trieste 44 (2012), 393-397.
  18. A. Dossavi-Yovo, F. Luca, and A. Togbe On the c-coordinates of Pell equations which are rep-digits, Publ. Math. Debrecen 88 (2016), 3-4. https://doi.org/10.5486/PMD
  19. C. Adegbindin, F. Luca, A. Togbe, Lucas numbers as sums of two rep digits, Lith. Math. J. 59 (2019), 295-304. https://doi.org/10.1007/s10986-019-09451-y
  20. B. M. M. de Weger, Algorithms for diophantine equations, Stichting Mathematisch Centrum, 1989.