Skip to main content
SpringerLink
Log in

k-Pell Numbers as Product of Two Repdigits

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

Let \(k\ge 2\). A generalization of the well-known Pell sequence is the k-Pell sequence. For this sequence the first k terms are \(0,\ldots ,0,1\) and each term afterwards is given by the linear recurrence

$$\begin{aligned} P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)}. \end{aligned}$$

In this manuscript, our main objective is to find all k-Pell numbers which are product of two repdigits. This generalizes a result of Erduvan and Keskin (Pell and Pell–Lucas numbers as products of two repdigits, submitted) regarding Pell numbers with the above property.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adegbindin, C., Luca, F., Togbé, A.: Pell and Pell-Lucas numbers as sums of two repdigits. Bull. Malays. Math. Sci. Soc. 43, 1253–1271 (2020)

    Article  MathSciNet  Google Scholar 

  2. Baker, A., Davenport, H.: The equations \(3x^2 - 2 = y^2\) and \(8x^2 - 7 = z^2\). Q. J. Math. Oxf. Ser. (2) 20, 129–137 (1969)

    Article  Google Scholar 

  3. Bravo, J.J., Herrera, J.L.: Repdigits in generalized Pell sequences. Arch. Math. 56(4), 249–262 (2020)

    MathSciNet  MATH  Google Scholar 

  4. Bravo, J.J., Herrera, J.L., Luca, F.: On a generalization of the Pell sequence. Math. Bohem. (accepted)

  5. de Weger, B.M.M.: Algorithms for Diophantine equations. PhD. Eindhoven University of Technology, Eindhoven (1989)

    MATH  Google Scholar 

  6. Dujella, A., Pethő, A.: A generalization of a theorem of Baker and Davenport. Q. J. Math. Oxf. Ser. (2) 49(195), 291–306 (1998)

    Article  MathSciNet  Google Scholar 

  7. Erduvan, F., Keskin, R.: Fibonacci and Lucas numbers as products of two repdigits. Turk. J. Math. 43, 2142–2153 (2019)

    Article  MathSciNet  Google Scholar 

  8. Erduvan, F., Keskin, R.: Pell and Pell–Lucas numbers as products of two repdigits (submitted)

  9. Faye, B., Luca, F.: Pell and Pell-Lucas numbers with only one distinct digits. Ann. Math. 45, 55–60 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Kiliç, E.: On the usual Fibonacci and generalized order-\(k\) Pell numbers. Ars Combin. 109, 391–403 (2013)

    MathSciNet  Google Scholar 

  11. Kiliç, E.: The Binet formula, sums and representations of generalized Fibonacci \(p\)-numbers. Eur. J. Combin. 29, 701–711 (2008)

    Article  MathSciNet  Google Scholar 

  12. Kiliç, E., Taćsci, D.: The generalized Binet formula, representation and sums of the generalized order-\(k\) Pell numbers. Taiwan. J. Math. 10(6), 1661–1670 (2006)

    Article  MathSciNet  Google Scholar 

  13. Matveev, E.M.: An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers, II. Izv. Math. 64(6), 1217–1269 (2000)

    Article  MathSciNet  Google Scholar 

  14. Normenyo, B., Luca, F., Togbé, A.: Repdigits as sums of three Pell numbers. Period. Math. Hung. 77(2), 318–328 (2018)

    Article  MathSciNet  Google Scholar 

  15. Normenyo, B., Luca, F., Togbé, A.: Repdigits as sums of four Pell numbers. Bol. Soc. Mat. Mex. 25, 249–266 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Institute of Science and Technology, University Center of Mila, Mila, Algeria

    Salah Eddine Rihane

Authors
  1. Salah Eddine Rihane
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rihane, S.E. k-Pell Numbers as Product of Two Repdigits. Mediterr. J. Math. 19, 61 (2022). https://doi.org/10.1007/s00009-022-01983-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-022-01983-x

Keywords

Mathematics Subject Classification

Search

Navigation