Skip to main content
SpringerLink
Log in

Fibonacci Numbers of the Form k2+k+2

  • Chapter
Book cover Applications of Fibonacci Numbers

Abstract

Cohn (see [1]) determined all the Fibonacci and the Lucas numbers which are squares and London & Finkelstein (see [3]) found all the Fibonacci and the Lucas numbers which are cubes. Luo Ming (see [4], [5], [6], [7]) has determined all the Fibonacci and the Lucas numbers which are either triangular or pentagonal and Williams (see [9]) found all the Fibonacci numbers which are of the form k 2 +1 for some integer k. Stark (see [2]), or [8]) asks which Fibonacci numbers are half the difference (or sum) of two cubes.

This work was partially supported by the Alexander von Humboldt Foundation.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Cohn, J.H.E. “On square Fibonacci numbers.” J. London Math. Soc., Vol. 39 (1964): pp. 537–540.

    Article  MathSciNet  MATH  Google Scholar 

  2. Guy, R.K. Unsolved Problems in Number Theory. Springer-Verlag, New York, (1981): p. 106.

    MATH  Google Scholar 

  3. London, H. and Finkelstein, R. “On Fibonacci and Lucas numbers which are perfect powers.” The Fibonacci Quarterly, Vol. 7 (1969): pp. 476–481, 487. (Errata The Fibonacci Quarterly, Vol. 8 (1970): p. 248).

    Google Scholar 

  4. Ming, L. “On triangular Fibonacci numbers.” The Fibonacci Quarterly, Vol. 27 (1989): pp. 98–108.

    MathSciNet  MATH  Google Scholar 

  5. Ming, L. “On triangular Lucas numbers.” Applications of Fibonacci Numbers, Volume 4. Edited by G.E. Bergum, A.F. Horadam and A.N. Philippou. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991: pp. 98–108.

    Google Scholar 

  6. Ming, L. “Pentagonal numbers in the Fibonacci sequence.” Applications of Fibonacci Numbers. Volume 6. Edited by G.E. Bergum, A.F. Horadam and A.N. Philippou. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994: pp. 349–354.

    Google Scholar 

  7. Ming, L. “Pentagonal numbers in the Lucas sequence.” Portugaliae Mat., Vol. 53 (1996): pp. 325–329.

    MATH  Google Scholar 

  8. Stark, H.M. Problem 23, Summer Institute on Number Theory, Stony Brook, 1969.

    Google Scholar 

  9. Williams, H.C. “On Fibonacci numbers of the form k 2 + 1.” The Fibonacci Quarterly, Vol. 13 (1975): pp. 213–214.

    MathSciNet  MATH  Google Scholar 

Download references

Authors
  1. Florian Luca
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics and Computer Science, Wake Forest University, Winston-Salem, North Carolina, USA

    Fredric T. Howard

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Luca, F. (1999). Fibonacci Numbers of the Form k2+k+2. In: Howard, F.T. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4271-7_24

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-4271-7_24

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5851-3

  • Online ISBN: 978-94-011-4271-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation