Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T03:10:05.113Z Has data issue: false hasContentIssue false
  • English
  • Français

On factorials expressible as sums of at most three Fibonacci numbers

Part of: Diophantine equations

Published online by Cambridge University Press:  05 August 2010

Florian Luca  and
Samir Siksek
Florian Luca
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, CP 58089, Morelia, Michoacán, México ([email protected])
Samir Siksek
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we determine all the factorials that are a sum of at most three Fibonacci numbers.

Keywords

diophantine equationFibonacci numbersfactorials

MSC classification

Secondary: 11D85: Representation problems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 53 , Issue 3 , October 2010 , pp. 747 - 763
Copyright
Copyright © Edinburgh Mathematical Society 2010

References

1. Berend, D. and Harmse, J. E., On polynomial-factorial Diophantine equations, Trans. Am. Math. Soc. 358(4) (2006), 1741–1779.CrossRefGoogle Scholar
2. Bilu, Yu, Hanrot and P. M. Voutier, G., Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math. 539 (2001), 75–122.Google Scholar
3. Bollman, M., Sums of consecutive factorials in the Fibonacci sequence, Congr. Numer. 194 (2009), 77–83.Google Scholar
4. Borevich, Z. I. and Shafarevich, I. R., Number theory (Academic Press, 1966).Google Scholar
5. Bosma, W., Cannon, J. and Playoust, C., The Magma algebra system I: the user language, J. Symb. Computat. 24 (1997), 235–265 (see also www.maths.usyd.edu.au)Google Scholar
6. Brocard, H., Question 166, Nouv. Corresp. Math. 2 (1876), 287.Google Scholar
7. Bugeaud, Y., Lucas, F., Mignotte, M. and Siksek, S., Perfect powers from products of terms in Lucas sequences, J. Reine Angew. Math. 611 (2007), 109–129.Google Scholar
8. Bugeaud, Y., Mignotte, M. and Siksek, S., Classical and modular approaches to exponential Diophantine equations, I, Fibonacci and Lucas perfect powers, Annals Math. 163 (2006), 969–1018.CrossRefGoogle Scholar
9. Cassels, J. W. S., Local fields, London Mathematical Society Student Texts, Volume 3 (Cambridge University Press, 1986).Google Scholar
10. Cipu, M., Luca, F. and Mignotte, M., Solutions of the Diophantine equation aux + bvy + cwz = n!, Annals Sci. Math. Québec 31 (2007), 127–137.Google Scholar
11. Erdʺos, P. and Obláth, R., Über diophantische Gleichungen der Form n! = xp ± yp und n! ± m! = xp, Acta Sci. Math. (Szeged) 8 (1937), 241–255.Google Scholar
12. Grossman, G. and Luca, F., Sums of factorials in binary recurrence sequences, J. Number Theory 93(2) (2002), 87–107.Google Scholar
13. Guy, R. K., Unsolved problems in number theory, 3rd edn (Springer, 2004).Google Scholar
14. Knott, R., Fibonacci numbers and the golden section, www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/.Google Scholar
15. Luca, F., The number of nonzero digits of n!, Can. Math. Bull. 45(1) (2002), 115–118.CrossRefGoogle Scholar
16. Luca, F. and Stӑnicӑ, P., F1F2F3F4F5F6F8F10F12 = 11!, Portugaliae Math. 63 (2006), 251–260.Google Scholar
17. Ramanujan, S., Question 469, J. Indian Math. Soc. 5 (1913), 59.Google Scholar
18. Rosser, J. B. and Schoenfeld, L., Sharper bounds for the Chebyshev functions θ(x) and Ψ(x), Math. Comp. 29 (1975), 243–269.Google Scholar