Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
On Nearly Linear Recurrence Sequences Read chapter Read first chapter

2017 | OriginalPaper | Chapter

On the Monoid Generated by a Lucas Sequence

Authors : Clemens Heuberger, Stephan Wagner

Published in: Number Theory – Diophantine Problems, Uniform Distribution and Applications

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

A Lucas sequence is a sequence of the general form \(v_{n} = (\phi ^{n} -\overline{\phi }^{n})/(\phi -\overline{\phi })\), where ϕ and \(\overline{\phi }\) are real algebraic integers such that \(\phi +\overline{\phi }\) and \(\phi \overline{\phi }\) are both rational. Famous examples include the Fibonacci numbers, the Pell numbers, and the Mersenne numbers. We study the monoid that is generated by such a sequence; as it turns out, it is almost freely generated. We provide an asymptotic formula for the number of positive integers ≤ x in this monoid, and also prove Erdős–Kac type theorems for the distribution of the number of factors, with and without multiplicity. While the limiting distribution is Gaussian if only distinct factors are counted, this is no longer the case when multiplicities are taken into account.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Quasi-Equivalence of Heights and Runge’s Theorem
next chapter Measures of Pseudorandomness: Arithmetic Autocorrelation and Correlation Measure
Literature
1.
T.M. Apostol, Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics (Springer, New York/Heidelberg, 1976)
2.
Y. Bilu, G. Hanrot, P.M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers. J. Reine Angew. Math. 539, 75–122 (2001). With an appendix by M. Mignotte (2002j:11027)
3.
4.
5.
P.D.T.A. Elliott, Probabilistic Number Theory. II. Central Limit Theorems. Grundlehren der Mathematischen Wissenschaften, vol. 240 (Springer, Berlin/New York, 1980)
6.
P. Erdős, M. Kac, The Gaussian law of errors in the theory of additive number theoretic functions. Am. J. Math. 62, 738–742 (1940)MathSciNetCrossRefMATH
7.
P. Erdős, J. Lehner, The distribution of the number of summands in the partitions of a positive integer. Duke Math. J. 8, 335–345 (1941)MathSciNetCrossRefMATH
8.
P. Flajolet, H. Prodinger, Register allocation for unary-binary trees. SIAM J. Comput. 15(3), 629–640 (1986) (87j:68052)
9.
P. Flajolet, P. Grabner, P. Kirschenhofer, H. Prodinger, R.F. Tichy, Mellin transforms and asymptotics: digital sums. Theor. Comput. Sci. 123, 291–314 (1994)MathSciNetCrossRefMATH
10.
11.
W.M.Y. Goh, E. Schmutz, The number of distinct part sizes in a random integer partition. J. Comb. Theory Ser. A 69(1), 149–158 (1995)MathSciNetCrossRefMATH
12.
J. Knopfmacher, Abstract Analytic Number Theory (North-Holland/American Elsevier, Amsterdam/Oxford/New York, 1975)MATH
13.
J. Knopfmacher, W.-B. Zhang, Number Theory Arising from Finite Fields. Monographs and Textbooks in Pure and Applied Mathematics, vol. 241 (Marcel Dekker, New York, 2001)
14.
F. Luca, Š. Porubský, The multiplicative group generated by the Lehmer numbers. Fibonacci Quart. 41(2), 122–132 (2003)MathSciNetMATH
15.
16.
17.
F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (eds.), NIST Handbook of Mathematical Functions (Cambridge University Press, New York, 2010)MATH
18.
19.
20.
E.T. Whittaker, G.N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, 1996). Reprint of the fourth edition (1927)
Metadata
Title
On the Monoid Generated by a Lucas Sequence
Authors
Clemens Heuberger
Stephan Wagner
Copyright Year
2017
Book
Number Theory – Diophantine Problems, Uniform Distribution and Applications

Print ISBN: 978-3-319-55356-6

Electronic ISBN: 978-3-319-55357-3

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-55357-3_14

Premium Partner

    Image Credits