On the Laxton group

  • Miho AokiEmail author
  • Masanari Kida


In 1969, Laxton defined a multiplicative group structure on the set of rational sequences satisfying a fixed linear recurrence of degree two. He also defined some natural subgroups of the group, and determined the structures of their quotient groups. Nothing has been known about the structure of Laxton’s whole group and its interpretation. In this paper, we redefine his group in a natural way and determine the structure of the whole group, which clarifies Laxton’s results on the quotient groups. This definition makes it possible to use the group to show various properties of such sequences.


Laxton groups Linear recurrence sequences Quadratic fields 

Mathematics Subject Classification

11B37 11B39 11R11 


Author's contributions


The authors would like to thank the referee for useful suggestions.


  1. 1.
    Ballot, C.: Density of prime divisors of linear recurrences. Mem. Am. Math. Soc. 115(551), viii+102 (1995)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Carmichael, R. D.: On the numerical factors of the arithmetic forms \(\alpha ^n \pm \beta ^n\), Ann. Math. 2(15), 30–70 (1913, 1914)Google Scholar
  3. 3.
    Hall, M.: An isomorphism between linear recurring sequences and algebraic rings. Trans. Am. Math. Soc. 44(2), 196–218 (1938)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Laxton, R.R.: On groups of linear recurrences. I. Duke Math. J. 36, 721–736 (1969)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Laxton, R.R.: On groups of linear recurrences. II. Elements of finite order. Pacif. J. Math. 32, 173–179 (1970)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Lucas, E.: Théorie des fonctions numériques simplement périodiques, Am. J. Math. 1, 184–240 and 289–321 (1878)Google Scholar
  7. 7.
    Ribenboim, P.: The New Book of Prime Number Records. Springer, New York (1996)CrossRefGoogle Scholar
  8. 8.
    Koshy, T.: Fibonacci and Lucas Numbers with Applications, Pure and Applied Mathematics. Wiley, New York (2001)zbMATHGoogle Scholar
  9. 9.
    Silverman, J.H.: The Arithmetic of Elliptic Curves, GTM 106. Springer, New York (1986)CrossRefGoogle Scholar
  10. 10.
    Suwa, N: Geometric aspects of Lucas sequences, I, to appear in Tokyo J. Math.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Interdisciplinary Faculty of Science and EngineeringShimane UniversityMatsueJapan
  2. 2.Department of Mathematics, Faculty of Science Division ITokyo University of ScienceTokyoJapan

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