Skip to main content
SpringerLink
Log in

On the primitive divisors of the recurrent sequence \(u_{n+1}=(4\rm{cos}^2(2\pi/7)-1)\it{u}_{n}-u_{n-\rm{1}}\) with applications to group theory

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

Consider the sequence of algebraic integers un given by the starting values u0 = 0, u1 = 1 and the recurrence \(u_{n+1}=(4\rm{cos}^2(2\pi/7)-1)\it{u}_{n}-u_{n-\rm{1}}\). We prove that for any n ∉ {1, 2, 3, 5, 8, 12, 18, 28, 30} the n-th term of the sequence has a primitive divisor in \(\mathbb{Z}[2\rm{cos}(2\pi/7)]\). As a consequence we deduce that for any suffciently large n there exists a prime power q such that the group PSL2(q) can be generated by a pair x, y with \(x^2=y^3=(xy)^7=1\) and the order of the commutator [x, y] is exactly n. The latter result answers in affrmative a question of Holt and Plesken.

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. Bilu Y, Hanrot G, Voutier P M. Existence of primitive divisors of Lucas and Lehmer numbers: With an appendix by M. Mignotte. J Reine Angew Math, 2001, 539: 75–122

    MATH  Google Scholar 

  2. Birkhoff G D, Vandiver H S. On the integral divisors of a n - b n. Ann of Math (2), 1904, 5: 173–180

    Article  MathSciNet  Google Scholar 

  3. Borevich Z I, Shafarevich I R. Number Theory. London: Academic Press, 1966

    MATH  Google Scholar 

  4. Bosma W, Cannon J, Playoust C. The Magma algebra system, I: The user language. J Symbolic Comput, 1997, 24: 235–265

    Article  MathSciNet  MATH  Google Scholar 

  5. Burnside W. Note on the simple group of order 504. Math Ann, 1889, 52: 174–176

    Article  MathSciNet  MATH  Google Scholar 

  6. Carmichael P D. On the numerical factors of the arithmetic forms α n ± β n. Ann of Math (2), 1913, 15: 30–70

    Article  MathSciNet  Google Scholar 

  7. Conder M D E. Hurwitz groups: A brief survey. Bull Amer Math Soc (NS), 1990, 23: 359–370

    Article  MathSciNet  MATH  Google Scholar 

  8. Conder M D E. A question of Graham Higman concerning quotients of the (2, 3, 7) triangle group. J Algebra, 1991, 141: 275–286

    Article  MathSciNet  MATH  Google Scholar 

  9. Conder M D E. An update on Hurwitz groups. Groups Complex Cryptol, 2010, 2: 35–49

    Article  MathSciNet  MATH  Google Scholar 

  10. Coxeter H S M, Moser W O J. Generators and Relations for Discrete Groups. Berlin: Springer-Verlag, 1972

    Book  MATH  Google Scholar 

  11. Edjvet M. An example of an infinite group. In: Discrete Groups and Geometry. London Mathematical Society Lecture Note Series, vol. 173. Cambridge: Cambridge University Press, 1992, 66–74

    MathSciNet  MATH  Google Scholar 

  12. Fricke R. Ueber den arithmetischen Charakter der zu den Verzweigungen (2, 3, 7) und (2, 4, 7) gehörenden Dreiecksfunctionen. Math Ann, 1893, 41: 443–468

    Article  MATH  Google Scholar 

  13. Fricke R. Ueber eine einfache Gruppe von 504 Oprationen. Math Ann, 1899, 52: 321–339

    Article  MathSciNet  MATH  Google Scholar 

  14. Holt D F, Plesken W. A cohomological criterion for a finitely presented group to be infinite. J Lond Math Soc (2), 1992, 45: 469–480

    Article  MathSciNet  MATH  Google Scholar 

  15. Holt D F, Plesken W, Souvignier B. Constructing a representation of the group (2, 3, 7, 11). J Symbolic Comput, 1997, 24: 489–492

    Article  MathSciNet  MATH  Google Scholar 

  16. Howie J, Thomas R M. The groups (2, 3, p; q); asphericity and a conjecture of Coxeter. J Algebra, 1993, 154: 289–309

    Article  MathSciNet  MATH  Google Scholar 

  17. Leech J. Generators for certain normal subgroups of (2, 3, 7). Math Proc Cambridge Philos Soc, 1965, 61: 321–332

    Article  MathSciNet  MATH  Google Scholar 

  18. Leech J. Note on the abstract group (2, 3, 7; 9). Math Proc Cambridge Philos Soc, 1966, 62: 7–10

    Article  MathSciNet  MATH  Google Scholar 

  19. Macbeath A M. Generators of the linear fractional groups. Proc Sympos Pure Math, 1969, 12: 14–32

    Article  MathSciNet  MATH  Google Scholar 

  20. Postnikova L P, Schinzel A. Primitive divisors of the expression a n - b n in algebraic number fields (in Russian). Mat Sb, 1968, 75: 171–177; English translation, Math USSR-Sb, 1968, 4: 153–159

    MathSciNet  MATH  Google Scholar 

  21. Rockett A M, Szüsz P. Continued Fractions. Singapore-New Jersey-London-Hong Kong: World Scientific, 1992

    Book  MATH  Google Scholar 

  22. Schinzel A. Primitive divisors of the expression A n - B n in the algebraic number fields. J Reine Angew Math, 1974, 268/269: 27–33

    MATH  Google Scholar 

  23. Sims C C. On the group (2, 3, 7; 9). Notices Amer Math Soc, 1964, 11: 687–688

    Google Scholar 

  24. Stewart C L. Primitive divisors of Lucas and Lehmer sequences. In: Transcendence Theory: Advances and Applications. New York: Academic Press, 1977, 79–92

    Google Scholar 

  25. Strambach K, Völklein H. On linearly rigid tuples. J Reine Angew Math, 1999, 510: 57–62

    MathSciNet  MATH  Google Scholar 

  26. Takeuchi K. Arithmetic triangle groups. J Math Soc Japan, 1977, 29: 91–106

    Article  MathSciNet  MATH  Google Scholar 

  27. Takeuchi K. Commensurability classes of arithmetic triangle groups. J Fac Sci Univ Tokyo, 1977, 24: 201–212

    MathSciNet  MATH  Google Scholar 

  28. Tamburini M C, Vsemirnov M. Hurwitz groups and Hurwitz generation. Handb Algebra, 2006, 4: 385–426

    Article  MathSciNet  MATH  Google Scholar 

  29. Voutier P M. Primitive divisors of Lucas and Lehmer sequences, II. J Théor Nombres Bordeaux, 1996, 8: 251–274

    Article  MathSciNet  MATH  Google Scholar 

  30. Voutier P M. Primitive divisors of Lucas and Lehmer sequences, III. Math Proc Cambridge Philos Soc, 1998, 123: 407–419

    Article  MathSciNet  MATH  Google Scholar 

  31. Vsemirnov M. The groups G 2(p), p ⩾ 5 as quotients of (2, 3, 7, 2p). Transform Groups, 2006, 11: 295–304

    Article  MathSciNet  MATH  Google Scholar 

  32. Vsemirnov M, Mysovskikh V, Tamburini M C. Triangle groups as subgroups of unitary groups. J Algebra, 2001, 245: 562–583

    Article  MathSciNet  MATH  Google Scholar 

  33. Zsigmondy K. Zur Theorie der Potenzreste. Monatsh Math, 1892, 3: 265–284

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the Government of the Russian Federation (Grant No. 14.Z50.31.0030). The author is grateful to an anonymous referee, whose comments and suggestions helped to improve the presentation of the paper.

Author information

Authors and Affiliations

  1. St. Petersburg Department of V.A. Steklov Institute of Mathematics, St. Petersburg, 191023, Russia

    Maxim Vsemirnov

  2. Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, 198504, Russia

    Maxim Vsemirnov

Authors
  1. Maxim Vsemirnov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maxim Vsemirnov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vsemirnov, M. On the primitive divisors of the recurrent sequence \(u_{n+1}=(4\rm{cos}^2(2\pi/7)-1)\it{u}_{n}-u_{n-\rm{1}}\) with applications to group theory. Sci. China Math. 61, 2101–2110 (2018). https://doi.org/10.1007/s11425-017-9347-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-017-9347-3

Keywords

MSC(2010)

Search

Navigation