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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

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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.

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References

  1. 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

  2. 2

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

  3. 3

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

  4. 4

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

  5. 5

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

  6. 6

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

  7. 7

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

  8. 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

  9. 9

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

  10. 10

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

  11. 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

  12. 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

  13. 13

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

  14. 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

  15. 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

  16. 16

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

  17. 17

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

  18. 18

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

  19. 19

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

  20. 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

  21. 21

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

  22. 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

  23. 23

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

  24. 24

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

  25. 25

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

  26. 26

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

  27. 27

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

  28. 28

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

  29. 29

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

  30. 30

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

  31. 31

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

  32. 32

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

  33. 33

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

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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.

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Correspondence to Maxim Vsemirnov.

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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

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Keywords

  • recurrent sequences
  • primitive divisors
  • Hurwitz groups

MSC(2010)

  • 11B37
  • 20F05