Skip to main content

Linear Recurrence Sequences

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1819)

Abstract

The best known linear recurrence sequence is the Fibonacci Sequence 1,1,2,3,5,8,..., where each term (after the first two terms) is a sum of the two preceding terms. We may extend it to become a “doubly infinite” sequence

$$ ..., - 8,5, - 3,2, - 1,1,0,1,1,2,3,5,8,... . $$

.

Keywords

  • Exponential Type
  • Irreducible Factor
  • Fibonacci Sequence
  • Acta Arith
  • Irreducible Element

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Bavencoffe et J.-P. Bézivin. Une Famille Remarquable de Suites Recurrentes Linéaires. Monatsh. f. Math. 120 (1995), 189–203.

    CrossRef  MATH  Google Scholar 

  2. F. Beukers. The zero multiplicity of ternary sequences. Compositio Math. 77 (1991), 165–177.

    MATH  MathSciNet  Google Scholar 

  3. F. Beukers and H. P. Schlickewei. The equation x + y = 1 in finitely generated groups. Acta Arith. 78 (1996), 189–199.

    MATH  MathSciNet  Google Scholar 

  4. E. Bombieri, J. Mueller and U. Zannier. Equations in one variable over function fields. Acta Arith. 99 (2001), 27–39.

    MATH  MathSciNet  Google Scholar 

  5. S. David et P. Philippon. Minorations des hauteurs normalisées des sous-variétés des tores. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 489–543. Errata, ibid. 29 (2000), 729–731.

    MATH  MathSciNet  Google Scholar 

  6. E. Dobrowolski. On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith. 34 (1979), 391–401.

    MATH  MathSciNet  Google Scholar 

  7. P. Erdős, C. L. Stewart and R. Tijdeman. Some diophantine equations with many solutions. Compositio Math. 66 (1988), 37–56.

    MathSciNet  Google Scholar 

  8. G. R. Everest and A. J. van der Poorten. Factorisation in the ring of exponential polynomials. Proc. Amer. Math. Soc. 125 (1997), 1293–1298.

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. J.-H. Evertse. On equations in S-units and the Thue-Mahler equation. Invent. Math. 75 (1984), 561–584.

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. J.-H. Evertse. On sums of S-units and linear recurrences. Compositio Math. 53 (1984), 225–244.

    MATH  MathSciNet  Google Scholar 

  11. J.-H. Evertse. The number of solutions of decomposable form equations. Invent. Math. 122 (1995), 559–601.

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. J.-H. Evertse. The number of solutions of linear equations in roots of unity. Acta Arith. 89 (1999), 45–51.

    MATH  MathSciNet  Google Scholar 

  13. J.-H. Evertse and H. P. Schlickewei. A quantitative version of the Absolute Subspace Theorem. J. reine angew. Math. 548 (2002), 21–127.

    MATH  MathSciNet  Google Scholar 

  14. J.-H. Evertse, H. P. Schlickewei and W. M. Schmidt. Linear equations in variables which lie in a multiplicative group. Ann. of Math. (2) 155 (2002), 807–836.

    MATH  MathSciNet  Google Scholar 

  15. E. Gourin. On Irreducible Polynomials in Several Variables Which Become Reducible When the Variables Are Replaced by Powers of Themselves. Trans. Amer. Math. Soc. 32 (1930), 485–501.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. S. Lang. Integral points on curves. Publ. Math. I.H.E.S. 6 (1960), 27–43.

    Google Scholar 

  17. M. Laurent. Équations exponentielles polynômes et suites réecurrentes linéaires. Journées arithmétiques de Besançon. Astérisque 147–148 (1987), 121–139, 343–344. Équations exponentielles polynômes et suites récurrentes linéaires, II. J. Number Theory 31 (1989), 24–53.

    MathSciNet  Google Scholar 

  18. C. Lech. A note on recurring series. Ark. Math. 2 (1953), 417–421.

    CrossRef  MATH  MathSciNet  Google Scholar 

  19. K. Mahler. Zur Approximation algebraischer Zahlen I. Über den grössten Primteiler binärer Formen. Math. Ann. 107 (1933), 691–730.

    CrossRef  MATH  MathSciNet  Google Scholar 

  20. K. Mahler. Eine arithmetische Eigenschaft der Taylor-Koeffizienten rationaler Funktionen. Proc. Akad. Wetensch. Amsterdam 38 (1935), 50–60.

    MATH  Google Scholar 

  21. K. Mahler. On the Taylor coefficients of rational functions. Proc. Camb. Phil. Soc. 52 (1956) 39–48. Addendum 53, 544.

    MATH  MathSciNet  Google Scholar 

  22. H. B. Mann. On linear relations between roots of unity. Mathematika 12 (1965), 107–117.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. A. J. van der Poorten. Solution de la conjecture de Pisot sur le quotient de Hadamard de deux fractions rationnelles. C. R. Acad. Sci. Paris 306 (1988), Série I, 97–102.

    MATH  Google Scholar 

  24. A. J. van der Poorten. Some facts that should be better known, especially about rational functions. Macquarie Mathematics Reports No. 88-0022 (1988).

    Google Scholar 

  25. A. J. van der Poorten and H. P. Schlickewei. Additive relations in fields. J. Austral. Math. Soc. (Ser. A) 51 (1991), 154–170.

    CrossRef  MATH  MathSciNet  Google Scholar 

  26. J. F. Ritt. A factorization theory for functions \( \sum\nolimits_{i = 1}^n {a_i e^{\alpha _i x} } \) . Trans. Amer. Math. Soc. 29 (1929), 584–596.

    CrossRef  MathSciNet  Google Scholar 

  27. C. A. Rogers. A note on coverings and packings. J. London Math. Soc. 25 (1950), 327–331.

    CrossRef  MATH  MathSciNet  Google Scholar 

  28. R. S. Rumely. Notes on van der Poorten’s proof of the Hadamard quotient theorem. I, II. Séminaire de Théorie des Nombres, Paris 1986–87, 349–382, 383–409. Progr. Math. 75, Birkhäuser, 1988.

    MathSciNet  Google Scholar 

  29. R. S. Rumely and A. J. van der Poorten. A note on the Hadamard kth root of a rational function. J. Austral. Math. Soc. (Ser. A) 43 (1987), 314–327.

    MATH  MathSciNet  Google Scholar 

  30. A. Schinzel. Selected Topics on Polynomials. The University of Michigan Press, 1982.

    Google Scholar 

  31. H. P. Schlickewei. S-unit equations over number fields. Invent. Math. 102 (1990), 95–107.

    CrossRef  MATH  MathSciNet  Google Scholar 

  32. H. P. Schlickewei. Equations ax + by = 1. Manuscript (1996).

    Google Scholar 

  33. H. P. Schlickewei. Equations in roots of unity. Acta Arith. 76 (1996), 99–108.

    MATH  MathSciNet  Google Scholar 

  34. H. P. Schlickewei. Lower Bounds for Heights in Finitely Generated Groups. Monatsh. f. Math. 123 (1997), 171–178.

    CrossRef  MATH  MathSciNet  Google Scholar 

  35. H. P. Schlickewei. The multiplicity of binary recurrences. Invent. Math. 129 (1997), 11–36.

    CrossRef  MATH  MathSciNet  Google Scholar 

  36. H. P. Schlickewei and W. M. Schmidt. On polynomial-exponential equations. Math. Ann. 296 (1993), 339–361.

    CrossRef  MATH  MathSciNet  Google Scholar 

  37. H. P. Schlickewei and W. M. Schmidt. Linear equations in members of recurrence sequences. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), 219–246.

    MATH  MathSciNet  Google Scholar 

  38. H. P. Schlickewei and W. M. Schmidt. Equations au n = bu k m satisfied by members of recurrence sequences. Proc. Amer. Math. Soc. 118 (1993), 1043–1051.

    CrossRef  MATH  MathSciNet  Google Scholar 

  39. H. P. Schlickewei and W. M. Schmidt. The number of solutions of polynomial-exponential equations. Compositio Math. 120 (2000), 193–225.

    CrossRef  MATH  MathSciNet  Google Scholar 

  40. H. P. Schlickewei, W. M. Schmidt and M. Waldschmidt. Zeros of linear recurrence sequences. Manuscripta Math. 98 (1999), 225–241.

    CrossRef  MATH  MathSciNet  Google Scholar 

  41. W. M. Schmidt. Heights of points on subvarieties of \( \mathbb{G}_m^n \) . London Math. Soc. Lecture Note Series 235 (1996), 157–187.

    Google Scholar 

  42. W. M. Schmidt. The zero multiplicity of linear recurrence sequences. Acta Math. 182 (1999), 243–282.

    CrossRef  MATH  MathSciNet  Google Scholar 

  43. W. M. Schmidt. Zeros of linear recurrences. Publicationes Math. Debrecen 56 (2000), 609–630.

    MATH  Google Scholar 

  44. T. N. Shorey and R. Tijdeman. Exponential Diophantine Equations. Cambridge Tracts in Mathematics 87, Cambridge Univ. Press, 1986.

    Google Scholar 

  45. Th. Skolem. Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und diophantischer Gleichungen. (8. Skand. Mat. Kongr. Stockholm 1934) (1935), 163–188.

    Google Scholar 

  46. P. Voutier. An effective lower bound for the height of algebraic numbers. Acta Arith. 74 (1996), 81–95.

    MathSciNet  Google Scholar 

  47. U. Zannier. (Unpublished 1998 manuscript on polynomial-exponential equations where some α i/α j is transcendental).

    Google Scholar 

  48. U. Zannier. A proof of Pisot’s dth root conjecture. Ann. of Math. (2) 151 (2000), 375–383.

    CrossRef  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Schmidt, W.M. (2003). Linear Recurrence Sequences. In: Amoroso, F., Zannier, U. (eds) Diophantine Approximation. Lecture Notes in Mathematics, vol 1819. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44979-5_4

Download citation

  • DOI: https://doi.org/10.1007/3-540-44979-5_4

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40392-0

  • Online ISBN: 978-3-540-44979-9

  • eBook Packages: Springer Book Archive