Skip to main content
SpringerLink
Log in

On the Period Length Modulo p of the Numerators of Convergents for the Square Root of a Prime Number p

  • Conference paper
  • First Online:
Mathematical Modeling and Supercomputer Technologies (MMST 2022)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1750))

Abstract

In this paper, we investigate the properties of the sequence of numerators of convergents for the square root of a prime number. It is proved that the period length L of this sequence modulo p is equal to l,  2l or 4l,  where l is the period length of the continued fraction for the square root of the prime p. Namely, if the remainder of dividing p by 8 is equal to 7,  then \(L=l;\) if the remainder of dividing p by 8 is equal to 3,  then \(L=2l;\) if the remainder of dividing p by 4 is equal to 1,  then \(L=4l.\)

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • 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

Institutional subscriptions

References

  1. Bloom, D.M.: Periodicity in generalized Fibonacci sequences. Amer. Math. Monthly 72(8), 856–861 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brent, R.P.: On the periods of generalized Fibonacci sequences. Math. Comput. 63(207), 389–401 (1994)

    Article  MATH  Google Scholar 

  3. Davenport, H.: The Higher Arithmetic: An Introduction to the Theory of Numbers. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  4. Engstrom, H.T.: On sequences defined by linear recurrence relations. Trans. Am. Math. Soc. 33(1), 210–218 (1931)

    Article  MathSciNet  MATH  Google Scholar 

  5. Falcon, S., Plaza, A.: \(k\)-Fibonacci sequence modulo \(m.\) Chaos, Solitons Fractals 41 (1), 497–504 (2009)

    Google Scholar 

  6. Fulton, J.D., Morris, W.L.: On arithmetical functions related to the Fibonacci numbers. Acta Arith 16, 105–110 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  7. Golubeva, E.P.: Quadratic irrationalities with a fixed length of the expansion period into a continued fraction. Zap. Nauch. Sem. LOMI 196, 5–30 (1991). [in Russian]

    MATH  Google Scholar 

  8. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Oxford University Press, Oxford (1962)

    Google Scholar 

  9. Khinchin, A.Y.: Continued fractions (1978). In Russian

    Google Scholar 

  10. Khovansky, A.N.: Application of continued fractions and their generalizations to questions of approximate analysis, 206 (1956). In Russian

    Google Scholar 

  11. Lagrange, J.L.: Oeuvres de Lagrange. Gauthier Villars, Paris (1877)

    Google Scholar 

  12. Lang, S.: Introduction to Diophantine Approximations. Addison-Wesley, Boston (1966)

    MATH  Google Scholar 

  13. Matthews K.R.: The Diophantine equation \(x^2-Dy^2=N, D>0.\) Expositiones Math. 18, 323–331 (2000)

    Google Scholar 

  14. Nagell, T.: Introduction to Number Theory. Wiley, New York (1981)

    MATH  Google Scholar 

  15. Niven, I., Zuckerman, H.S., Montgomery, H.L.: An Introduction to the Theory of Numbers, 5th edn. Wiley, New York (1991)

    MATH  Google Scholar 

  16. Olds, C.D.: Continued Fractions. Random House, New York (1963)

    Book  MATH  Google Scholar 

  17. Perron, O.: Die Lehre von den Kettenbrüchen. - Stuttgart: Teubner. Bd. l. Elementare Kettenbrüche. - Reprograph. Nachdr. d. 3, verb. u. erw. Aufi. Stuttgart, p. 202 (1977)

    Google Scholar 

  18. Renault, M.: The Fibonacci Sequence Under Various Moduli. Master’s Thesis. Wake Forest University, p. 82 (1996)

    Google Scholar 

  19. Rockett, A.M., Szüsz, P.: Continued Fractions. World Scientific, New York (1992)

    Book  MATH  Google Scholar 

  20. Rosen, K.H.: Elementary Number Theory And Its Applications. Addison-Wesley, Boston (1984)

    MATH  Google Scholar 

  21. Sarma K.V.S.S.R.S.S., Avadhani P.S.: Public Key Cryptosystem based on Pell’s Equation Using The Gnu Mp Library. Int. J. Comput. Sci. Eng. 4, 739–743 (2012)

    Google Scholar 

  22. Wall, D.D.: Fibonacci series modulo \(m.\) Amer. Math. Monthly 67(6), 525–532 (1960)

    Google Scholar 

  23. Ward, M.: The characteristic number of a sequence of integers satisfying a linear recursion relation. Trans. Am. Math. Soc. 33(1), 153–165 (1931)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to anonymous reviewers for valuable comments.

Author information

Authors and Affiliations

  1. Lobachevsky State University of Nizhni Novgorod, Nizhni Novgorod, Russia

    S. V. Sidorov & P. A. Shcherbakov

Authors
  1. S. V. Sidorov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. A. Shcherbakov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. V. Sidorov .

Editor information

Editors and Affiliations

  1. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia

    Dmitry Balandin

  2. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia

    Konstantin Barkalov

  3. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia

    Iosif Meyerov

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Sidorov, S.V., Shcherbakov, P.A. (2022). On the Period Length Modulo p of the Numerators of Convergents for the Square Root of a Prime Number p. In: Balandin, D., Barkalov, K., Meyerov, I. (eds) Mathematical Modeling and Supercomputer Technologies. MMST 2022. Communications in Computer and Information Science, vol 1750. Springer, Cham. https://doi.org/10.1007/978-3-031-24145-1_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-24145-1_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-24144-4

  • Online ISBN: 978-3-031-24145-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation