On Some New Arithmetic Properties of the Generalized Lucas Sequences

Abstract

Some arithmetic properties of the generalized Lucas sequences are studied, extending a number of recent results obtained for Fibonacci, Lucas, Pell, and Pell–Lucas sequences. These properties are then applied to investigate certain notions of Fibonacci, Lucas, Pell, and Pell–Lucas pseudoprimality, for which we formulate some conjectures.

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

References

  1. 1.

    Andreescu, T., Andrica, D.: Number Theory. Structures, Examples, and Problems. Birkhäuser, Boston (2009)

    Google Scholar 

  2. 2.

    Andreescu, T., Andrica, D.: Quadratic Diophantine Equations. Developments in Mathematics. Springer, Berlin (2015)

    Google Scholar 

  3. 3.

    Andrejic, V.: On Fibonacci powers. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 17, 38–44 (2006)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Andrica, D., Bagdasar, O.: Recurrent Sequences: Key Results. Applications and Problems. Springer, Berlin (2020)

    Google Scholar 

  5. 5.

    Andrica, D., Crişan, V., Al-Thukair, F.: On Fibonacci and Lucas sequences modulo a prime and primality testing. Arab J. Math. Sci. 24(1), 9–15 (2018)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Borges, A., Catarino, P., Aires, A.P., Vasco, P., Campos, H.: Two-by-two matrices involving k-Fibonacci and k-Lucas sequences. Appl. Math. Sci. 8(34), 1659–1666 (2014)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Bruckman, P.S.: On the infinitude of Lucas pseudoprimes. Fibonacci Q. 32(2), 153–154 (1994)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Catarino, P., Vasco, P.: On dual \(k\)-Pell quaternions and octonions. Mediterr. J. Math. 14, 75 (2017)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Catarino, P.: A note involving two-by-two matrices of the \(k\)-Pell and \(k\)-Pell–Lucas sequences. Int. Math. Forum 8(32), 1561–1568 (2013)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Catarino, P., Vasco, P., Borges, A., Campos, H., Aires, A.P.: Sums, products and identities involving \(k\)-Fibonacci and \(k\)-Lucas sequences. JP J. Algebra Number Theory Appl. 32(1), 63–77 (2014)

    MATH  Google Scholar 

  11. 11.

    Crandall, R., Dilcher, K., Pomerance, C.: A search for Wieferich and Wilson primes. Math. Comput. 66(5), 433–449 (1997)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Crandall, R., Pomerance, C.: Prime Numbers: A Computational Perspective, 2nd edn. Springer, New York (2005)

    Google Scholar 

  13. 13.

    Falcon, S., Plaza, A.: On the Fibonacci \(k\)-numbers. Chaos Soliton Fract. 32(5), 1615–1624 (2007)

    Article  Google Scholar 

  14. 14.

    Falcon, S.: On the Lucas triangle and its relationship with the \(k\)-Lucas numbers. J. Math. Comput. Sci. 2, 425–434 (2012)

    MathSciNet  Google Scholar 

  15. 15.

    Falcon, S.: Relationships between some \(k\)-Fibonacci sequences. Appl. Math. 5, 2226–2234 (2014)

    Article  Google Scholar 

  16. 16.

    Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34(3), 596–615 (1987)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Halton, J.: Some properties associated with square Fibonacci numbers. Fibonacci Q. 5(4), 347–354 (1967)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Hosoya, H.: What can mathematical chemistry contribute to the development of mathematics? Int. J. Philos. Chem. 19(1), 87–105 (2013)

    Google Scholar 

  19. 19.

    Jancić, M.: On linear recurrence equations arising from compositions of positive integers. J. Int. Seq. 18, Article 15.4.7 (2015)

  20. 20.

    Jaroma, J.H.: Note on the Lucas–Lehmer test. Ir. Math. Soc. Bull. 54, 63–72 (2004)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Kiefer, J.: Sequential minimax search for a maximum. Proc. Am. Math. Soc. 4, 502–506 (1953)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Kiss, P., Phong, B.M., Lieuwens, E.: On Lucas pseudoprimes which are products of s primes. In: Philippou, A.N., Bergum, G.E., Horadam, A.F. (eds.) Fibonacci Numbers and Their Applications, vol. 1, pp. 131–139. Reidel, Dordrecht (1986)

    Google Scholar 

  23. 23.

    Knuth, D.E.: The Art of Computer Programming, vol. 3, 2nd edn. Addison Wesley, Boston (2003)

  24. 24.

    Koshy, T.: Fibonacci and Lucas Numbers with Applications. Wiley, Hoboken (2001)

    Google Scholar 

  25. 25.

    Lehmer, D. H.: An extended theory of Lucas’ functions. Ann. Math. 2nd Ser. 2(3), 419–448 (1930)

  26. 26.

    Lehmer, E.: On the infinitude of Fibonacci pseudoprimes. Fibonacci Q. 2(3), 229–230 (1964)

    MATH  Google Scholar 

  27. 27.

    Noonea, C.J., Torrilhonb, M., Mitsosa, A.: Heliostat field optimization: a new computationally efficient model and biomimetic layout. Sol. Energy 86(2), 792–803 (2012)

    Article  Google Scholar 

  28. 28.

    OEIS Foundation Inc.: The On-Line Encyclopedia of Integer Sequences. http://oeis.org. Accessed 15 Nov 2019

  29. 29.

    Rabago, J.F.T.: On \(k\)-Fibonacci numbers with applications to continued fractions. Proc. ICMAME 2015 J. Phys. Conf. Ser. 693, 012005 (2016)

  30. 30.

    Rotkiewicz, A.: Lucas and Frobenius pseudoprimes. Ann. Math. Sil. 17, 17–39 (2003)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Schuster, S., Fitchner, M., Sasso, S.: Use of Fibonacci numbers in lipidomics—enumerating various classes of fatty acids. Nat. Sci. Rep. 7, 39821 (2017)

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ovidiu Bagdasar.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Andrica, D., Bagdasar, O. On Some New Arithmetic Properties of the Generalized Lucas Sequences. Mediterr. J. Math. 18, 47 (2021). https://doi.org/10.1007/s00009-020-01653-w

Download citation

Keywords

  • Generalized Lucas sequences
  • Pell sequence
  • Pell–Lucas sequence
  • Legendre symbol
  • Pseudoprimality

Mathematics Subject Classification

  • 11A51
  • 11B39
  • 11B50