Skip to main content
SpringerLink
Log in

Positive integer solutions of some Diophantine equations in terms of integer sequences

  • Published:
Afrika Matematika Aims and scope Submit manuscript

Abstract

In this paper, we define some new number sequences, which we represent as \( (B_{n}),(b_{n}),(y_{n})\) and present relations of these new sequences with each other. Then, we give all positive integer solutions of some Diophantine equations in terms of these new sequences.

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. Adler, A., Cloury, J.E.: The Theory of Numbers: A Text and Source Book of Problems. Jones and Bartlett Publishers, Boston (1995)

    Google Scholar 

  2. Behera, A., Panda, G.K.: On the square roots of triangular numbers. Fibonacci Q. 37(2), 98–105 (1999)

    MathSciNet  MATH  Google Scholar 

  3. Kalman, D., Mena, R.: The Fibonacci numbers-exposed. Math. Mag. 76, 167–181 (2003)

    MathSciNet  MATH  Google Scholar 

  4. Karaatlı, O., Keskin, R.: On some Diophantine equations related to square Triangular and Balancing numbers. J. Algebra Number Theory Adv. Appl. 4(2), 71–89 (2010)

    MATH  Google Scholar 

  5. Keskin, R.: Solutions of some quadratic Diophantine equations. Comput. Math. Appl. 60(8), 2225–2230 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Keskin, R., Karaatlı, O.: Some new properties of balancing numbers and square Triangular numbers. J. Integer Seq. 15, 3 (2012) (Article 12.1.4)

  7. McDaniel, W.L.: Diophantine representation of Lucas sequences. Fibonacci Q. 33, 58–63 (1995)

    MathSciNet  MATH  Google Scholar 

  8. Melham, R.: Conics which characterize certain Lucas sequences. Fibonacci Q. 35, 248–251 (1997)

    MathSciNet  MATH  Google Scholar 

  9. Muskat, J.B.: Generalized Fibonacci and Lucas sequences and rootfinding methods. Math. Comput. 61, 365–372 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Nagell, T.: Introduction to Number Theory. Chelsea Publishing Company, New York (1981)

    MATH  Google Scholar 

  11. Panda, G.K., Ray, P.K.: Cobalancing numbers and cobalancers. Int. J. Math. Math. Sci. 8, 1189–1200 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Potter, D.C.D.: Triangular square numbers. Math. Gaz. 56, 109–110 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rabinowitz, S.: Algorithmic manipulation of Fibonacci identities. Appl. Fibonacci Numbers 6, 389–408 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ribenboim, P.: My Numbers, My Friends. Springer, New York (2000)

    MATH  Google Scholar 

  15. Walker, D.T.: On the Diophantine equation \( mX^{2}-nY^{2}=\pm 1,\). Am. Math. Mon. 74(5), 504–513 (1967)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Sakarya University, Sakarya, Turkey

    Refik Keskin

  2. Department of Mathematics, Bingöl University, Bingöl, Turkey

    Zafer Şiar

Authors
  1. Refik Keskin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zafer Şiar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zafer Şiar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Keskin, R., Şiar, Z. Positive integer solutions of some Diophantine equations in terms of integer sequences. Afr. Mat. 30, 181–194 (2019). https://doi.org/10.1007/s13370-018-0632-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13370-018-0632-y

Keywords

Mathematics Subject Classification

Search

Navigation