Skip to main content
Log in

On the Diophantine equation \(Cx^{2}+D=2y^{q}\)

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

Let C and D denote positive integers such that \(CD>1\). In this paper we investigate the solvability of the Diophantine equation \(Cx^{2}+D=2y^{q}\), in positive integers xy and odd prime number q where \(CD\not \equiv 3 \pmod 4\) and CD is squarefree.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Abu Muriefah, F.S., Luca, F., Siksek, S., Tengely, Sz: On the Diophantine equation \(x^{2}+C=2y^{n}\). Int. J. Number Theory 5(6), 1117–1128 (2009)

    Article  MathSciNet  Google Scholar 

  2. Bilu, Y., Hanrot, G., Voutier, P.M.: Existence of primitive divisors of Lucas and Lehmer numbers. J. Reine Angew. Math. 539, 75–122 (2001)

    MathSciNet  MATH  Google Scholar 

  3. Cesaro, E.: Solution to problem \(570 \) (proposed by Gelin). Nouv. Corresp. Math. 6, 423–24 (1880)

    Google Scholar 

  4. Cohn, J.H.E.: Perfect Pell powers. Glasgow Math. J. 38, 19–20 (1996)

    Article  MathSciNet  Google Scholar 

  5. Debnath, L.: A short history of the Fibonacci and golden numbers with their applications. Internat. J. Math. Edu. Sci. Tech. 42(3), 337–367 (2011)

    Article  MathSciNet  Google Scholar 

  6. Keskin, R., Yosma, Z.: On Fibonacci and Lucas numbers of the form \(Cx^{2}\). J. Int. Seq. 14, 1–12 (2011)

    MATH  Google Scholar 

  7. Ljunggren, W.: On the Diophantine equation \(Cx^{2}+D=2y^{n}\). Math. Scand. 18, 69–86 (1966)

    Article  MathSciNet  Google Scholar 

  8. McDaniel, W.L.: The G. C. D. in Lucas sequences and Lehmer number sequences. Fibonacci Q. 29(1991), 24–30 (1988)

    MathSciNet  MATH  Google Scholar 

  9. Pink, I., Tengely, S.: Full powers in arithmetic progressions. Publ. Math. Debr. 57, 535–545 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Robbins, N.: Fibonacci numbers of the form \(Cx^{2}\), where \( C < 1000\). Fibonacci Q. 28, 306–315 (1990)

    MATH  Google Scholar 

  11. Schinzel, A., Tijdeman, R.: On the Diophantine equation \(y^{m}=P(x)\). Acta Arith. 31, 199–204 (1976)

    Article  MathSciNet  Google Scholar 

  12. Tengely, S.: On the Diophantine equation \(x^{2}+a^{2}=2y^{n}\). Indag. Math. 15, 291–304 (2004)

    Article  MathSciNet  Google Scholar 

  13. Tengely, S.: On the Diophantine equation \(x^{2}+q^{2}=2y^{p}\). Acta Arith. 127, 71–86 (2007)

    Article  MathSciNet  Google Scholar 

  14. Zhu, H., Le, M., Togbe, A.: On the exponential Diophantine equation \(x^{2}+p^{2m}=2y^{n}\). Bull. Aust. Math. Soc. 86, 303–314 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Fadwa S. Abu Muriefah.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghanmi, N., Abu Muriefah, F.S. On the Diophantine equation \(Cx^{2}+D=2y^{q}\). Ramanujan J 53, 389–397 (2020). https://doi.org/10.1007/s11139-019-00165-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-019-00165-w

Keywords

Mathematics Subject Classification

Navigation