On integer solutions of x4 + y4 – 2z4 = 0

Mustafa Asci and Esref Gurel
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 19, 2013, Number 1, Pages 25—36
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Authors and affiliations

Mustafa Asci
Department of Mathematics, Science and Arts Faculty
Pamukkale University, Kınıklı Denizli, Turkey
* Corresponding author

Esref Gurel
Department of Mathematics, Science and Arts Faculty
Pamukkale University, Kınıklı Denizli, Turkey

Abstract

In this study we define and study the Gaussian Jacobsthal and Gaussian Jacobsthal Lucas polynomials. We give generating function, Binet formula, explicit formula, Q matrix, determinantal representations and partial derivation of these polynomials. By defining these Gaussian polynomials for special cases GJn(1) is the Gaussian Jacobsthal numbers, Gjn(1) is the Gaussian Jacobsthal Lucas numbers defined in [2].

Keywords

  • Jacobsthal polynomials
  • Jacobsthal Lucas polynomials
  • Gaussian Fibonacci numbers

AMS Classification

  • 11A07
  • 11A41
  • 11A51
  • 11B50
  • 11B65
  • 11B75

References

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  2. Asci M., Gurel E. “Gaussian Jacobsthal and Gaussian Jacobsthal Lucas Numbers” (to appear in Ars. Comb).
  3. Berzsenyi, G. Gaussian Fibonacci numbers. Fibonacci Quarterly, Vol. 15, 1977, No. 3, 233–236.
  4. Good, I. J. Complex Fibonacci and Lucas numbers, continued fractions, and the square root of the golden ratio. Fibonacci Quarterly, Vol. 31, 1993, No. 1, 7–20.
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  6. Harman, C. J. Complex Fibonacci numbers. Fibonacci Quarterly, Vol. 19, 1981, No. 1, 82–86.
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  10. Horadam, A. F. Complex Fibonacci Numbers and Fibonacci Quaternions. American Math. Monthly, Vol. 70, 1963, 289–291.
  11. Horadam, A. F., E. M. Horadam. Roots of recurrence-generated polynomials. Fibonacci Quarterly, Vol. 20, 1982, No. 3, 219–226.
  12. Jordan, J. H. Gaussian Fibonacci and Lucas numbers. Fibonacci Quarterly, Vol. 3, 1965, 315–318.
  13. Pethe, S., A. F. Horadam. Generalized Gaussian Fibonacci numbers. Bull. Austral. Math. Soc., Vol. 33, 1986, No. 1, 37–48.

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Cite this paper

APA

Asci, M., & Gurel E. (2013). Gaussian Jacobsthal and Gaussian Jacobsthal Lucas polynomials. Notes on Number Theory and Discrete Mathematics, 19(1), 25-36.

Chicago

Asci, Mustafa, and Esref Gurel. “Gaussian Jacobsthal and Gaussian Jacobsthal Lucas Polynomialss.” Notes on Number Theory and Discrete Mathematics 19, no. 1 (2013): 25-36.

MLA

Asci, Mustafa, and Esref Gurel. “Gaussian Jacobsthal and Gaussian Jacobsthal Lucas Polynomials.” Notes on Number Theory and Discrete Mathematics 19.1 (2013): 25-36. Print.

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