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Year 2025, Volume: 38 Issue: 2, 891 - 897, 01.06.2025

Refik Keskin , Kübra Ged , Fatih Erduvan

https://doi.org/10.35378/gujs.1378010

Abstract

References

  • [1] Keskin, R., “Solutions of some quadratic Diophantine equations”, Computer and Mathematics with Applications, 60(8): 2225-2230, (2010). DOI: 10.1016/j.camwa.2010.08.012
  • [2] Keskin, R., Güney, Duman, M., “Positive Integer Solutions of Some Pell Equations”, Palestine Journal of Mathematics, 8: 213-226, (2019).
  • [3] Keskin, R., Güney, Duman, M., “Positive Integer Solutions of Some Second-Order Diophantine equations”, Annale Ştiintifice ale Universitatii, “Alexandru Ioan Cuza”, din Iaşi Matematica, (New Series), 3: 649-664, (2017).
  • [4] Ribenboim, P., “My Numbers, My Friends: Popular Lectures on Number Theory”, Springer Verlag, 1-50, (2000).

On solutions of some Diophantine equations

Year 2025, Volume: 38 Issue: 2, 891 - 897, 01.06.2025

Refik Keskin , Kübra Ged , Fatih Erduvan

https://doi.org/10.35378/gujs.1378010

Abstract

Let k and p be prime numbers with p>2. In this note, we find all positive integer solutions to the Diophantine equations

x^2-kxy-y^2=±k,
x^2-(k^2+4) y^2=±4k,
x^2-2pxy-y^2=±2p,
x^2-kxy-y^2=±k(k^2+4)

in terms of generalized Fibonacci and Lucas numbers.

Keywords

Diophantine equations Generalized Fibonacci numbers Generalized Lucas numbers

References

  • [1] Keskin, R., “Solutions of some quadratic Diophantine equations”, Computer and Mathematics with Applications, 60(8): 2225-2230, (2010). DOI: 10.1016/j.camwa.2010.08.012
  • [2] Keskin, R., Güney, Duman, M., “Positive Integer Solutions of Some Pell Equations”, Palestine Journal of Mathematics, 8: 213-226, (2019).
  • [3] Keskin, R., Güney, Duman, M., “Positive Integer Solutions of Some Second-Order Diophantine equations”, Annale Ştiintifice ale Universitatii, “Alexandru Ioan Cuza”, din Iaşi Matematica, (New Series), 3: 649-664, (2017).
  • [4] Ribenboim, P., “My Numbers, My Friends: Popular Lectures on Number Theory”, Springer Verlag, 1-50, (2000).
There are 4 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Mathematics
Authors

Refik Keskin 0000-0003-2547-2082

Kübra Ged 0000-0001-5445-9798

Fatih Erduvan 0000-0001-7254-2296

Early Pub Date February 4, 2025
Publication Date June 1, 2025
Submission Date October 18, 2023
Acceptance Date January 2, 2025
Published in Issue Year 2025 Volume: 38 Issue: 2

Cite

APA Keskin, R., Ged, K., & Erduvan, F. (2025). On solutions of some Diophantine equations. Gazi University Journal of Science, 38(2), 891-897. https://doi.org/10.35378/gujs.1378010
AMA Keskin R, Ged K, Erduvan F. On solutions of some Diophantine equations. Gazi University Journal of Science. June 2025;38(2):891-897. doi:10.35378/gujs.1378010
Chicago Keskin, Refik, Kübra Ged, and Fatih Erduvan. “On Solutions of Some Diophantine Equations”. Gazi University Journal of Science 38, no. 2 (June 2025): 891-97. https://doi.org/10.35378/gujs.1378010.
EndNote Keskin R, Ged K, Erduvan F (June 1, 2025) On solutions of some Diophantine equations. Gazi University Journal of Science 38 2 891–897.
IEEE R. Keskin, K. Ged, and F. Erduvan, “On solutions of some Diophantine equations”, Gazi University Journal of Science, vol. 38, no. 2, pp. 891–897, 2025, doi: 10.35378/gujs.1378010.
ISNAD Keskin, Refik et al. “On Solutions of Some Diophantine Equations”. Gazi University Journal of Science 38/2 (June 2025), 891-897. https://doi.org/10.35378/gujs.1378010.
JAMA Keskin R, Ged K, Erduvan F. On solutions of some Diophantine equations. Gazi University Journal of Science. 2025;38:891–897.
MLA Keskin, Refik et al. “On Solutions of Some Diophantine Equations”. Gazi University Journal of Science, vol. 38, no. 2, 2025, pp. 891-7, doi:10.35378/gujs.1378010.
Vancouver Keskin R, Ged K, Erduvan F. On solutions of some Diophantine equations. Gazi University Journal of Science. 2025;38(2):891-7.