On the Diophantine equation x2 - kxy + y2 - 2n = 0

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Keskin R., ŞİAR Z., Karaatli O.

Czechoslovak Mathematical Journal, vol.63, no.3, pp.783-797, 2013 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 63 Issue: 3
  • Publication Date: 2013
  • Doi Number: 10.1007/s10587-013-0052-y
  • Journal Name: Czechoslovak Mathematical Journal
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.783-797
  • Keywords: Diophantine equation, generalized Fibonacci number, generalized Lucas number, Pell equation
  • Bilecik Şeyh Edebali University Affiliated: Yes

Abstract

In this study, we determine when the Diophantine equation x 2-kxy+y 2-2n = 0 has an infinite number of positive integer solutions x and y for 0 ≤ n ≤ 10. Moreover, we give all positive integer solutions of the same equation for 0 ≤ n ≤ 10 in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation x 2 - kxy + y 2 - 2n = 0. © 2013 Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic.