Emin, Ahmet2025-01-292025-01-29202422195688https://pjm.ppu.edu/sites/default/files/papers/PJM_13%284%29_2024_65_to_74_0.pdfhttps://hdl.handle.net/20.500.14619/15017In this study, the positive solutions of the Diophantine equation D: x2 −( σ2 + 4) y2 − (2σ − 2) x −(2σ4 + 8σ2) y − σ6 − 4σ4 + σ2 − 2σ − 3 = 0 on the set Z are investigated, along with some recurrence relations that provide the relationships among these solutions. In addition, solutions of the Diophantine equation D in terms of generalized Fibonacci and Lucas sequences are examined. Furthermore, we search for the solutions of this equation over finite fields Fp where p is prime and p > 5. Finally, an example is given that satisfies our results.eninfo:eu-repo/semantics/openAccessDiophantine equationgeneralized Fibonacci and Lucas sequencesPell equationrecurrence relationArticle2-s2.0-8521530962012764Q4126613