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    On positive integer solutions of a special diophantine equation
    (Palestine Polytechnic University, 2024) Emin, Ahmet
    In 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.
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    ON THE DIOPHANTINE EQUATION L2 m +L2n=2a
    (Publ House Bulgarian Acad Sci, 2024) Emin, Ahmet
    This study researches numbers that are powers of two and can be represented as the sum of the squares of any two Lucas numbers. We apply Baker's theory of linear forms in logarithms of algebraic numbers, combined with a variation of the Baker-Davenport reduction method, to solve the Diophantine equation L2m 2 m + L2n 2 n = 2 a , where m, n and a are positive integers, as presented in this study.