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Keywords:
Diophantine equation; Pell equation; generalized Fibonacci number; generalized Lucas number
Diophantine equation; Pell equation; generalized Fibonacci number; generalized Lucas number
Summary:
In this study, we determine when the Diophantine equation $x^{2}-kxy+y^{2}-2^{n}=0$ has an infinite number of positive integer solutions $x$ and $y$ for $0\leq n\leq 10.$ Moreover, we give all positive integer solutions of the same equation for $0\leq n\leq 10$ in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation $x^{2}-kxy+y^{2}-2^{n}=0$.
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