Geodesic
On the Diophantine equation $x^{2}-kxy+y^{2}-2^{n}=0$
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 3, pp. 783-797
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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 $x^{2}-kxy+y^{2}-2^{n}=0$.
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 $x^{2}-kxy+y^{2}-2^{n}=0$.
DOI : 10.1007/s10587-013-0052-y
Classification : 11B37, 11B39, 11B50, 11B99
Keywords: Diophantine equation; Pell equation; generalized Fibonacci number; generalized Lucas number 
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Keskin, Refik; Şiar, Zafer; Karaatlı, Olcay. On the Diophantine equation $x^{2}-kxy+y^{2}-2^{n}=0$. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 3, pp. 783-797. doi: 10.1007/s10587-013-0052-y

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