Necessary and sufficient condition for compactness of the embedding operator.

Ramm, Alexander G.

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On the Diophantine equation $x^{2} - k x y + y^{2} - 2^{n} = 0$

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In this study, we determine when the Diophantine equation $x^{2} - k x y + y^{2} - 2^{n} = 0$ 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 $x^{2} - k x y + y^{2} - 2^{n} = 0$ .