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If I understand correctly, your question is the following: suppose that for a given positive integer $d$ the equation $$\displaystyle x^2 - dy^2 = c \text{ } (\ast)$$ has a solution in integers $x,y$ for some integer $c$. Then does there exist an infinite family of solutions generated by $A^k (x,y)^T$ for some $A \in \text{GL}_2(\mathbb{Z})$ having infinite ...

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