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I was looking at how to generate infinitely many solutions to the Pell's equation $x^2-dy^2 = 1$, where $d$ is a square-free positive integer. On https://en.wikipedia.org/wiki/Pell%27s_equation#Generalized_Pell's_equation it is also said that $x^2-dy^2 = 4$ can have infinitely many solutions if one solution is found.

How would I generate infinitely many solutions for $x^2-dy^2 = 4$?

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  • $\begingroup$ Note that $$x^2-dy^2=4 \Rightarrow \left(\frac{x}{2}\right)^{2}-d\left(\frac{y}{2}\right)^{2}=1$$ $\endgroup$
    – Mourad
    Commented Oct 27, 2020 at 2:30
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    $\begingroup$ I think you mean infinitely many solutions $\endgroup$ Commented Oct 27, 2020 at 2:39

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If $a^2 - db^2 = \alpha$ and $u^2 - dv^2 = \beta$, then $(au + dbv)^2 - d(av + bu)^2 = \alpha\beta$.

This gives you infinitely many solutions to $x^2 - dy^2 = 4$, once you have one solution to $x^2 - dy^2 = 4$ and one solution to $x^2 - dy^2 = 1$ (the latter always exists).

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