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I am assigned to find solutions to the Pell-type equation $x^2-2y^2=-1$. So far, I only have $(7,5), (41,29)$ and $(239,169)$.

My question is, is there a general formula to find all its solutions? Are there infinitely many solutions? If it's possible to write Mathematica code that would find its solutions, what would the code look like?

Thanks a lot.

kate
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Jr Antalan
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    Work in $\mathbb{Z}[\sqrt{2}]$. The solution $(7,5)$ means that $N(7+5\sqrt{2})=-1$. Therefore, all elements of the form $(7+5\sqrt{2})^{m}$, where $m$ is an odd integer will generate solutions. So yes, there are infinitely many solutions. To get all solutions just raise the fundamental unit: $(1+\sqrt{2})^m$ to odd powers $m$. – tc1729 May 05 '15 at 22:58
  • See also Pell's equation. – Peter Phipps May 05 '15 at 23:02

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Starting with $(1,1),$ and your examples, new $(x,y)$ solutions follow $$ x_{n+2} = 6 x_{n+1} - x_n, $$ $$ y_{n+2} = 6 y_{n+1} - y_n. $$ This follows from $$ (x_{n+1}, y_{n+1} ) = (3 x_n + 4 y_n \, , \; \; 2 x_n + 3 y_n) $$ which is easier to confirm.

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Will Jagy
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