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We know the integer solutions of Pell's equation $$a^2-2b^2=\pm1$$ correspond to the units of $\textbf{Z}[\sqrt{2}]$. How can we prove this?

Willie Wong
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bsdshell
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  • I've edited your post for grammar and formatting. Please check to see that I didn't change the meaning. – KReiser Dec 06 '13 at 09:23
  • See http://math.stackexchange.com/questions/590775/need-a-proofreading-why-all-the-units-are-satisfied-a2-2b2-pm1-for-mathb?rq=1. – Dietrich Burde Dec 06 '13 at 09:52

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Probably you mean if $(a,b)$ is a solution of Pell's equation then $a+b\sqrt{2}$ is a unit of $\mathbb{Z}\sqrt{2}$. This directly follows from the fact$$(a+b\sqrt{2})(a-b\sqrt{2})=\pm 1$$

pritam
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  • Yes, see http://math.stackexchange.com/questions/590775/need-a-proofreading-why-all-the-units-are-satisfied-a2-2b2-pm1-for-mathb?rq=1. – Dietrich Burde Dec 06 '13 at 09:54