For [itex]\alpha[/itex] and [itex]\beta[/itex] odd integers and X,Y integers, we have the following (by collecting terms):(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

(\alpha + \beta \sqrt{5})^n = x + y \sqrt{5}

[/tex]

My question is how do we know that x and y are both divisible by exactly [itex]2^{n-1}[/itex]? (no more and no less 2's in each)

I can show this with integer ring theory, but I wanted a more concrete and direct way to show this.

I'm looking for some inherrent numerical property rather than a proof by induction, if there is one that is easy to understand.

Thanks.

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# Question related to Pell's Equation

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