Why $\phi^n$ is nearly an integer | Colin

This article is one of those ‘half-finished thoughts’ put together late at night. Details are missing, and — in a spirit of collaboration — I’d be glad if you wanted to fill them in for me.

The estimable @onthisdayinmath (Pat in real life) recently posted about nearly-integers, and remarked that $\phi^n$ is very close to a whole number for moderately large values of $n$ (for instance, $\pi^{20} \approx 15\ 126.\ 999\ 887$). Aha!, I thought: I know why that is!

It’s all to do with binomial expansions: in particular, those of $\phi^n$ and $\phi^{-n}$, which are closely related1:

$\phi^n =…

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