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 RomanM 2021-12-01 16:37

Approximation of r by m^(1/n)

[TEX]r\pm\epsilon=\sqrt[n]{m}[/TEX]
where r - real root of polynomial P(r), order >=6, and m, n - integers, and P(r+/-eps)<1
P.S. I'm suspect that the simpler the question look like, the less likely it is to get an answer

 Dr Sardonicus 2021-12-01 17:28

[QUOTE=RomanM;594283][TEX]r\pm\epsilon=\sqrt[n]{m}[/TEX]
where r - real root of polynomial P(r), order >=6, and m, n - integers, and P(r+/-eps)<1
P.S. I'm suspect that the simpler the question look like, the less likely it is to get an answer[/QUOTE]It depends on what you're given first.

If r = x[sub]1[/sub] is a Pisot number (an algebraic integer > 1 whose algebraic conjugates x[sub]2[/sub],... x[sub]n[/sub] all have absolute value less than 1) then for positive integer k, the sums

$$S_{k}\;=\;\sum_{i=1}^{n}x_{i}^{k}$$

are all rational integers, and all the terms except the first tend to 0 as k increases without bound. Thus

$$r\;\approx\;$$S_{k}$$^{\frac{1}{k}}$$

becomes an increasingly good approximation as k increases.

The simplest case is with the polynomial P(x) = x^2 - x - 1. The sums are the Lucas numbers.

So the k[sup]th[/sup] root of the k[sup]th[/sup] Lucas number has limiting value equal to the root r > 1 of P(x) = 0.

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