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Old 2021-12-01, 17:28   #2
Dr Sardonicus
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Feb 2017

2·29·103 Posts

Originally Posted by RomanM View Post
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
It depends on what you're given first.

If r = x1 is a Pisot number (an algebraic integer > 1 whose algebraic conjugates x2,... xn all have absolute value less than 1) then for positive integer k, the sums


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


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 kth root of the kth Lucas number has limiting value equal to the root r > 1 of P(x) = 0.
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