20211201, 16:37  #1 
Jun 2021
3×17 Posts 
Approximation of r by m^(1/n)
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 Last fiddled with by RomanM on 20211201 at 16:42 Reason: *** 
20211201, 17:28  #2  
Feb 2017
Nowhere
2^{2}·1,523 Posts 
Quote:
If r = x_{1} is a Pisot number (an algebraic integer > 1 whose algebraic conjugates x_{2},... x_{n} 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 k^{th} root of the k^{th} Lucas number has limiting value equal to the root r > 1 of P(x) = 0. 

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