 mersenneforum.org Approximation of r by m^(1/n)
 Register FAQ Search Today's Posts Mark Forums Read 2021-12-01, 16:37 #1 RomanM   Jun 2021 2×52 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 2021-12-01 at 16:42 Reason: ***   2021-12-01, 17:28   #2
Dr Sardonicus

Feb 2017
Nowhere

73·17 Posts Quote:
 Originally Posted by RomanM 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.  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post greenskull Homework Help 8 2021-10-07 23:49 mathPuzzles Math 8 2017-05-04 10:58 ixfd64 Math 1 2007-11-26 05:57

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