A counting method for finding rational approximates to arbitrary order roots of integers

Abstract

It is shown that for finding rational approximates to m'th root of any integer to any accuracy one only needs the ability to count and to distinguish between m different classes of objects. To every integer N can be associated a 'replacement rule' that generates a word W* from another word W consisting of symbols belonging to a finite 'alphabet' of size m. This rule applied iteratively on almost any initial word W0, yields a sequence of words {Wi} such that the relative frequency of different symbols in the word Wi approaches powers of the m'th root of N as i tends to infinity