Notes

Chapter 4: Systems Based on Numbers

Section 2: Elementary Arithmetic


General powers [of numbers]

It has been known in principle since the 1930s that Mod[h^n, 1] is uniformly distributed in the range 0 to 1 for almost all values of h. However, no specific value of h for which this is true has ever been explicitly found. (Some attempts to construct such values were made in the 1970s.) Exceptions are known to include so-called Pisot numbers such as GoldenRatio, Sqrt[2] + 1 and Root[#^3-#-1&, 1] (the numerically smallest of all Pisot numbers) for which Mod[h^n, 1] becomes 0 or 1 for large n. Note that Mod[x h^n, 1] effectively extracts successive digits of x in base h (see pages 149 and 919).


From Stephen Wolfram: A New Kind of Science [citation]