If we're talking about fractional bases, allow me to introduce you to a conjecture of Leonard J. Schulman and me: Consider writing zero in base 3/2. That is, for i=0,1,2... let a_i be an integer, and let 0 = \sum_{i=0}^\infty a_i (2/3)^i . Furthermore, assume that a_i is in {+1,0,-1} for each i, and that a_0 = 1. Thus the first term is 1, and the remaining terms need to cancel it out. Conjecture: all initial subsequences have a nonzero density of nonzero coefficients. That is, there is a constant C > 0 such that, for every n, at least Cn of the coefficients a_1, a_2, ... a_n are nonzero. I am equally interested in any nearby conjecture: e.g. replace 3 with 5 or 7, let a_i belong to {-2,-1,0,1,2}, and so on. If you can prove this, it will have nice consequences for a construction in computer science called tree codes: see http://arxiv.org/abs/1308.6007 - Cris On Oct 13, 2013, at 12:24 PM, Hans Havermann <gladhobo@teksavvy.com> wrote:
Keith F. Lynch: "No, the digits of seventeen *in base pi*, 120.220021101020230020003… This of course never repeats or terminates."
MathWorld's article on 'base' cautions that "the representation of a given integer in an irrational base may be nonunique" (but this is true also of integer bases: 1=.999… in base ten). More relevant perhaps is the issue of normalcy. My understanding of the concept is that it cannot be applied to irrational bases. What then is the frequency distribution of the digits (0, 1, 2, 3) of seventeen in base pi? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun