Re: A problem in Pi digits as Lattice space filling

*To*: mathgroup at smc.vnet.net*Subject*: [mg93843] Re: A problem in Pi digits as Lattice space filling*From*: Roger Bagula <rlbagula at sbcglobal.net>*Date*: Thu, 27 Nov 2008 05:28:17 -0500 (EST)*References*: <ggj7co$j2i$1@smc.vnet.net>

> > > A friend and long time Mathematica programmer came up with this: Clear[f, n] f[0] = 1; f[n_Integer /; n > 0] := Module[{cnt, m, p}, cnt = 0; m = Table[0, Evaluate[Apply[Sequence, Table[{10}, {n}]]]]; p = Partition[RealDigits[Pi, 10, 20*10^n][[1]], n, 1]; For[i = 1, i <= 20*10^n - n + 1, i++, If[m[[Apply[Sequence, p[[i]] + 1]]]++ == 0, If[++cnt == 10^n, Print[i]; Break[]]]; ] ]; Table[f[n], {n, 0, 4}] From In[9]:= 33 From In[9]:= 606 From In[9]:= 8554 From In[9]:= 99847 There my machine gives up in real time... I pretty much trust this and my older clunky programs. When run on a faster machine the answers are: starting at f[0]=1; 1,33,606,8554,99847,1369561, 14118308 the hyperbolic probability limit: Limit[10^n/f[n],n->Infinity] appears to be zero from these numbers. That result would suggest that the Pi digits are really not normal at all. But with just these few numbers it is hard to say. Since the BBP approach is actually based on the normal character of the Pi digits ( one of the assumptions necessary for a PSLQ digit solve to actually work! I'm not certain of the theory, but that is what the papers say.). I'm not willing to argue the theory of the point, as I really don't much understand it. The idea of the space filling character of the Pi digits came from them being normal. If the lattice diverges as it seems to toward Infinity, that would imply that the digits of Pi aren't really normal, I think. As a result the BBP Pi digits base 16 forms are not derived correctly? Or at least there is a question, now, involved. Me, I'm an experimenter and not an axiomatic theoretical type. But in any case the question of the space filling limit seems important to my mind. Respectfully, Roger L. Bagula 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :http://www.geocities.com/rlbagulatftn/Index.html alternative email: rlbagula at sbcglobal.net