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>

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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