[FOM] Roth's Theorem; Liouville numbers

Stephen G Simpson simpson at math.psu.edu
Tue Apr 18 16:38:52 EDT 2006

I thank Jacques Carette and Timothy Chow for their comments.

The problem I had in mind was, of course, to prove that the nth *base
10* digit of pi is primitive recursive as a function of n.  If I had
known about the BBP formula, I would have specified base 10.

Nevertheless, it isn't clear (to me at least) that the BBP formula by
itself shows that the nth digit of pi is primitive recursive, even in
base 16.  It seems to me that you still encounter the same old
problem, namely the possibility of long runs of 15's.  The only way I
know to overcome such problems is to use a theorem such as Mahler's,
saying that pi cannot be closely approximated by rationals.

Concerning the possibility of BBP-type formulas for bases other than
16, the web page http://mathworld.wolfram.com/BBP-TypeFormula.html
contains the following remark:

  Borwein and Galway have recently shown that there is no non-binary
  BBP-type formula for pi, although this does not rule out a
  completely different scheme for digit-extraction algorithms in other
  bases (Bailey 2002).

-- Steve

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