[FOM] Roth's Theorem; Liouville numbers

Jacques Carette carette at mcmaster.ca
Tue Apr 18 14:03:05 EDT 2006

Stephen G Simpson wrote:

>Earlier this semester, in an introductory course on foundations of
>mathematics, I naively assigned students the following problem:
>   Prove that the function f(n) = the nth digit of pi is primitive
>   recursive.
Note that, as far as *effectivity* is concerned, what base is used when 
asking the above question matters greatly.  By the BBP formula (after 
Borwein-Baily-Plouffe, see 
http://mathworld.wolfram.com/BBPFormula.html), in certain bases this 
question is ``easy'', while it is much harder in other bases.  [ie 
computing digits in certain bases is much easier than in other bases]

In particular, it appears that Pi is nice in various powers-of-2 bases, 
but no known formula exists in other bases.

>Can anyone here supply an alternative proof that doesn't involve such
>heavy number theory?
In base 16, there should be an easy proof using the BBP formula and the 
so-called "digit extraction" algorithms.

Using many of the different formulas for Pi found at 
http://mathworld.wolfram.com/PiFormulas.html should allow for a wealth 
of different (completely constructive) ways to approximate Pi.  The 
speed of convergence of the Chudnovsky's formula is indeed impressive.


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