[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.
Jacques
More information about the FOM
mailing list