http://qs321.pair.com?node_id=77935


in reply to the secret of PI

If someone could be able to give a proof that \pi's digits follow a casual distribution, you could find in it every possible finite sequence of digits. Thus, you could find on \pi the source for Perl 6, all nodes of PerlMonks (even those that are not written yet) and so on...

Furthermore, such a true casual number exists: we are able to define it and we know some of its properties, but it is demonstrated that we can't compute it. This number is called \Omega, and it's defined as the probability that an Universal Turing Machine halts given random input.

\Omega has other interesting properties. If we could know \Omega we would be able to solve the halting problem for every Turing Machine, finding a solution to, for example, the Goldbach's conjecture and Collatz's game.

If you want to read more about this mystic number and related topics, visit the home page of this wise man.

Replies are listed 'Best First'.
Re: Mystic numbers (Re: the secret of PI)
by japhy (Canon) on May 04, 2001 at 16:43 UTC
    I like the Infinity engine (found here). It uses the notion that:
    • after one second, do X
    • after half a second, do X
    • after a quarter of a second, do X
    • after an eight of a second, do X
    • ...
    That way, after 2 seconds, we have executed "X" an infinite number of times.

    japhy -- Perl and Regex Hacker
[reply]

      This sounds like a variant on my patented Time Extensible Processor. The TEP projects your task far enough into the future so that it is done now. It has an interesting side effect, however. If the tasks ever stop running, it indicates the end of the universe. But because of how time and space work, you can simply pick up the machine and move it a few inches to the left or right to resume processing. At least, until that time/space thread ends.

      --Chris

      e-mail jcwren
[reply]
Re (tilly) 1: Mystic numbers (Re: the secret of PI)
by tilly (Archbishop) on May 04, 2001 at 19:20 UTC
    The usual name for this is a "normal number". It is unknown whether Pi is normal, but it is strongly suspected to be.
[reply]

      Not as if it's much consolation, but a statistical analysis shows that both pi and e1 have each digit (0..9) appear one tenth of the time. It's not a rigorous mathematical proof of normality, but it shows that such a proof shouldn't be able to be disproved, which is important in itself...

      nuf evah,
      jynx

[reply]
Re: Mystic numbers (Re: the secret of PI)
by Anonymous Monk on Aug 19, 2003 at 23:27 UTC
    Pi is
[reply]
      Pi is (normal)
      Can you provide a proof for that?
[reply]

In Section Meditations