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Re: [PrimeNumbers] A amusing idle speculation

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  • David Cleaver
    ... The number of such blocks is limited, albeit to a very large (by human standards) number. If we only had 3 blocks in base 2^24, there would be (2^24)^3
    Message 1 of 2 , Jul 25, 2003
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      xed wrote:
      > Folks;
      > And now for something completely different...
      > (As the Pythons were wont to say)
      > When searching for visual patterns in base-k
      > representations of the primes (patterns formed
      > by an n by m display of the base-k form of the
      > prime number in question) it occurs to me you're
      > thinking too small.
      > So far, various folks have discussed small
      > patterns (32 x 32, etc.) formed by base-2
      > representations of primes. That's all well and
      > good as far as it goes...but what if we think
      > bigger?
      > Suppose instead of base 2 we represent
      > the prime number in the form base 2^24 and
      > suppose instead of small blocks we visualize
      > the base 2^24 representation as a 720 x
      > 480 block. (Yes, you can see where this is
      > heading..)
      > This happens to be the format used by DVDs to
      > represent single frames from movies (in NTSC
      > -- for those of you in Europe, the sequences
      > would be represented in base 2^24 as 720 x
      > 576 blocks).
      > Q: Do there exist infinitely many such Truecolor
      > 720 x 480 pictures encoded in the infinity of primes?
      > Or is the number of such blocks limited?

      The number of such blocks is limited, albeit to a very large (by human
      standards) number. If we only had 3 "blocks" in base 2^24, there would
      be (2^24)^3 different possibilities, or 2^72 ~= 4.7x10^21. Now, with
      the American encoding of these pictures there are (2^24)^(720x480) =
      (2^24)^(345600) = 2^8294400 ~= 10^2496863.196 which is indeed a big
      number, but still finite. In the European encoding scheme we have that
      there are (2^24)^(720x576) = (2^24)^(414720) = 2^9953280 ~=
      10^2996235.835 So, the total number of possible pictures encoded this
      way, in either system, is less than a 3 million digit base-10 number.

      Now, while not all of these numbers (pictures) are primes, I would
      venture a guess that there are alot that are. And, personally, I think
      it would be amusing to see which frames of some movies already out there
      are prime. I also think it'd be neat to find out if any of those frames
      have a simple formula to describe them.

      -David C.
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