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The binary rythme of the primes

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  • Mark Underwood
    Hi all There are lots of ways to reduce the symphony of the primes to just two different notes. For instance, any prime over 3 can be expressed as 6t - 1 or 6t
    Message 1 of 1 , Jun 30, 2003
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      Hi all

      There are lots of ways to reduce the symphony of the primes to just
      two different notes. For instance, any prime over 3 can be expressed
      as 6t - 1 or 6t +1. If a prime happened to be of the form 6t - 1 we
      would call it an 'a'. If a prime was of the form 6t + 1 we would call
      it a 'b'.

      The primes in sequence take on the following expression, starting
      with the primes 7, 11, 13 and so on up to 131:

      b a b a b a a b b a b a a a b b a b b a a b a b a b a b a

      See the pattern? (hehe) And yes, the group abba is hidden in the
      primes!

      Another more interesting method to reduce the primes to just two
      different notes is the following. Goldbach's conjecture states
      (something like) that any even number over 3 (?) is the sum of two
      primes. Since every prime over 3 is an even number plus (and minus)
      one, then any prime p plus one is the sum of two primes. Also that
      same prime p minus one is the sum of two primes. We now select two
      primes which add up to p + 1 and two primes which add up to p -1.
      Since there may be many such pairs, we chose the two primes which are
      closest to each other. (I should clarify that this closeness is not
      simply the difference between primes but how close the primes are in
      sequence.) For instance, let p = 47 ;

      p + 1 = 48 = 19 + 29.

      19 and 29 are the two closest primes (two primes apart) which add up
      to 47 + 1 so we choose them. However,

      p - 1 = 46 = 23 + 23.

      23 and 23 are the two closest primes (zero primes apart) which add up
      to 47 - 1. Since 47 - 1 produces a prime pair which is closer than
      the prime pair produced by 47 + 1, then 47 -1 is the winner. If we
      let 'a' always correspond to -1 and 'b' always correspond to + 1,
      then 47's winning letter is 'a' and 47 would be represented as
      an 'a'. (BTW, it is easy to show that a and b never tie for primes
      over 5, so there is always a clear winner.)

      As another example, if we consider the prime 7, then
      7 + 1 = 3 + 5
      7 - 1 = 3 + 3.

      7 - 1 is the winner, so 'a' is the winner and we will represent 7 as
      an 'a'.

      The primes starting with 7, 11, 13, and so on up to 131 are
      represented thusly:

      a a b b a a b a b b a a a a b b b b a a b a a b a a a b b

      Hmmm, this sequence appears more chunky than the previous sequence.

      To compare them together:

      b a b a b a a b b a b a a a b b a b b a a b a b a b a b a
      a a b b a a b a b b a a a a b b b b a a b a a b a a a b b

      Doesn't seem to be going anywhere, but it was fun. :) There don't
      appear to be no easy beat to this rythme :)

      Mark
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