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Some Prime Twin Philosophy

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  • mwiner_stock
    I was exposed to this problem first in university. My professor said that he needed to see a base case and a method from getting from k to k+1 (induction).
    Message 1 of 1 , Mar 27, 2003
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      I was exposed to this problem first in university. My professor said
      that he needed to see a base case and a method from getting from k to
      k+1 (induction). What I couldn't express to him at the time was that
      no such function existed because the creation of primes was and is
      the only true random function that I've ever been exposed to. To
      have a method to go from k to k+1 breaks that assumption of
      randomness.

      The numbers 2 and 5 are very nice numbers because they predictably
      destroy, in the sieve, all even numbers and all 5 ending numbers.
      Nice of them. But that's the end of the predictability.

      3 is a crucial number in that it defines the first infinite pattern
      of possibilities for primes. The pattern of opportunities for prime
      twins that the number 3 creates is well known. In the sieve of odd
      numbers it looks like
      100100100100100100..., were 1 is a multiple of 3 and 0 isn't (hence
      an opportunity for a prime)

      Standard combinatorics yields that for any prime p, the there will be
      opportunities for prime twins in the resulting pattern (proof is
      trivial, but can be provided).

      Ok, let's have some fun now: Assuming from the above that for any
      prime number p, the pattern (sieve) has opportunities for prime
      twins... prime twins will occur whenever the prime twin opportunity
      lies between p and p^2. If only we could prove that such an
      opportunity will always eventually lie between some n and n^2 where n
      is prime. We can't. But don't get dejected.

      We can't because of something magical... the pattern produced by
      infinitely adding new primes to the sieves is RANDOM. Not, pseudo
      random, but really random. Up until recently I only believed
      in 'patterns sufficiently complex to avoid simple categorization'.
      This process is really random!

      The pattern produced by 3 is simple: 100 (size=3). The pattern
      produced by 3,5 is more complicated 110100100101100 (size=3*5=15) but
      still predictable. 3,5,7 is more complicated still. As we continue
      to add primes to the pattern (producing longer, more complicated
      patterns) at any given step, the resulting pattern is more
      complicated than the previous p-1 pattern were p-1 is the previous
      prime. As we continue, and add primes to the pattern without bound,
      we produce the a random pattern, one with infinite complexity and
      zero predictability.

      Assume you swallow the previous argument (that the patterns of primes
      and prime twins et all are random, btw it took me 5 years to swallow
      this argument) if the patterns of prime twin opportunities are random
      then so too is the distance of the prime twin opportunity from it's
      largest prime contributor p. So take a pattern pat1 where the first
      prime opportunity is beyond c^2 and c is the largest prime
      contributor to that pattern. Well we won't get a prime twin then,
      but forunately the algorithm continues. We have n->inf iterations of
      this algorithm until the pattern becomes random.

      In a random process you can't guarantee me that the first prime
      opportunity will NEVER come between c and c^2 and therefore it will
      eventually come there. Unless you can describe p such that after
      that prime p, no prime twin opportunities exist. Since there is no
      such p that removes all prime twin opportunities, and the process of
      producing new sieves in iterations without bounds produces random
      patterns, there will always be a prime twin opportunity which
      eventually lies between c and c^2.

      The same randomness which caused our difficulty in proving this
      problem now then solves it. How do you like that?
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