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RE Recasting sieve for primes into prime generator

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  • Jose Ramón Brox
    ... From: Mark Underwood Hi Kermit, Jose Perhaps the finer details of the results are interesting, but surely the main result
    Message 1 of 2 , Dec 29, 2005
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      ----- Original Message -----
      From: "Mark Underwood" <mark.underwood@...>

      Hi Kermit, Jose

      Perhaps the finer details of the results are interesting, but surely
      the main result of prime number generation is self evident?

      Essentially what is being said is this. Define a set X of numbers
      x0, x1, x2, x3, xm, ....

      such that
      2* xm +1 is always composite and includes all composites.

      If we take any y which is not in the set X, then by definition
      2*y + 1 cannot be composite, namely it must be prime.

      ----------------------------------------

      Yes, Mark, the main result is self evident.

      But in my opinion, it has some interest. It's somewhat like saying "erase all the
      pentagonal numbers and the remaining numbers n all generate the prime numbers with 2*n+1".
      Well, these numbers have two parameters ( m,n ) and therefore they are much less rare than
      pentagonals (in fact, it's easy to know their density using the PNT), but you see my
      point: a "simple" formula is involved.

      It's a different way to get the same result from always, and I think we can totally
      automatize them, get the pairs totally ordered with a simple algorithm or a simple set of
      rules and then get some conclusions about them. For example:

      If we could prove that for a fixed n0 we can always find a pair (m,n), such that the next
      pair (m',n') bigger than it has D(m',n') >= 2+D(m,n), then we would have proven the twin
      primes conjecture.

      In general, note that if (m',n') is the next pair for (m,n) in this order, then we know
      that the odd numbers between 2*D(m,n)-1 and 2*D(m',n')-1 are all prime. The interesting
      thing is this order, in my opinion.

      Of course, I can be totally mistaken and this order could be impossible of determining
      with simple conditions, and to work on it worth nothing; or it could be obvious but
      without real practical uses or untrivial information. While I arrive to one of these
      conclusions, I'll keep thinking about this subject in my free time (well, among other
      subjects as well :P).

      Jose
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