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A sieve for twin primes

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  • Kermit Rose
    To sieve for all the twin primes less than 6000, fill an array, J, of size 1000 with the integers 1 to 1000. For k = 1 to 1000 J(k) = 0 next k Then for m = 1
    Message 1 of 3 , Dec 30, 2005
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      To sieve for all the twin primes less than 6000,

      fill an array, J, of size 1000 with the integers 1 to 1000.


      For k = 1 to 1000
      J(k) = 0
      next k


      Then for m = 1 to 120, and n = m to 120

      calculate

      d0 = 6 m n - m - n

      d1 = 6 m n + m - n

      d2 = 6 m n - m + n

      d3 = 6 m n + m + n


      if d0 is between 1 and 1000, set J(d0) = 0

      if d1 is between 1 and 1000, set J(d1) = 0

      if d2 is between 1 and 1000, set J(d2) = 0

      if d3 is between 1 and 1000, set J(d3) = 0


      Then for any positive integer, G, still in the array,


      6G-1 and 6G+1 are prime.


      Illustration for smaller range.


      6 * 1 * 1 - 1 - 1 = 4

      1, 2, and 3 are skipped.

      so 6 * 1 -1 = 5 and 6 * 1 + 1 = 7 are twin primes.

      6*2-1 = 11 and 6*2+1=13 are twin primes

      6*3-1 = 17 and 6 * 3 + 1 = 19 are twin primes.


      6*1*1+1-1 = 6

      5 is skipped.


      so 6 * 5 -1 and 6 * 5 + 1 are twin primes.


      6 * 1*1 + 1 + 1 = 8

      7 is skipped.

      6 * 7 -1 = 41 and 6 * 7 + 1 = 43 are twin primes.


      etc.



      Kermit
      kermit@...
    • Joseph Moore
      That particular sieve has been around since (at least) January of 2000 when it was posted by Maria Suzuki in the American Mathematical Monthly journal. Which
      Message 2 of 3 , Dec 30, 2005
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        That particular sieve has been around since (at least)
        January of 2000 when it was posted by Maria Suzuki in
        the American Mathematical Monthly journal.

        Which types of primes do you get when you replace the
        6 with other numbers? E.g., 4mn +/- m +/- n. Can
        other prime sieves be written in this (or a similar)
        form successfully?

        Joseph.


        --- Kermit Rose <kermit@...> wrote:

        > To sieve for all the twin primes less than 6000,
        >
        > fill an array, J, of size 1000 with the integers 1
        > to 1000.
        >
        >
        > For k = 1 to 1000
        > J(k) = 0
        > next k
        >
        >
        > Then for m = 1 to 120, and n = m to 120
        >
        > calculate
        >
        > d0 = 6 m n - m - n
        >
        > d1 = 6 m n + m - n
        >
        > d2 = 6 m n - m + n
        >
        > d3 = 6 m n + m + n
        >
        >
        > if d0 is between 1 and 1000, set J(d0) = 0
        >
        > if d1 is between 1 and 1000, set J(d1) = 0
        >
        > if d2 is between 1 and 1000, set J(d2) = 0
        >
        > if d3 is between 1 and 1000, set J(d3) = 0
        >
        >
        > Then for any positive integer, G, still in the
        > array,
        >
        >
        > 6G-1 and 6G+1 are prime.
        >
        >
        > Illustration for smaller range.
        >
        >
        > 6 * 1 * 1 - 1 - 1 = 4
        >
        > 1, 2, and 3 are skipped.
        >
        > so 6 * 1 -1 = 5 and 6 * 1 + 1 = 7 are twin primes.
        >
        > 6*2-1 = 11 and 6*2+1=13 are twin primes
        >
        > 6*3-1 = 17 and 6 * 3 + 1 = 19 are twin primes.
        >
        >
        > 6*1*1+1-1 = 6
        >
        > 5 is skipped.
        >
        >
        > so 6 * 5 -1 and 6 * 5 + 1 are twin primes.
        >
        >
        > 6 * 1*1 + 1 + 1 = 8
        >
        > 7 is skipped.
        >
        > 6 * 7 -1 = 41 and 6 * 7 + 1 = 43 are twin primes.
        >
        >
        > etc.
        >
        >
        >
        > Kermit
        > kermit@...
        >
        >





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      • Kermit Rose
        From: Joseph Moore Date: 12/30/05 14:51:01 To: Kermit Rose; primenumbers@yahoogroups.com Subject: Re: [PrimeNumbers] A sieve for twin primes That particular
        Message 3 of 3 , Jan 2, 2006
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          From: Joseph Moore
          Date: 12/30/05 14:51:01
          To: Kermit Rose; primenumbers@yahoogroups.com
          Subject: Re: [PrimeNumbers] A sieve for twin primes

          That particular sieve has been around since (at least)
          January of 2000 when it was posted by Maria Suzuki in
          the American Mathematical Monthly journal.

          Which types of primes do you get when you replace the
          6 with other numbers? E.g., 4mn +/- m +/- n. Can
          other prime sieves be written in this (or a similar)
          form successfully?

          Joseph.


          *****************

          From Kermit Rose
          kermit@...


          4 m n + m + n and

          4 m n - m - n

          sieves out all numbers which generate 4 D + 1.


          4 m n + m - n
          and
          4 m n - m + n

          sieves out all numbers which generate 4 D - 1.


          In general

          A m n + m + n and

          A m n - m - n

          sieves out all numbers which generate 4 D + 1.

          [Non-text portions of this message have been removed]
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