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knowing when both 6k -1 and 6k + 1 are prime

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  • Kermit Rose
    Message: 2 Date: Sun Jun 4, 2006 10:31 am (PDT) From: develator81 develator81@gmail.com Subject: Trying to find when 6k+1 returns a prime I started with this
    Message 1 of 1 , Jun 4 6:49 PM
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      Message: 2
      Date: Sun Jun 4, 2006 10:31 am (PDT)
      From: "develator81" develator81@...
      Subject: Trying to find when 6k+1 returns a prime

      I started with this question: It is possible to find a number "k"
      that: 6k+1 returns a prime, in a way that we are sure of it
      primality, with no need of testing it?
      I'm speaking of "k" that are non zero positives integers.

      I studied the numbers of the form: 6k+1 and 6k+1
      I separated these numbers into three categories:
      a) 6k+1 or 6k-1 that are primes
      B) 6k+1 that aren't primes
      c) 6k-1 that aren't primes.
      In the categories "B" and "c", I wonder why those "k" don't return a
      prime. Is there any rule for them?


      Yes.

      You have apparently derived one simple rule.

      Here is another.


      Suppose 6 k - 1 is composite.

      Then for some r1 and s1 both > 1,

      6 k - 1 =

      Write r1 = 6 r2 + r3
      s1 = 6 s2 + s3

      r3 s3 = -1 = 5 mod 6.

      Possible values of r3 and s3

      r3 s3
      1 5
      5 1

      Because of the symmetry, we can write

      r1 = 6 r2 -1
      s1 = 6 s2 + 1

      r1 s1 = (6 r2 -1) (6 s2 + 1) = 36 r2 s2 + 6 r2 - 6 s2 - 1 = 6 ( 6 r2 s2 + r2
      - s2) - 1

      So if k is NOT a number of either of the forms

      6 r2 s2 + r2 - s2,

      6 r2 s2 - r2 _ s2

      for both r2 and s2 > 0,

      then 6 k - 1 is prime.

      Likewise,

      if k is NOT a number of either of the forms

      6 r2 s2 - r2 - s2,

      6 r2 s2 + r2 + s2,

      Then 6 k + 1 is prime.


      Combining these,

      If k is NOT a number of any of the four forms,

      6 r2 s2 - r2 - s2,

      6 r2 s2 - r2 + s2,

      6 r2 s2 + r2 - s2,

      6 r2 s2 + r2 + s2

      then

      6 k -1 and 6 k + 1 are twin primes.




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