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Trying to find when 6k+1 returns a prime

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  • develator81
    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
    Message 1 of 3 , Jun 4, 2006
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      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.
      This is how I started and follow my research:
      1)
      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? After I looked at them for a
      while I realized this:

      CAT B: 6k+1

      Let P be a prime greater than 3
      If: k = (P^2-1)/6 + P.j then 6k+1 is composite, where "j" is
      every integer greater or equal to zero.


      Then I obtained this:

      Let k be a number of this form:
      i) k= (P^2-1)/6 – a where P is the nth prime number,
      and 0<a<(P^2-1)/6

      If:
      k – (Q^2-1)/6 is not a multiple of Q; then 6k+1 is prime with
      this considerations: Q is every prime that is lesser to P and
      greater than 3, so we need to do the test for the validity
      of "k" "m" times.

      I also noticed this:
      If we took a k, P and Q with the conditions stated in i) then we
      have that:

      If: k - (Q^2-1)/6 is multiple of Q, being Q every prime that is
      lesser to P and greater than 3 then 6k+1 is multiple of every Q that
      divides: k - (Q^2-1)/6

      CAT C: 6k-1

      Let P be a prime greater than 3, and Q the next prime after P with
      this considerations: Q-P is not a multiple of six.
      Then we have:
      If: k = (P.Q + 1)/6 + P.j then 6k-1 is composite, where "j" is
      every integer greater or equal to zero.
    • Bernardo Boncompagni
      ... Indeed you can. k=1, k=2 and k=3 are good examples! Bernardo Boncompagni ________________________________ Wars not make one great Yoda When s who A
      Message 2 of 3 , Jun 4, 2006
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        develator81 dice:

        > 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?

        Indeed you can. k=1, k=2 and k=3 are good examples!

        Bernardo Boncompagni

        ________________________________

        "Wars not make one great"
        Yoda

        When's who
        A timeline of history makers
        http://whenswho.redgolpe.com

        Factorization of
        special form numbers
        http://factors.redgolpe.com
        ________________________________
      • LALGUDI BALASUNDARAM
        Hi, Please refer to my communication of Oct.17,2004 to Yahoo prime number group on the subject of Prime Structure where I have discussed the occurrence of
        Message 3 of 3 , Jun 5, 2006
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          Hi,
          Please refer to my communication of Oct.17,2004 to Yahoo prime number group on the subject of "Prime Structure" where I have discussed the occurrence of
          6*n -1 and 6*n + 1 type primes and the governable factors for the appearance of such primes in prime sequence. In the same communication I have discussed the impact of such prime representation on the many intriguing prime properties like presence or absence of twin primes, prime formulae for prime generation, computation of prime pairs satisfying Goldbach Conjecture etc. If you need further information or wish to discuss the topic plese feel free to email me.

          L.J.balasundaram

          develator81 <develator81@...> wrote: 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.
          This is how I started and follow my research:
          1)
          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? After I looked at them for a
          while I realized this:

          CAT B: 6k+1

          Let P be a prime greater than 3
          If: k = (P^2-1)/6 + P.j then 6k+1 is composite, where "j" is
          every integer greater or equal to zero.


          Then I obtained this:

          Let k be a number of this form:
          i) k= (P^2-1)/6 – a where P is the nth prime number,
          and 0<a<(P^2-1)/6

          If:
          k – (Q^2-1)/6 is not a multiple of Q; then 6k+1 is prime with
          this considerations: Q is every prime that is lesser to P and
          greater than 3, so we need to do the test for the validity
          of "k" "m" times.

          I also noticed this:
          If we took a k, P and Q with the conditions stated in i) then we
          have that:

          If: k - (Q^2-1)/6 is multiple of Q, being Q every prime that is
          lesser to P and greater than 3 then 6k+1 is multiple of every Q that
          divides: k - (Q^2-1)/6

          CAT C: 6k-1

          Let P be a prime greater than 3, and Q the next prime after P with
          this considerations: Q-P is not a multiple of six.
          Then we have:
          If: k = (P.Q + 1)/6 + P.j then 6k-1 is composite, where "j" is
          every integer greater or equal to zero.







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