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Re: [PrimeNumbers] probability of (2*p1*p2) + 1 being prime

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  • Phil Carmody
    ... Well, it s just like any arbitrary number of the same size except that it s even, it s not divisible by p1, and it s not divisible by p2. Therefore there s
    Message 1 of 6 , Sep 9, 2007
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      --- jtrjtrjtr2001 <jtrjtrjtr2001@...> wrote:
      > Hi,
      >
      > Let p2 be a large prime. We find another large random prime p1, such that
      > y = (2*p1*p2) + 1.
      >
      > Is there any way, one could quantify the probability of y being a prime?

      Well, it's just like any arbitrary number of the same size except that
      it's even, it's not divisible by p1, and it's not divisible by p2.
      Therefore there's a prime density boost of (2/1) * (p1/(p1-1)) * (p2/(p2-1))
      Those final two factors are effectively 1.

      Phil

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    • Jens Kruse Andersen
      ... Phil meant y is odd, but another factor must also be considered. If q is a random odd prime other than p1 and p2, then q does not divide y-1. q must divide
      Message 2 of 6 , Sep 9, 2007
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        Phil Carmody wrote:
        > --- jtrjtrjtr2001 <jtrjtrjtr2001@...> wrote:
        >> Let p2 be a large prime. We find another large random prime p1, such that
        >> y = (2*p1*p2) + 1.
        >>
        >> Is there any way, one could quantify the probability of y being a prime?
        >
        > Well, it's just like any arbitrary number of the same size except that
        > it's even, it's not divisible by p1, and it's not divisible by p2.
        > Therefore there's a prime density boost of (2/1) * (p1/(p1-1)) *
        > (p2/(p2-1))
        > Those final two factors are effectively 1.

        Phil meant y is odd, but another factor must also be considered.
        If q is a random odd prime other than p1 and p2, then q does
        not divide y-1. q must divide one of the q-1 numbers from
        y to y+q-2, and the chance that q divides y increases from
        1/q to 1/(q-1).
        So the chance that q does not divide changes from
        (q-1)/q to (q-2)/(q-1), which corresponds to multiplying the
        chance by ((q-2)/(q-1)) / ((q-1)/q) = q*(q-2)/(q-1)^2.

        The situation is just like twin primes:
        If p is a large prime then p+2 is odd. If q is a random odd
        prime other than p then q does not divide p, so the chance
        of q dividing p+2 increases from 1/q to 1/(q-1).

        The twin prime constant C_2 = 0.66016... is the product
        over all odd primes q of q*(q-2)/(q-1)^2
        See for example http://mathworld.wolfram.com/TwinPrimesConstant.html

        A random large number n has chance 1/log(n) of being prime.
        Our y is odd and also taking C_2 into consideration changes
        the chance to 2*C_2/log(y).

        --
        Jens Kruse Andersen
      • Jack Brennen
        ... And y is not congruent to 1 mod 3, or to 1 mod 5, or to 1 mod 7, etc. Which should make y somewhat more likely to be divisible by these small numbers...
        Message 3 of 6 , Sep 9, 2007
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          Phil Carmody wrote:
          > --- jtrjtrjtr2001 <jtrjtrjtr2001@...> wrote:
          >> Hi,
          >>
          >> Let p2 be a large prime. We find another large random prime p1, such that
          >> y = (2*p1*p2) + 1.
          >>
          >> Is there any way, one could quantify the probability of y being a prime?
          >
          > Well, it's just like any arbitrary number of the same size except that
          > it's even, it's not divisible by p1, and it's not divisible by p2.
          > Therefore there's a prime density boost of (2/1) * (p1/(p1-1)) * (p2/(p2-1))
          > Those final two factors are effectively 1.
          >

          And y is not congruent to 1 mod 3, or to 1 mod 5, or to 1 mod 7, etc.

          Which should make y somewhat more likely to be divisible by these small
          numbers... For instance, half of a random sample of y values should be
          divisible by 3.

          So I think that the actual density boost would be 2 times the twin prime
          constant (~ 0.66) which would make these y numbers about 1.32 times
          more likely to be prime than arbitrary numbers of the same size.

          Jack
        • Phil Carmody
          ... Good catch, Jack! Phil () ASCII ribbon campaign () Hopeless ribbon campaign / against HTML mail / against gratuitous bloodshed
          Message 4 of 6 , Sep 14, 2007
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            --- Jack Brennen <jb@...> wrote:
            > Phil Carmody wrote:
            > > --- jtrjtrjtr2001 <jtrjtrjtr2001@...> wrote:
            > >> Hi,
            > >>
            > >> Let p2 be a large prime. We find another large random prime p1, such that
            > >> y = (2*p1*p2) + 1.
            > >>
            > >> Is there any way, one could quantify the probability of y being a prime?
            > >
            > > Well, it's just like any arbitrary number of the same size except that
            > > it's even, it's not divisible by p1, and it's not divisible by p2.
            > > Therefore there's a prime density boost of (2/1) * (p1/(p1-1)) *
            > (p2/(p2-1))
            > > Those final two factors are effectively 1.
            > >
            >
            > And y is not congruent to 1 mod 3, or to 1 mod 5, or to 1 mod 7, etc.

            Good catch, Jack!

            Phil

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          • Jens Kruse Andersen
            ... All 4 above posts were mailed September 9 but it took 5 days to deliver the posts by Jack and I. We are far apart and the posts showed up the same minute
            Message 5 of 6 , Sep 14, 2007
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              September 9 jtrjtrjtr2001 wrote:
              > y = (2*p1*p2) + 1.

              Phil Carmody wrote:
              > > Well, it's just like any arbitrary [odd] number of the same size

              I wrote:
              > If q is a random odd prime other than p1 and p2,
              > then q does not divide y-1

              Jack Brennen wrote:
              > And y is not congruent to 1 mod 3, or to 1 mod 5,
              > or to 1 mod 7, etc.

              All 4 above posts were mailed September 9 but it took 5 days to
              deliver the posts by Jack and I. We are far apart and the posts
              showed up the same minute so it seems like a Yahoo problem.
              http://tech.groups.yahoo.com/group/primeform/message/8788 was delayed
              from August 31 to September 5. Maybe the problem from
              http://tech.groups.yahoo.com/group/yg-alerts/message/24 has not been
              completely fixed yet. I'm currently posting through
              http://tech.groups.yahoo.com/group/primenumbers/ where I haven't seen
              a long delay.

              --
              Jens Kruse Andersen
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