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Re: prime form

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  • Phil Carmody
    Apologies for the aborted attempt at a post. Tab followed by either space or return (too quick to know what happened) seems to be a lethal key
    Message 1 of 77 , Apr 26, 2005
      Apologies for the aborted attempt at a post. 'Tab' followed by either
      'space' or 'return' (too quick to know what happened) seems to be a
      lethal key combination.

      From: "mcnamara_gio" <mcnamara_gio@...>
      >
      > What do you think about this sequence a(n)=n^2+7n-1. Its terms are
      > usually prime. I have calculated that 72% of a(n) so that n<500 is
      > prime. 81% of a(n) is prime when n<5000. 85.6% of a(n) is prime when
      > n<50000 and 88.5% of a(n) is prime when n<500000. I am going to find
      > more prime terms in this sequence. What do you think about it?

      2, 3, and 5 can never be factors. This boosts density by a factor of
      (2/1)*(3/2)*(5/4) over arbitrary ranges. However, 7,11, 13 and 17 both
      divide 2 of the p possible residues. This decreases density by a factor
      of (5/6)*(9/10)*(11/12)*(15/16) over arbitrary ranges.

      Looking at primes up to 10000, the density boost is almost exactly 2.75.
      This is pretty feeble compared with Euler's famous trinomials.

      Run this script in Pari/GP:

      rnorm=1.0
      rthis=1.0
      forprime(p=2,10000,roots=polrootsmod(x^2+7*x-1,p)~;rnorm*=(p-1)/p;rthis*=(p-#roots)/p;print(p"
      "rthis" "rnorm" "roots))
      print(rthis/rnorm);

      Research the Euler trinomials, and try the above script on them too, to see why
      I say 2.75 is pretty feeble.

      Phil


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    • Phil Carmody
      Apologies for the aborted attempt at a post. Tab followed by either space or return (too quick to know what happened) seems to be a lethal key
      Message 77 of 77 , Apr 26, 2005
        Apologies for the aborted attempt at a post. 'Tab' followed by either
        'space' or 'return' (too quick to know what happened) seems to be a
        lethal key combination.

        From: "mcnamara_gio" <mcnamara_gio@...>
        >
        > What do you think about this sequence a(n)=n^2+7n-1. Its terms are
        > usually prime. I have calculated that 72% of a(n) so that n<500 is
        > prime. 81% of a(n) is prime when n<5000. 85.6% of a(n) is prime when
        > n<50000 and 88.5% of a(n) is prime when n<500000. I am going to find
        > more prime terms in this sequence. What do you think about it?

        2, 3, and 5 can never be factors. This boosts density by a factor of
        (2/1)*(3/2)*(5/4) over arbitrary ranges. However, 7,11, 13 and 17 both
        divide 2 of the p possible residues. This decreases density by a factor
        of (5/6)*(9/10)*(11/12)*(15/16) over arbitrary ranges.

        Looking at primes up to 10000, the density boost is almost exactly 2.75.
        This is pretty feeble compared with Euler's famous trinomials.

        Run this script in Pari/GP:

        rnorm=1.0
        rthis=1.0
        forprime(p=2,10000,roots=polrootsmod(x^2+7*x-1,p)~;rnorm*=(p-1)/p;rthis*=(p-#roots)/p;print(p"
        "rthis" "rnorm" "roots))
        print(rthis/rnorm);

        Research the Euler trinomials, and try the above script on them too, to see why
        I say 2.75 is pretty feeble.

        Phil


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        /\ against HTML mail /\ against gratuitous bloodshed

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