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Re: [PrimeNumbers] Re: Carmichael question

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  • Markus Frind
    The website richard made reference to lists all Carmichaels under 10^16 listed and all 3-Carmichaels
    Message 1 of 77 , Nov 2, 2002
      The website richard made reference to lists all Carmichaels under 10^16
      listed and all 3-Carmichaels < 10^18 listed.
      http://www.chalcedon.demon.co.uk/rcam.html

      richard_heylen would it be possible for us to see your code? I can throw
      a few fast machines at this for a while...


      At 05:54 PM 11/2/2002 +0000, paulunderwooduk wrote:
      >--- In primenumbers@y..., "richard_heylen" <richard_heylen@y...>
      >wrote:
      > > It's all rather hand-
      > > wavey and the implementation sucks but it trots out the Carmichaels
      > > everyone else has found, in a couple of minutes on a P3-800.
      > >
      >
      >Are your methods transferable to n^3-n-1? If so, how long would it
      >take to do to n=10^13 or n=10^14 in this case?
      >
      >Paul
      >
      >
      >
      >Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
      >The Prime Pages : http://www.primepages.org/
      >
      >
      >
      >Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
    • 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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