Loading ...
Sorry, an error occurred while loading the content.

Re: Carmichael question

Expand Messages
  • paulunderwooduk
    ... 10^16 ... I have this expensively computed data already ;-) [SNIP] ... Here n^3-n-1 is 10^39 or 10^42. Paul
    Message 1 of 77 , Nov 2, 2002
    • 0 Attachment
      --- In primenumbers@y..., Markus Frind <flames@u...> wrote:
      > 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

      I have this expensively computed data already ;-)

      [SNIP]
      > >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

      Here "n^3-n-1" is 10^39 or 10^42.

      Paul
    • 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
      • 0 Attachment
        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


        () ASCII ribbon campaign () Hopeless ribbon campaign
        /\ against HTML mail /\ against gratuitous bloodshed

        [stolen with permission from Daniel B. Cristofani]

        __________________________________________________
        Do You Yahoo!?
        Tired of spam? Yahoo! Mail has the best spam protection around
        http://mail.yahoo.com
      Your message has been successfully submitted and would be delivered to recipients shortly.