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

Re: Carmichael question

Expand Messages
  • richard_heylen
    I ve found another Carmichael of the form n^2-n-1 It is n=151068195602 so n^2-n-1=22821599722293063946801 It has the 8 factor factorisation
    Message 1 of 77 , Nov 1, 2002
      I've found another Carmichael of the form n^2-n-1

      It is n=151068195602 so n^2-n-1=22821599722293063946801

      It has the 8 factor factorisation
      19.29.181.911.1051.1801.7561.17551

      The lcm of all of these factors minus 1 is
      2^3.3^3.5^2.7.13 and this of course divides n^2-n-2

      Now this should shed some light on the why the Carmichael tends to
      end in 1. It's because the prime factors of the Carmichael tend to be
      a smooth number plus one. 5 is a very useful small prime for
      rendering numbers smooth. From Richard Pinch's website it appears
      that only about a tenth of Carmichael numbers in the 10^15 range that
      don't have 5 as a factor don't end in a 1.

      Richard
    • 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


        () 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.