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Carmichael question

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  • Markus Frind
    Given the following series, x^2 - x - 1 can this ever form a Carmichael Number? Note that any divisor of x^2 - x - 1 has the following properties. 1.
    Message 1 of 77 , Oct 28, 2002
      Given the following series, x^2 - x - 1 can this ever form a Carmichael
      Number?

      Note that any divisor of x^2 - x - 1 has the following properties.
      1. divisor + 1 = a+b
      2. a*b mod divisor = divisor - 1.

      Example 1
      31 + 1 = 13+19 and 13*19 MOD 31 = 30
      This also means that 31 divides 13^2-13-1 and 19^2-19-1

      Example 2
      59+ 1 = 26+34and 26*34 MOD 59 = 58
      This also means that 59 divides 26^2-26-1 and 34^2-34-1

      If a Carmichael can be formed from this series what is the smallest
      possible one?

      Markus
    • 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 7:06 AM
        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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