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Re: Carmichael question

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  • David Broadhurst
    Congrats on a neat method, Richard. ... How did you rediscover Jack s 675557402^2-675557402-1 = 456377802721432201 = 29*211*281*22669*11708611 where 22669-1 =
    Message 1 of 77 , Nov 3, 2002
      Congrats on a neat method, Richard.

      However one things puzzles me:

      > it trots out the Carmichaels
      > everyone else has found,
      > in a couple of minutes on a P3-800.

      How did you rediscover Jack's

      675557402^2-675557402-1 =
      456377802721432201 =
      29*211*281*22669*11708611

      where

      22669-1 = 2^2*3*1889
      11708611-1 = 2*3*5*23*71*239

      are not so smooth?

      The seeds {211,281} are too small
      to generate a result from the
      residue class of n=675557402 modulo
      m=211*281*lcm(210,280)=49804440

      With n/m>13, you would have needed 14 attempts at
      n=k*m+r
      on _each_ of your 128 CRT residue classes
      no so?

      Note that Jack's Carmichael has only 18 digits,
      whereas your 3 new finds have 23 digits.

      I suspect that there are quite a
      few n^2-n-1 Carmichaels below 23 digits
      that cannot be found by a double
      loop of residue classes from seeds {p,q}
      with both p-1 and q-1
      as smooth as your are hoping.

      But still, yours is a really neat way to
      find a subset of the targets!

      Here's another 23-digit find:

      151245190802^2-151245190802-1 =
      22875107740582140212401 =
      271*1151*3221*6841*36541*91081

      I think this is #10 and the largest yet?

      Best regards

      David
    • 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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