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Re: 3-, 4-, and 5-factor Carmichael families: not terribly common

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  • WarrenS
    ... --another remarkable thing about this family by Chernick is that its change the sign of 1 version N = (6 x - 1) (12 x - 1) (18 x - 1) is another kind of
    Message 1 of 6 , Dec 13, 2012
      --- In primenumbers@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
      >
      > (6 x + 1) (12 x + 1) (18 x + 1)
      > is a Carmichael number if all three factors are simultaneously
      > prime and where x>0 is integer.

      --another remarkable thing about this family by Chernick is that
      its "change the sign of 1" version
      N = (6 x - 1) (12 x - 1) (18 x - 1)
      is another kind of interesting pseudoprime if all three factors are simultaneously prime and where x>0 is integer.

      Namely, every prime P dividing N has property that P+1 divides N+1;
      this causes certain Lucas-like primality tests to always think N is a prime (unless they find a factor).

      ----

      If N enjoyed both this property and the usual Carmichael property simultaneously, then N would be (in my terminology) a "2-generalized Carmichael."

      Broadhurst provided a list by Robert Gerbicz 1 April 2009,
      which it turns out was not a joke, of 144153 numbers N.
      Also, Broadhurst provided his own list of 246 numbers N.
      Unfortunately none of the members of either list
      satisfy my criteria for being 2-generalized Carmichael --
      that is, I demand that if P is prime and P|N, then (P+1)|(N+1) and (P-1)|(N-1).
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