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Re: Carmichael numbers of order 2

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  • David Broadhurst
    ... Richard Finch calls these unusually strong Lucas-Carmichael-minus (uLC-) numbers, with p^2-1|N-1, for every prime p|N. See:
    Message 1 of 12 , Apr 3 7:09 AM
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      --- In primenumbers@yahoogroups.com,
      "David Broadhurst" <d.broadhurst@...> wrote:

      > "What are the first 3 Carmichael numbers of order 2?"
      >
      > 443372888629441
      > 39671149333495681
      > 842526563598720001

      Richard Finch calls these
      "unusually strong Lucas-Carmichael-minus" (uLC-) numbers,
      with p^2-1|N-1, for every prime p|N. See:
      http://www.chalcedon.demon.co.uk/rgep/p20.pdf

      I found the third of these by mining Richard's file of
      Carmichael numbers between 10^17 than 10^18.
      The first two had already been noted in
      http://www.chalcedon.demon.co.uk/rgep/cartable.html

      Richard classified
      582920080863121 = 41 * 53 * 79 * 103 * 239 * 271 * 509
      as a "strong", but not "unusually strong",
      Lucas-Carmichael-minus number
      since in this case both p-1 and p+1
      divide N-1, for each prime p|N, but p^2-1
      does not divide N-1 in the cases p = 79, 239, 271,
      where p^2 = 1 mod 2^5
      whilst N = 17 mod 2^5.

      If we ask merely that p-1 and (p+1)/2 divide N-1,
      then the following 3 numbers also occur, for N < 10^18:
      28295303263921
      894221105778001
      2013745337604001
      making 7 in all, of which only

      842526563598720001
      = 17 * 61 * 71 * 89 * 197 * 311 * 769 * 2729

      occurs for 10^18 > N > 10^17.

      Finally, I remark that Richard found precisely one
      "unusually strong Lucas-Carmichael-plus" (uLC+) number,
      with p^2-1|N+1, for prime p|N and N < 10^13, namely

      79397009999 = 23 * 29 * 41 * 43 * 251 * 269.

      David
    • David Broadhurst
      Oh dear, another silly typo: I meant Richard Pinch, of course.
      Message 2 of 12 , Apr 3 7:21 AM
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        Oh dear, another silly typo: I meant Richard Pinch, of course.
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