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Re: k-generalized Carmichael numbers

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  • WarrenS
    ... --None of the three numbers listed in http://oeis.org/A175531 qualifies as a 2-generalized Carmichael number by my definition. Howe s definition on page
    Message 1 of 6 , Dec 12, 2012
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      > > http://alumnus.caltech.edu/~however/talks/FortCollins.pdf
      >
      > http://oeis.org/A175531 gives an update
      >
      > David

      --None of the three numbers listed in http://oeis.org/A175531
      qualifies as a "2-generalized Carmichael number" by my definition.

      Howe's definition on page 16 of his lecture slides is different from
      and I'm not even sure it is related to mine.
      I don't understand Howe's motivation, although I guess it must be
      spiritually similar to mine.

      I suspect a 2-generalized Carmichael exists with N<10^100.
    • djbroadhurst
      ... Yes. His was more interesting to me, in the context of pseudoprimality: https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1006&L=nmbrthry&T=0&F=&S=&P=51
      Message 2 of 6 , Dec 12, 2012
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        --- In primenumbers@yahoogroups.com,
        "WarrenS" <warren.wds@...> wrote:
        > Howe's definition on page 16 of his lecture slides is different

        Yes. His was more interesting to me, in the context
        of pseudoprimality:

        https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1006&L=nmbrthry&T=0&F=&S=&P=51

        http://tech.groups.yahoo.com/group/primenumbers/message/21461

        It is easy to construct larger solutions. For example
        http://tech.groups.yahoo.com/group/primenumbers/message/19971
        provides a link to 246 solutions in
        http://physics.open.ac.uk/~dbroadhu/cert/erdos2.out
        with n = 1 mod p^2-1, for each prime p|n.
        For 144153 such solutions, see the 12 MB file
        http://physics.open.ac.uk/~dbroadhu/cert/carmrob2.txt
        obtained by Robert Gerbicz on 1 April 2009

        David
      • WarrenS
        Ouch, there was a typo in the definition. Here is the message back again with typo corrected (k-1 changed to k): It seems the following concept ought to be
        Message 3 of 6 , Dec 12, 2012
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          Ouch, there was a typo in the definition. Here is the message back again
          with typo corrected (k-1 changed to k):

          It seems the following concept ought to be important:
          "k-Generalized Carmichael numbers."

          DEFINITION:
          If N is composite and squarefree and:
          for all primes p that divide N,
          p-1 happens to divide N-1,
          and:
          for each j=2,3,4,...,k,
          for all primes p that divide N,
          (p^j-1)/(p-1)
          happens to divide (N^j-1)/(N-1),
          then call N a "k-generalized Carmichael number."

          If you make k large enough as a function of N, then I guess
          k-generalized Carmichael numbers must stop existing.
          QUESTION: how large do you need?
          As an initial guess, if k>(logN)*(some positive constant)
          that is enough to stop N from being a
          k-Generalized Carmichael number.

          QUESTION:
          Can you find any k-generalized Carmichaels for k=2,3 or 4?
          [I believe my computer showed none exist when N<10^16.]
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