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Re: Lucas super-pseudoprime puzzle

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  • djbroadhurst
    ... Indeed :-) Here my Chinese speed-up is huge. Please note that I counted *all* the (q,n) pairs with square-free non-Carmichael n, coprime to 6, and more
    Message 1 of 46 , Nov 2, 2010
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      --- In primenumbers@yahoogroups.com,
      "mikeoakes2" <mikeoakes2@...> wrote:

      > if(fac>3
      > then the run times collapse drastically

      Indeed :-) Here my Chinese speed-up is huge.

      Please note that I counted *all* the (q,n) pairs with
      square-free non-Carmichael n, coprime to 6, and more
      than 3 prime divisors. You stopped when you found
      just one q, for a given n, not so?
      So you may have missed the dramatic existence of
      more than 70,000 (q,n) pairs here:

      > n=7056721 [7, 1; 47, 1; 89, 1; 241, 1] is ok

      We *already* knew that it was "ok", since q=1 comes from
      > "Mike Oakes" Chebyshev NMBRTHRY

      What I did not know (but very soon found)
      is that for n = 7056721 there are

      > precisely 75383 integers q
      > such that n > q > 0 and V(p,q,n) = p mod n,
      > for every integer p, namely those with
      > q = 1 mod 47 and kronecker(q,241) > -1

      I still haven't fully digested that very interesting kronecker.
      Why does 241 care about the kronecker, while 7 and 89 do not?

      David
    • djbroadhurst
      ... http://physics.open.ac.uk/~dbroadhu/cert/dbmo116.out gives my 116, in the format [n, factors, number of solutions] With n
      Message 46 of 46 , Nov 9, 2010
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        --- In primenumbers@yahoogroups.com,
        "mikeoakes2" <mikeoakes2@...> wrote:

        > > My revised count up to 2*10^10 is 116.
        > My (original) count up to 2*10^10 was 105.
        > So it must have missed 11, i.e. a bigger proportion.

        http://physics.open.ac.uk/~dbroadhu/cert/dbmo116.out
        gives my 116, in the format [n, factors, number of solutions]

        With n < 2*10^10, the record-holder for the number of solutions is
        [2214495361, [13, 17, 23, 29, 83, 181], 147407]
        which googles quite nicely, linking to
        http://www.cs.rit.edu/usr/local/pub/pga/fibonacci_pp

        David
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