Loading ...
Sorry, an error occurred while loading the content.

Re: Lucas super-pseudoprimes for Q <> 1

Expand Messages
  • djbroadhurst
    ... Proposition: For every integer pair (p,k), we have V(p, (23*k+11)*3103, 4638985) = p mod 4638985 Proof [using the Sun Tzu Suan Ching]:
    Message 1 of 46 , Nov 1, 2010
    • 0 Attachment
      --- In primenumbers@yahoogroups.com,
      "djbroadhurst" <d.broadhurst@...> wrote:

      > > 1080905, 1739089, 1992641, 2110159, found in only a few minutes
      > A few minutes later these turned up: 4013569, 4638985

      Proposition: For every integer pair (p,k), we have
      V(p, (23*k+11)*3103, 4638985) = p mod 4638985

      Proof [using the Sun Tzu Suan Ching]:
      v(p,q,n,m)=2*polcoeff(lift(Mod((p+x)/Mod(2,m),x^2+4*q-p^2)^n),0);
      n=4638985;
      t(p,k,m)=lift(v(p,(23*k+11)*3103,n,m)-p);
      s(m)=sum(p=1,m,sum(k=1,m,t(p,k,m)));
      if(issquarefree(n),fordiv(n,m,if(isprime(m),print1(s(m)" "))));
      0 0 0 0 0

      Comment: That gives 65 (q,n) pairs, beating Mike's 37 pairs. But
      http://tech.groups.yahoo.com/group/primenumbers/message/21953&var=0
      asked, on All-Hallows-Even, for more than 70,000 such pairs,
      with n not Carmichael.

      David Broadhurst, Hallowmas, 2010
    • 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
      • 0 Attachment
        --- 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
      Your message has been successfully submitted and would be delivered to recipients shortly.