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Re: qfbsolve

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  • Aldrich
    ... Hi Kermit This is interesting material, but it does not really answer any of my questions. I ll check to see if any of references are more illuminating.
    Message 1 of 5 , Sep 23, 2010
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      --- In primenumbers@yahoogroups.com, Kermit Rose <kermit@...> wrote:
      >
      > On 9/22/2010 10:09 AM, primenumbers@yahoogroups.com wrote:
      > >
      > > There is 1 message in this issue.
      > >
      > > Topics in this digest:
      > >
      > > 1.1. Re: "wriggly" probable primes
      > > From: Aldrich
      > >
      > >
      > > Message
      > > ________________________________________________________________________
      > > 1.1. Re: "wriggly" probable primes
      > > Posted by: "Aldrich" aldrich617@... aldrich617
      > > Date: Wed Sep 22, 2010 1:38 am ((PDT))
      > >
      > >
      > >
      > > --- In primenumbers@yahoogroups.com, "djbroadhurst"<d.broadhurst@> wrote:
      > >>
      >
      > >> Exercise: Find two pairs of positive integers (x,y) such that
      > >> 4065702994722252685573484796054334194691713593576645739409115721859519
      > >> = 5x^2 + 5xy + y^2
      > >>
      > >> Comment: Pari-GP's "qfbsolve" enables a solution in two minutes.
      > >> Devotees of "issquare", like Aldrich, may take considerably longer.
      > >>
      > >> David
      > >>
      > >
      > > How does "qfbsolve" work? Will it enable fast solutions for all
      > > A, or just special cases? If it fails to work is A then proved
      > > prime?
      > >
      > > Aldrich
      > >
      > >
      >
      > Aldrich, here is a partial answer to your question.
      >
      > Bill Allombert is the author of qfbsolve.
      >
      > http://pari.math.u-bordeaux.fr/archives/pari-dev-0311/msg00004.html
      >
      > Hello PARI-dev,
      >
      > I have added a new function qfbsolve.
      >
      > qfbsolve(Q,p):
      >
      > Solve the equation Q(x,y) = p over the integers, where Q is a
      > imaginary
      > binary quadratic form and p a prime number.
      >
      > Return [x,y] as a two-components vector, or zero if there is no
      > solution.
      > Note that this functions return only one solution and not all the solutions.
      >
      > This is a preliminary implementation. I plan to allow non prime p
      > and real binary quadratic.
      >
      > This use a random polynomial time algorithm similar to cornacchia but
      > probably due to Gauss, using the reduction of quadratic form.
      >
      > Cheers,
      > Bill.
      >

      Hi Kermit

      This is interesting material, but it does not really answer
      any of my questions. I'll check to see if any of references
      are more illuminating.

      Aldrich
    • djbroadhurst
      ... The Cornacchia-Smith algorithm is given by Henri Cohen as CCANT 1.5.2 and by also by Crandall and Pomerance 2.3.12. David
      Message 2 of 5 , Oct 2, 2010
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        --- In primenumbers@yahoogroups.com,
        "Aldrich" <aldrich617@...> wrote:

        > This is interesting material, but it does not really answer
        > any of my questions. I'll check to see if any of references
        > are more illuminating.

        The Cornacchia-Smith algorithm is given by Henri Cohen
        as CCANT 1.5.2 and by also by Crandall and Pomerance 2.3.12.

        David
      • djbroadhurst
        ... Exercise: Find two pairs of positive integers (x,y) such that 5*x^2 + 5*x*y + y^2 = (137^137 + 1992)*(137^137 + 3464) Comment: This exercise may solved in
        Message 3 of 5 , Oct 2, 2010
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          --- In primenumbers@yahoogroups.com,
          "djbroadhurst" <d.broadhurst@...> wrote:

          > The Cornacchia-Smith algorithm is given by Henri Cohen
          > as CCANT 1.5.2 and by also by Crandall and Pomerance 2.3.12.

          Exercise: Find two pairs of positive integers (x,y) such that
          5*x^2 + 5*x*y + y^2 = (137^137 + 1992)*(137^137 + 3464)

          Comment: This exercise may solved in 10 milliseconds.

          David
        • Aldrich
          ... Looks like a special case, a put-up job as the posties say. quibble, quibble. a.
          Message 4 of 5 , Oct 5, 2010
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            --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
            >
            >
            >
            > --- In primenumbers@yahoogroups.com,
            > "djbroadhurst" <d.broadhurst@> wrote:
            >
            > > The Cornacchia-Smith algorithm is given by Henri Cohen
            > > as CCANT 1.5.2 and by also by Crandall and Pomerance 2.3.12.
            >
            > Exercise: Find two pairs of positive integers (x,y) such that
            > 5*x^2 + 5*x*y + y^2 = (137^137 + 1992)*(137^137 + 3464)
            >
            > Comment: This exercise may solved in 10 milliseconds.
            >
            > David
            >

            Looks like a special case, a put-up job as the posties
            say. quibble, quibble.

            a.
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