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21856Re: qfbsolve

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  • Aldrich
    Sep 23, 2010
      --- 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
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