- On 9/22/2010 10:09 AM, primenumbers@yahoogroups.com wrote:
>

Aldrich, here is a partial answer to your question.

> 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

>

>

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. - --- In primenumbers@yahoogroups.com, Kermit Rose <kermit@...> wrote:
>

Hi Kermit

> 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.

>

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 - --- In primenumbers@yahoogroups.com,

"Aldrich" <aldrich617@...> wrote:

> This is interesting material, but it does not really answer

The Cornacchia-Smith algorithm is given by Henri Cohen

> any of my questions. I'll check to see if any of references

> are more illuminating.

as CCANT 1.5.2 and by also by Crandall and Pomerance 2.3.12.

David - --- In primenumbers@yahoogroups.com,

"djbroadhurst" <d.broadhurst@...> wrote:

> The Cornacchia-Smith algorithm is given by Henri Cohen

Exercise: Find two pairs of positive integers (x,y) such that

> as CCANT 1.5.2 and by also by Crandall and Pomerance 2.3.12.

5*x^2 + 5*x*y + y^2 = (137^137 + 1992)*(137^137 + 3464)

Comment: This exercise may solved in 10 milliseconds.

David - --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
>

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

>

>

> --- 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

>

say. quibble, quibble.

a.