Gerry
Posts:
464
Registered:
7/13/06
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Re: x^2 - Ay^2 =1
Posted:
May 24, 2007 1:58 AM
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On May 24, 6:00 am, Gerry <Gerry...@gmail.com> wrote:
> On May 24, 3:19 am, "Philippe 92" <nos...@free.invalid> wrote:
>
>
>
>
>
> > Gerry wrote :
>
> > <snip unrelated quoting>
>
> > > Hi all,
>
> > > i know this is not directly related to the posted question but would
> > > like to ask your opinion on the following :
>
> > > for the values A=421,541,661 i checked for a rational solution for the
> > > Pell equation
> > > with d=A*N^2/4 such that x^2=d*y^2+1 where N is a certain square and
> > > found:
> > > (The solutions (A1,x1,y1) are considered as the fundamental solutions
> > > x1^2=A1*y1^2+1.)
>
> > > A=421
> > > d=197970713725/4 , x^2=39192403491994020520729/4 , y=444939
>
> > Why did you specifically choose N = 21685 ? from scratch ?
> > How did you get your rational solution ?
> > Why divided by 4 ?
>
> > If I choose N = 123, I get :
> > d = 6369309/4, x = 238 digits, y = 234 digits.
>
> > and if I choose N = 6 (why 6 ?), I get
> > x = ... 3879474045914926879468217167061449 / 2 = x1/2 !!!
>
> > > A1=421,
> > > x1=3879474045914926879468217167061449,
> > > y1=189073995951839020880499780706260
>
> > > [...]
>
> > > Is it easier to look for the (d,x,y) versus (A1,x1,y1) solution?
> > > (x,y seem to be much smaller)
>
> > It depends on the choice of N...
> > However you really solved u^2 - (A*N^2)*y^2 = 4 (as your x^2 is u^2/4)
> > solving x^2 - A*y^2 = k is generally harder than solving
> > ordinary Pell's equation = 1, just because you have to solve for more
> > cases : the = 1 equation has allways only one fundamental solution,
> > the = k may have several fundamental solutions, or even none at all.
>
> > > Is it possible to derive the solution (A1,x1,y1) from (d,x,y)?
>
> > I don't think so.
>
> > > (the gcd(y1,numerator(y*d/A))>1 seems to be always true)
>
> > and ???
>
> > Regards.
>
> > --
> > Philippe C., mail : chephip+n...@free.fr
> > site :http://chephip.free.fr/ (recreational mathematics)
>
> Hi Philippe
>
> ok i cleaned up the examples and found some relationships
> The solutions (D,X,Y) of the Pell Equation X^2=D*Y^2+1
> and the solutions (d,x,y) of the pell equation x^2=d*y^2+4
> with d=D*N^2
> seem to be related by X/x+1=Y/(y*N) , d=x+2, x=y^2+2
>
> Example 1:
> D=421
> X=3879474045914926879468217167061449
> Y=189073995951839020880499780706260
> N=21685
> d=197970713725
> x=197970713723 =y^2+2=197970713723
> y=444939
> X/x+1=Y/(y*N)=19596201745997010260364
> X^2
> =15050318872927532189776674998850090215109143585106390842393741979601
> D*Y^2+1
> =15050318872927532189776674998850090215109143585106390842393741979601
> x^2 =39192403491994020520729
> d*y^2+4=39192403491994020520729
>
> Example 2:
> D=541
> X=3707453360023867028800645599667005001
> Y=159395869721270110077187138775196900
> N=60037
> d=1950002780629
> x=1950002780627 =y^2+2=1950002780627
> y=1396425
> X/x+1=Y/(y*N)=1901255422226515943256564
> X^2
> =13745210416752261392240910232373075990720198494336152094568176869359010001
> D*Y^2+1
> =13745210416752261392240910232373075990720198494336152094568176869359010001
> x^2 =3802510844453031886513129
> d*y^2+4=3802510844453031886513129
>
> Example 3:
> D=661
> X=16421658242965910275055840472270471049
> Y=638728478116949861246791167518480580
> N=69605
> d=3202449832525
> x=3202449832523 =y^2+2=3202449832523
> y=1789539
> X/x+1=Y/(y*N)=5127842464913295374272764
> X^2
> =26967085944877022742373944206285355610692339100498164619615014484434716040?1
> D*Y^2+1
> =26967085944877022742373944206285355610692339100498164619615014484434716040?1
> x^2 =10255684929826590748545529
> d*y^2+4=10255684929826590748545529
>
> any comments are appreciated
>
> Regards
>
> Gerry- Hide quoted text -
>
> - Show quoted text -
Hi Phillipe,
>Why did you specifically choose N = 21685 ? from scratch ?
>How did you get your rational solution ?
>Why divided by 4 ?
The answer is d=y^2+4=DN^2=x+2
So it seems that the fundamental solutions
for a certain D for X^2=DY^2+1 are related to
the solutions of y^2+4=DN^2.
and the solution (x,y,d) can thus generate (X,Y,D)
Regards
Gerry
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