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Re: x^2 - Ay^2 =1
Posted:
Jun 4, 2007 3:02 AM
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Vincenzo Librandi wrote :
> Vincenzo Librandi wrote:
>
>> For Pell's equation x^2-Ay^2=1
>> let n>=h (h>0) then
>> For A=h^2*n^2+1 ==> y=2n*sqrt(h), x=2n^2*h^2+1;
>> For A=h^2*n^2-1 ==> y=2n*sqrt(h), x=2n^2*h^2-1;
>
> For Pell's equation x^2-Ay^2=1
> let n>=h (h>0); w>=1; n=0 (mod.w), then
> For A=h^2*n^2+w =>y=(2n*sqrt(h))/w, x=((2n^2*h^2)/w))+1;
> For A=h^2*n^2-w =>y=(2n*sqrt(h))/w, x=((2n^2*h^2)/w))-1;
>
> Vincenzo Librandi
> vincenzo.librandweoz@alice.it
y^2 = 4n^2*h/w^2
A*y^2 + 1 = 4*h^3*n^4/w^2 + 4*h*n^2/w + 1
x^2 = 4*h^4*n^4/w^2 + 4*h^2*n^2/w + 1
should be y=2n*h/w, not y=2n*sqrt(h)/w
However, "n=0 mod w" should be replaced.
That is replace formally any n by n*w
A = h^2*w^2*n^2 + w, x = 2w*h^2*n^2 + 1, y = 2h*n, for any n>0,
same for the other one.
Regards.
--
Philippe C., mail : chephip+news@free.fr
site : http://chephip.free.fr/ (recreational mathematics)
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