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Topic: x^2 - Ay^2 =1
Replies: 125   Last Post: Jun 28, 2007 2:32 PM

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 Philippe 92 Posts: 614 Registered: 1/25/05
Re: x^2 - Ay^2 =1
Posted: May 17, 2007 6:24 AM

Vincenzo Librandi wrote :
>
> Into Pell's equation: x^2-Ay^2=1
> If A= 25n^2-14n+2 (with n integer >=0) then
> y=250n-70, and x=1250n^2-700n+99

may be OK only for n>=1

> ===================================
>
> Into Pell's equation: x^2-Ay^2=1
> If A= 25n^2-36n+13 (with n integer >=0) then
> y=250n-180, and x=1250n^2-1800n+649

may be OK only for n != 1

> ==============================
>
> Into Pell's equation: x^2-Ay^2=1
> If A= 25n^2-2n (with n integer >=1) then
> y=250n-10, and x=1250n^2-100n+1

False.
The fundamental solution is
X0 = 25n - 1, Y0 = 5
the CF expansion is [5n-1,1,3,1,10n-2,1,3,1,10n-2...]

> ===================================
>
> Into Pell's equation: x^2-Ay^2=1
> If A= 25n^2+2n (with n integer >=1) then
> y=250n+10, and x=1250n^2+100n+1

False
The fundamental solution is
X0 = 25n+1, Y0 = 5
The CF expansion is [5n,5,10n,5,10n...]

> ===================================
>
> To continue......

... this endless list of specific cases.
No finite set of quadratic forms can cover the full set of integers.

Regards.

--
Philippe C., mail : chephip+news@free.fr
site : http://chephip.free.fr/ (recreational mathematics)

Date Subject Author
5/10/07 Vincenzo Librandi
5/10/07 Philippe 92
5/10/07 Vincenzo Librandi
5/10/07 Philippe 92
5/11/07 Vincenzo Librandi
5/11/07 Philippe 92
5/13/07 Iain Davidson
5/12/07 Vincenzo Librandi
5/17/07 Philippe 92
5/12/07 Vincenzo Librandi
5/13/07 Gerry
5/14/07 Philippe 92
5/14/07 Vincenzo Librandi
5/17/07 Philippe 92
5/14/07 Vincenzo Librandi
5/14/07 Iain Davidson
5/15/07 Philippe 92
5/16/07 Vincenzo Librandi
5/17/07 Philippe 92
5/16/07 Gerry
5/16/07 Philippe 92
5/16/07 Vincenzo Librandi
5/17/07 Philippe 92
5/17/07 Vincenzo Librandi
5/17/07 Iain Davidson
5/17/07 Philippe 92
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5/19/07 Philippe 92
5/19/07 Vincenzo Librandi
5/19/07 Philippe 92
5/20/07 Vincenzo Librandi
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5/20/07 Iain Davidson
5/20/07 Philippe 92
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5/26/07 Vincenzo Librandi
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5/26/07 Phil Carmody
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