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Bill Allombert on Tue, 19 Dec 2023 16:38:55 +0100
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Re: Pell's equations and beyond
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- To: pari-users@pari.math.u-bordeaux.fr
- Subject: Re: Pell's equations and beyond
- From: Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>
- Date: Tue, 19 Dec 2023 16:38:49 +0100
- Delivery-date: Tue, 19 Dec 2023 16:38:56 +0100
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- In-reply-to: <eed101e340e93e47fae82524ee3b93f9@stamm-wilbrandt.de>
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On Mon, Dec 18, 2023 at 07:31:56PM +0100, hermann@stamm-wilbrandt.de wrote:
> I stumbled over
> https://math.stackexchange.com/a/3341210
>
> on finding solution for Pell's equation x^2-D*y^2=1 for D=61.
>
> Then I implemented my pell.gq (bottom) that did the job for any D.
>
> Then I found this 2008 posting from Karim:
> https://pari.math.u-bordeaux.fr/archives/pari-users-0811/msg00001.html
>
>
> 1) How do these commands from Karim's posting reveal x and y?
>
> ? quadunit(61)
The discriminant of x^2-61 is 4*61, not 61, so you should do
? u=quadunit(4*61)
%3 = 29718+3805*w
? norm(u)
%4 = -1
So here the unit norm is -1, so this solves x^2-61*y^2 = -1
To solve for 1, square it:
? u^2
%5 = 1766319049+226153980*w
Cheers,
Bill