Re: Pell's equations and beyond

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

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