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## Re: [PrimeNumbers] Digest Number 3717

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• ... http://www.math.ou.edu/~kmartin/nti/chap5.pdf if a^2 - d b^2 = 1 Then 1 = (a^2 - d b^2)^2 = ( (a - b
Message 1 of 1 , Jun 19, 2013
On 6/13/2013 10:45 AM, primenumbers@yahoogroups.com wrote:
> ________________________________________________________________________
> 1. some questions about algebraic factoring in the field of adjoined sq
> Posted by: "bhelmes_1" bhelmes@... bhelmes_1
> Date: Wed Jun 12, 2013 5:27 am ((PDT))
>
> 1. Question
> What is the fastest way to solve the equation of Pell.
>

http://www.math.ou.edu/~kmartin/nti/chap5.pdf
<http://www.math.ou.edu/%7Ekmartin/nti/chap5.pdf>

if a^2 - d b^2 = 1

Then
1
= (a^2 - d b^2)^2

= ( (a - b sqrt(d)) (a + b sqrt(d)) )^2

= ( ( a + b sqrt(d))^2 ( a - sqrt(d))^2 )

= ( ( a^2 + d b^2) + (2 a b) sqrt(d)) ) ( (a^2 + d b^2) - (2 a b)
sqrt(d)) )

= ( (a^2 + d b^2)^2 - d ( 2 a b)^2)

(a1^2 - d b1^2) ( a2^2 - d b2^2) = (a1 a2 + d b1 b2)^2 - d (a1 b2 + b1 a2)^2

These recursively formulas quickly generate solutions to Pell's equations.
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