Loading ...
Sorry, an error occurred while loading the content.

Re: [PrimeNumbers] Digest Number 3717

Expand Messages
  • Kermit Rose
    ... 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
    • 0 Attachment
      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.
    Your message has been successfully submitted and would be delivered to recipients shortly.