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

Re: Pell Equation

Expand Messages
  • djbroadhurst
    typo: I meant d=3 mod 4
    Message 1 of 5 , Jun 23, 2013
      typo: I meant d=3 mod 4

      --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
      >
      >
      >
      > --- In primenumbers@yahoogroups.com,
      > Kermit Rose <kermit@> wrote:
      >
      > > if
      > > a1^2 - d b1^2 = 1
      > > a2^2 - d b2^2 = 1
      > > then
      > > (a1^2 + d b1^2)^2 - (2 d a1 b1)^2 = 1
      > > (a1 a2 + d b1 b2)^2 - d (a1 b2 + b1 a2)^2 = 1
      >
      > Indeed. Yet it is both clearer and more efficient to
      > take powers of a unit. I illustrated this for positive
      > d=4 mod 3, where one takes uses quadunit(4*d),
      > with positive norm. The extension to other
      > discriminants is straightforward. Moreover
      > "quadunit" yields the building block that Kermit
      > assumed to have been found, by undisclosed methods :-)
      >
      > David
      >
    • Maximilian Hasler
      ... no : the 3rd equation is wrong. M. [Non-text portions of this message have been removed]
      Message 2 of 5 , Jun 23, 2013
        On Sun, Jun 23, 2013 at 7:34 AM, djbroadhurst <d.broadhurst@...>wrote:

        > **
        > Kermit Rose <kermit@...> wrote:
        > > if
        > > a1^2 - d b1^2 = 1
        > > a2^2 - d b2^2 = 1
        > > then
        > > (a1^2 + d b1^2)^2 - (2 d a1 b1)^2 = 1
        > > (a1 a2 + d b1 b2)^2 - d (a1 b2 + b1 a2)^2 = 1
        >
        > Indeed.
        >

        no : the 3rd equation is wrong.

        M.


        [Non-text portions of this message have been removed]
      • Kermit Rose
        On 6/24/2013 12:07 PM, Maximilian Hasler ... (a1^2 + d b1^2)^2 - (2 d a1 b1)^2 = 1 should have been (a1^2 + d b1^2)^2 - d (2 a1 b1)^2 = 1 Kermit
        Message 3 of 5 , Jun 24, 2013
          On 6/24/2013 12:07 PM,

          "Maximilian Hasler"

          wrote:
          > ________________________________________________________________________
          > 1a. Re: Pell Equation
          > Posted by: "Maximilian Hasler" maximilian.hasler@... maximilian_hasler
          > Date: Sun Jun 23, 2013 12:37 pm ((PDT))
          >
          >
          >> **
          >> Kermit Rose <kermit@...> wrote:
          >>> if
          >>> a1^2 - d b1^2 = 1
          >>> a2^2 - d b2^2 = 1
          >>> then
          >>> (a1^2 + d b1^2)^2 - (2 d a1 b1)^2 = 1
          >>> (a1 a2 + d b1 b2)^2 - d (a1 b2 + b1 a2)^2 = 1
          >>
          > no : the 3rd equation is wrong.
          >
          > M.
          >

          (a1^2 + d b1^2)^2 - (2 d a1 b1)^2 = 1

          should have been

          (a1^2 + d b1^2)^2 - d (2 a1 b1)^2 = 1


          Kermit
        Your message has been successfully submitted and would be delivered to recipients shortly.