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Find a solution for a Pell's equation

  1. Jun 24, 2011 #1
    Hi all,

    I'm trying to find the integer solutions for

    6n^2 -18n +16 = m^2

    using pell equation, any idea


    please help
    thanks
     
  2. jcsd
  3. Jun 24, 2011 #2

    disregardthat

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    Try to write 6n^2-18n in a different way to make the equation look like a standard pell equation.
     
  4. Jun 24, 2011 #3
    I tryed and found this form:

    3/2 (2n-3)^2 -m^2 = -5/2
     
  5. Jun 24, 2011 #4
    OR

    m^2 - 3/2 (2n-3)^2 = 5/2
     
  6. Jun 24, 2011 #5
    and by multiply the second form with 2/5 I get

    2/5 m^2 - 3/5 (2n-3)^2 = 1
     
  7. Jun 24, 2011 #6
  8. Jun 25, 2011 #7
    Just a few solutions (n, m): (0, 4), (1, 2), (2, 2), (3, 4), (8,16), (17,38), (66,158), (155, 376), (640, 1564), ...
     
  9. Jun 25, 2011 #8
    Some solutions with negative n: (-5,16), (-14, 38), (-63,158), (-152,376), (-637,1564)
     
  10. Jun 25, 2011 #9
    thanks, great help

    but how I know all the +solutions as large as I can.
    what steps you follow so you found these solutions?
    (no negativs solutions)
     
  11. Jun 25, 2011 #10
    I got the solutions with the help of a small computer program:
    It goes like this:

    For n = 0 to 999 do
    [itex]\space\space\space[/itex]w= 6*n[itex]^{2}[/itex]-18*n+16
    [itex]\space\space\space[/itex]If w is a perfect square i.e. w=m[itex]^{2}[/itex]
    [itex]\space\space\space\space[/itex] then Return (n,m)
    [itex]\space\space\space[/itex]end_If
    end_For

    ...sort of experimental math !
     
  12. Jun 26, 2011 #11


    You go on Alperton's site ( which I provided above in my first post ) and you enter your parameter to that your equation becomes:
    6 x2 - y2 - 18 x + 16 = 0

    and you click on solve it and it does it for you. Now under Modes ( below the solve it button ), you have two choices:
    1- just solution ( for when you are in a hurry )
    2- step by step ( for when you want to in fact learn something )

    if you choose step by step, you will be taught how to solve this kind of equation. But if you do not take the time to check every single post ( like mine above ), then you may miss some important stuff.
     
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