Results 1 to 3 of 3

Math Help - Pell's Equations

  1. #1
    Member
    Joined
    Apr 2009
    Posts
    190

    Pell's Equations

    Hi I haven't done Pell's Equations but i did read up the method to solve them. However, I'm not sure how i can show that these 2 equations each have infinite integers solutions for x and y.



    and

    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Aquafina View Post
    Hi I haven't done Pell's Equations but i did read up the method to solve them. However, I'm not sure how i can show that these 2 equations each have infinite integers solutions for x and y.



    and

    The idea is to show that given any solution, you can always find another solution where the numbers are bigger. More precisely, if (x,y) is a solution then there are positive integers a,b,c,d such that (ax+by,cx+dy) is a solution. The theory of Pell's equation provides systematic ways of finding these constants. But if you don't know the theory, you can often guess them by looking at the first few solutions.

    For the equation x^2-2y^2=1, the smallest solution is x=3, y=2. The next one is x=17, y=12. From that, you might guess that if (x,y) is a solution then so is (3x+4y,2x+3y). You can then confirm that conjecture by verifying that (3x+4y)^2 - 2(2x+3y)^2 = x^2-2y^2.

    Similarly for the other equation: the first two solutions are (5,2) and (49,20). If (x,y) is a solution then so is (5x+12y,2x+5y), because (5x+12y)^2 - 6(2x+5y)^2 = x^2-6y^2.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Apr 2009
    Posts
    190
    Thank you

    I have a similair question with linear Diophantine equations if you have the time to look at it:

    its on the thread Sum of Unknown..

    http://www.mathhelpforum.com/math-he...tml#post316217
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Pell's equation
    Posted in the Number Theory Forum
    Replies: 7
    Last Post: April 25th 2010, 06:22 PM
  2. Pell equations
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: December 8th 2009, 04:08 PM
  3. Pell's equation
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: June 4th 2009, 07:32 AM
  4. Pell equation
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: December 10th 2008, 01:00 PM
  5. Pell Equations
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 11th 2006, 09:07 AM

Search Tags


/mathhelpforum @mathhelpforum