Results 1 to 10 of 10

Math Help - Question to Pell Equation

  1. #1
    Member
    Joined
    Nov 2009
    Posts
    169

    Question to Pell Equation

    Let x,y,d \in \mathbb{N}. For fixed d > 0 we want to find solution pairs (x,y) to the equation x^2-dy^2=1.

    Assume we have already found the solution (x_1,y_1) so that any other solution (x,y) has y_1 < y.

    Show that any pair (x_k,y_k) satisfying x_k + y_k\sqrt{d} = (x_1 + y_1\sqrt{d})^k for k > 1 is another solution.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by EinStone View Post
    Let x,y,d \in \mathbb{N}. For fixed d > 0 we want to find solution pairs (x,y) to the equation x^2-dy^2=1.

    Assume we have already found the solution (x_1,y_1) so that any other solution (x,y) has y_1 < y.

    Show that any pair (x_k,y_k) satisfying x_k + y_k\sqrt{d} = (x_1 + y_1\sqrt{d})^k for k > 1 is another solution.

    x_k^2-dy_k^2=(x_k+y_k\sqrt{d})(x_k-y_k\sqrt{d})=(x_1^2-dy_1^2)^k=1^k=1

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Nov 2009
    Posts
    169
    Quote Originally Posted by tonio View Post
    x_k^2-dy_k^2=(x_k+y_k\sqrt{d})(x_k-y_k\sqrt{d})=(x_1^2-dy_1^2)^k=1^k=1
    Do you use (x_k-y_k\sqrt{d}) = (x_1 - y1\sqrt{d})^k and if yes, why can you do it?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by EinStone View Post
    Do you use (x_k-y_k\sqrt{d}) = (x_1 - y1\sqrt{d})^k and if yes, why can you do it?

    It follows from a+b\sqrt{d}=(\alpha+\beta\sqrt{d})(\gamma+\delta\s  qrt{d})\Longrightarrow  a-b\sqrt{d}=(\alpha-\beta\sqrt{d})(\gamma-\delta\sqrt{d}) and induction (you can prove this by comparing the free coefficient

    and the \sqrt{d}-coefficient in both sides, remembering that \{1\,,\,\sqrt{d}\} is a free basis of the free abelian group \mathbb{Z}[\sqrt{d}] ).

    Tonio
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Nov 2009
    Posts
    169
    ok nice. Sorry for the break, I had some spring break . Now I want to prove:

    Show that there are no other solutions except the ones proven above.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    Quote Originally Posted by EinStone View Post
    ok nice. Sorry for the break, I had some spring break . Now I want to prove:

    Show that there are no other solutions except the ones proven above.
    The proof can be found here.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Feb 2010
    From
    New Jersey, USA
    Posts
    36
    What was Pell's full name?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    Quote Originally Posted by pollardrho06 View Post
    What was Pell's full name?
    John Pell
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member
    Joined
    Nov 2009
    Posts
    169
    Hey thanks for the proof, but I never heard of a "convergent" and Im not too familiar with continued fractions either. It would we nice to see a proof which uses more elementary tools, so to say.

    Can you help me?
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    Quote Originally Posted by EinStone View Post
    Hey thanks for the proof, but I never heard of a "convergent" and Im not too familiar with continued fractions either. It would we nice to see a proof which uses more elementary tools, so to say.

    Can you help me?
    Unfortunately, my knowledge of Diophantine equations are limited. I don't know how to explain this without using continued fractions.
    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's equation
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: June 4th 2009, 07:32 AM
  3. simple Pell's Equation
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: May 3rd 2009, 09:42 PM
  4. Pell equation
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: December 10th 2008, 01:00 PM
  5. please help me with these two questions! Pell equation
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: December 25th 2006, 09:53 AM

Search Tags


/mathhelpforum @mathhelpforum