Results 1 to 2 of 2

Math Help - Diophantine Equation Proof

  1. #1
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    652
    Thanks
    2
    Awards
    1

    Diophantine Equation Proof

    Show that the diophantine equation x^4 - y^4 = z^2 has no solutions in nonzero integers using the method of infinite descent.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by Aryth View Post
    Show that the diophantine equation x^4 - y^4 = z^2 has no solutions in nonzero integers using the method of infinite descent.
    Assume that there is a positive integer solution, then pick one that has z to be minimum. We have that (x,z)=(y,z)=(y,z) = 1 since otherwise we would be able to cancel by a common factor, contradicting that z is mimimal. Now write z^2 + (y^2)^2 = (x^2)^2. This is a Pythagorean equation, there are two cases either y is even or odd. If odd then z=2ab,y^2=a^2-b^2,x^2=a^2+b^2 for positive integers a>b and (a,b)=1. But then a^4 - b^4 = (a^2-b^2)(a^2+b^2) = (xy)^2. Thus, we found another solution to this Diophantine equation which is even smaller then the original, thus we have a contradiction by infinite descent. Now you do the case when y is even.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] diophantine equation
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: January 11th 2011, 07:26 PM
  2. Diophantine equation
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: May 30th 2010, 01:44 PM
  3. Diophantine Proof Help using Pythagorean Triples
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: July 9th 2009, 08:09 PM
  4. diophantine equation
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: June 3rd 2009, 06:08 AM
  5. Diophantine equation
    Posted in the Number Theory Forum
    Replies: 6
    Last Post: December 14th 2006, 12:13 PM

Search Tags


/mathhelpforum @mathhelpforum