Results 1 to 4 of 4

Math Help - Fermat's Theorem

  1. #1
    Junior Member
    Joined
    Sep 2010
    From
    London
    Posts
    28

    Fermat's Theorem

    Hi,

    I have difficulties with below problem:

    Let p and \theta be primes with  \theta>p such that p is not a factor of \theta - 1. As \theta is a prime, we know that for any x\epsilon Z we can write x^{\theta-1} in the form x^{\theta-1} = k\theta+1 for some integer k. Furthermore, since the greatest common divisor of p and \theta-1 is 1, we can write 1=ap+b(\theta-1) for some integers a and b.
    - Show that every integer x can be written in the form x=y^p+j\theta for some integer y and some integer j.
    - Prove the First Case of Fermat's Last Theorem for the exponents 13,17 and 19. (That is, show that if there is a nonzero integer solution to x^p+y^p=z^p for p=13 then 13 is a factor of xyz. and so on.)
    Any help would be highly appreciated.
    Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,987
    Thanks
    1650

    Re: Fermat's Theorem

    Misread statement.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,987
    Thanks
    1650

    Re: Fermat's Theorem

    Quote Originally Posted by Gibo View Post
    Hi,

    I have difficulties with below problem:

    Let p and \theta be primes with  \theta>p such that p is not a factor of \theta - 1. As \theta is a prime, we know that for any x\epsilon Z we can write x^{\theta-1} in the form x^{\theta-1} = k\theta+1 for some integer k.
    Are your sure we know this? If, for example, \theta= 3 and x= 6, this says that 6^{3-1}= 6^2= 36= 3k+ 1. And I don't believe this is true.

    Furthermore, since the greatest common divisor of p and \theta-1 is 1, we can write 1=ap+b(\theta-1) for some integers a and b.
    - Show that every integer x can be written in the form x=y^p+j\theta for some integer y and some integer j.
    - Prove the First Case of Fermat's Last Theorem for the exponents 13,17 and 19. (That is, show that if there is a nonzero integer solution to x^p+y^p=z^p for p=13 then 13 is a factor of xyz. and so on.)
    Any help would be highly appreciated.
    Thank you.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Sep 2010
    From
    London
    Posts
    28

    Re: Fermat's Theorem

    Thank you very much for pointing that, I found it really helpful.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: October 26th 2012, 03:35 AM
  2. Fermatís Theorem
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: September 27th 2011, 06:52 PM
  3. Fermat's Little Theorem
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: April 12th 2011, 05:28 AM
  4. Replies: 4
    Last Post: January 10th 2011, 08:51 AM
  5. Fermat's little theorem
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: May 17th 2009, 06:28 AM

Search Tags


/mathhelpforum @mathhelpforum