Results 1 to 2 of 2

Thread: Primitive roots

  1. #1
    Member
    Joined
    May 2008
    From
    Melbourne Australia
    Posts
    217
    Thanks
    29

    Primitive roots

    Suppose g is a primitive root modulo p (a prime) and suppose m|p-1 (1<m<p-1). How many solutions are there to the congruence:

    $\displaystyle x^m-g \equiv 0 $ (mod p)

    My attempt:

    $\displaystyle m*ind_gx \equiv ind_gg $ (mod $\displaystyle \phi(p)$)
    $\displaystyle \therefore m*ind_gx \equiv p-1 \equiv 0$ (mod $\displaystyle p-1$)
    $\displaystyle \therefore ind_gx \equiv 0$ (mod $\displaystyle p-1$)
    $\displaystyle \therefore x \equiv 1$ (mod p)

    Hence g=1 which is a contradiction so there are no solutions.

    I don't believe my own answer! I have not used the statement that
    m|p-1 which tells me that g.c.d(m,$\displaystyle \phi(p))$ = m
    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 Kiwi_Dave View Post
    Suppose g is a primitive root modulo p (a prime) and suppose m|p-1 (1<m<p-1). How many solutions are there to the congruence:

    $\displaystyle x^m-g \equiv 0 $ (mod p)
    If there is a solution $\displaystyle x$ then $\displaystyle x\equiv g^y~ (p)$.
    Thus, $\displaystyle g^{my} \equiv g^1 ~ (p)$.
    Therefore, $\displaystyle my\equiv 1 ~ (p-1)$.
    Now $\displaystyle \gcd(m,p-1) =m$ and $\displaystyle m\not | 1$.
    Thus, there are no solutions to this congruence.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Primitive roots
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: Jul 10th 2011, 05:15 PM
  2. Primitive Roots
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: Nov 17th 2008, 08:09 PM
  3. Primitive roots
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Mar 15th 2007, 09:19 AM
  4. Primitive roots
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Nov 21st 2006, 07:05 AM
  5. Primitive Roots
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Nov 18th 2006, 01:43 PM

Search Tags


/mathhelpforum @mathhelpforum