Results 1 to 2 of 2

Math Help - Index Arithmetic Questions

  1. #1
    Member
    Joined
    Nov 2008
    Posts
    152

    Index Arithmetic Questions

    1) Show that if p is an odd prime and r is a primitive root of p, then ind_{r}(p-1) = (p-1)/2

    2) Prove that there are infinitely many primes of the form 8k+1.

    (Hint: Assume that p1, p2, ..., pn are the only primes of this form. Let Q = (2p1, p2....pn)^k + 1. Show that Q must have an odd prime factor different than p1, p2, ...., pn, and must be of the form 8k + 1.)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1
    For the first Q.

    r is a primitive root of p ,hence \text{ind}_r1=p-1

    Now, you should know that(theorem):

    p-1=\text{ind}_r(-1)(-1)\equiv \text{ind}_r(-1)+\text{ind}_r(-1)=2\text{ind}_r(-1)(\text{mod}(p-1)).


    Some better hints for 2:


    Prove:
    Let Q = (2p1, p2....pn)^4 + 1. Show that Q must have an odd prime factor different than  p1, p2, ...., pn, and must be of the form 8k + 1.)

    Using this:

    The odd prime divisors of n^4+1 are from the form of 8k+1
    Last edited by Also sprach Zarathustra; November 10th 2010 at 06:09 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Arithmetic Sequence - Series Questions
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: January 12th 2012, 08:54 PM
  2. Index Arithmetic
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: July 7th 2011, 12:57 AM
  3. Replies: 2
    Last Post: October 5th 2010, 01:57 PM
  4. Index Arithmetic
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: January 17th 2009, 02:27 PM
  5. Replies: 2
    Last Post: February 27th 2008, 06:30 AM

Search Tags


/mathhelpforum @mathhelpforum