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Math Help - Primitive Root questions

  1. #1
    Newbie
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    Sep 2010
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    Primitive Root questions

    Hi, I have 3 problems concerning primitive roots that I have been unable to work through. Any advice on how to solve these would be greatly appreciated!

    1. Show if g, h are primitive roots of p and p is odd, the least residue of gh is not a primitive root of p.

    2. Show 131071 = (2^17)-1 is prime.

    3. Show that k exists such that g^(k+1)=(g^k)+1 (mod p), where g is a primitive root of p and p is a prime.

    Thanks in advance!
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  2. #2
    Super Member PaulRS's Avatar
    Joined
    Oct 2007
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    Some ideas/hints:

    1) If g is a primitive root, what is g^{\tfrac{p-1}{2}}(\bmod.p)

    2) Well consider a prime q dividing n=2^{17}-1 , then the order of 2 mod q is ? . But the order of 2 mod q divides q-1 by Fermat's Little Theorem, this reduces the number of primes we have to check (to see if they divide n) in fact, remember that if n were not prime, there's a prime divisor which is no greater than \sqrt{n}.

    3) g^{k+1}-g^k\equiv{1}(\bmod.p) ...
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