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Math Help - Primitive roots

  1. #1
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    Primitive roots

    If g is a primitive root mod 13 then I can show that g^11 is also a primitive root.

    My question is how can I deduce that the product of all the primitive roots mod 13 are congruent to 1 (mod 13) from this result?

    Is there a corresponding result for the product of all the primitive roots mod 169?

    Thanks in advance.
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  2. #2
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    find the condition for k such that g^k is also a primitive root mod 13.
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    I'm aware of the condition for k to such that g^k is a primitive root mod 13. In fact, I have shown that g^11 is a primitive root mod 13.

    But my question is how can I use this information to deduce that the product of all the primitive roots mod 13 is congruent to 1 mod 13.

    I was then curious if a similar result holds for the product of all primitive roots mod 169.
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  4. #4
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    Quote Originally Posted by Cairo View Post
    I'm aware of the condition for k to such that g^k is a primitive root mod 13. In fact, I have shown that g^11 is a primitive root mod 13.

    But my question is how can I use this information to deduce that the product of all the primitive roots mod 13 is congruent to 1 mod 13.

    I was then curious if a similar result holds for the product of all primitive roots mod 169.
    Given that g is a primitive root of 13 , all the primitive roots are given by g^k, where (k,12)=1; so the primitive roots of 13 are g^1, g^5, g^7, and g^{11}.

    Then the product of all the primitive roots of 13 is congruent to g^{1+5+7+11}=g^{24} modulo 13 . By Fermat's Theorem, g^{24}=(g^{12})^2\equiv1\pmod {13}.

    The general result is the following: Let m be a positive integer which has a primitive root. Then the product of the primitive roots of m is congruent to 1 modulo m ; the exceptions are m=3, 4, and 6 .

    By the way, the integers m that have primitive roots are 1, 2, 4, p^k, and 2p^k, where p is an odd prime.
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  5. #5
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    Thank you both for your help with this.

    I managed to prove the first result in the end. I never thought to use Fermat's Theroem until It struck me at the last minute.

    I would like some clarification on your second point melese if that is possible?

    If g is a primitive root mod 169, then so is g+13. So how are the rest of the primitive roots obtained and how does their product become congruent to 1 mod 169?
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