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Thread: Number Theory: Primitive Root

  1. #1
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    Number Theory: Primitive Root

    I'm doing some maths revision, also kinda weak in Number Theory. I would appreciate if I could get step by step explanation for the following question:

    Find a primitive root for Z*19. Z is integer. Attaching the image since I can't add the Z symbol

    Number Theory: Primitive Root-question1.png

    The answer is 2,3,10,13,14,15

    I looked at some video's on primitive root, correct me if i'm wrong here, First I need to find the Totient, which is 18 since 19 is a prime (n-1), so how do I get 2,3,10,13,14,15 ?

    Note to moderator: I tried posting this question earlier, for some reason I cant see my post. Kindly delete this post if its duplicate.
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  2. #2
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    Re: Number Theory: Primitive Root

    Look at powers of each element.

    Powers of 2 generate the whole cyclic group. Powers of 3 also. Powers of 4 do not. $4^9\equiv 1\pmod{19}$.

    The problem even tells you to use trial and error.
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  3. #3
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    Re: Number Theory: Primitive Root

    By the way, you mentioned the totient. You only need to check powers that are proper factors of 18. So, you check 2,3,6,9. But since 2 and 3 both divide 6, you only need to check powers of 6 and 9. If an element raised to the 6th or 9th power (mod 19) gives 1, it is not a primitive root. If $g^6 \pmod{19}$ and $g^9\pmod{19}$ are both NOT 1, then $g$ is a primitive root.
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