# Thread: Number Theory: Primitive Root

1. ## 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

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.

2. ## 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.

3. ## 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.