b. Suppose $a$ is some primitive root of $19$ (it must exist for any prime!).
- What is the order of $a^2$, $a^3$, $a^4$, and $a^5($mod $19)$?
- What elements $a^k($mod $19)$, where $k =2, \ldots 18$ will also be primitive roots of $19$? (Formulate as an easy-to-use rule and justify.)
- Are these all primitive roots?
c. We are given that $3$ is a primitive root of $19$. Using (b), find all numbers from $2$ to $18$ which are the primitive roots of $19$. Explain.
Hint
You indicated that you knew the group of units $\Bbb Z_p^×$ is cyclic when $p$ is prime. (That's a feather in your cap.)
A root $a$ is primitive $\pmod{19}$ if and only if it has order $18$.
The order of $a^k$ is $\dfrac {18}{(18,k)}$ (a basic fact about cyclic groups. Here's a proof) . Thus for a primitive root we need $(18,k)=1$. There's $\varphi (18)=6$ of these.
So, given that $3$ is primitive, we have $3,3^5,3^7,3^{11},3^{13},3^{17}$. $\pmod {19}$, those are $3,15,2,10,14,13$.
For instance $\lvert a^2\rvert =\dfrac {18}{(18,2)}=\dfrac {18}2=9$.