number generator. how to show that a number is/is not a generator? 
I have this question. But I am real weak at generator and do not how to tackle this problem? Can anyone give me some guidance? thanks :) 
 A: $\mathbb{Z}_{23}^*$ is the cyclic group $C_{22}$, and a generator is simply a number $a$ such that $o(a)=22$ ($a$'s order), that is, the lowest number $n$ such that $a^n=1$ is $n=22$.
A basic theorem in group theory says that the order of an element in a finite group divides the order of a group; $C_{22}$ contains $22$ elements and $22=11\cdot2$ so the order of all elements, including $3$ and $5$ is either $2$, $11$ or $22$.
To show that $5$ is a generator, simply calculate $5^2$ and $5^{11}$ (just multiply and save intermediate values - because you're working modulo $23$, the number won't get very big) and show that neither are $0$, and then you can conclude that $\{5,5^2,\ldots,5^{22}\}=\mathbb{Z}_{23}^*$.
To show that $3$ is not a generator, show that either $3^2 = 1$ or $3^{11}=1$, and then you will be able to compute the values of $\{3,3^2,\ldots,3^{22}\}$ - at some point you'll get to $1$ and then you can stop because the numbers will just repeat.
A: Hint: The questions are related. So start in the beginning: Find the sets $\{5,5^2,\ldots,5^{22}\}$ and $\{1,3,3^2,\ldots,3^{22}\}$. When you got this right, the question for the generators is not too hard any more.
A: I got it. 
For 5, it is basically 5^1 mod 23, 5^2 mod 23 ... You will notice the result is unique. No duplication of value. 
For 3, there is duplication of value. And there is a clear pattern of number repeating itself after a certain power. 
