I am trying to solve following problem. I have done the entire problem, so I'm not asking anyone to do the problem for me. But I need some confirmation on whether or not the very last part of my argument is valid.
Problem: Write an xy-table for $y ≡ (−2)^x$ mod $23 $ to verify $−2$ is a primitive root. Using the substitution $u ≡ (−2)^x$ mod $23$, rewrite the congruence $11u^7 ≡ 1$ mod $23$ as an equivalent congruence of the form $ax + b ≡ 0$ mod $n$. Then, solve the equivalent congruence you found and use it to derive the solution to the original congruence $11u^7$ ≡ $1$ mod $23$.
First, I had no trouble creating the table, nor any difficulty arguing that $-2$ is a primitive root. Then, when I rewrote the congruence $11u^7 ≡ 1$ mod $23$, and using the given substitution, I found that $91x + 9 ≡ 0$ mod $22$.
Finally, I determined that $91x + 9 ≡ 0$ mod $22$ can be simplified down to $x ≡ 19$ mod $22$.
Everything below this bolded text is where asking whether or not my argument is valid and legitimate:
I said that $x ≡ 19$ mod $22$, which implies that $x ≡ 19$ mod $23$. Thus, $u ≡ (-2)^{19} ≡ 20$ mod $23$. EDIT: Thanks to Bill Dubuque, I realized this implication is completely wrong!
If someone could verify the validity of this last step, that would be greatly appreciated.
Thank you.
