Insights · Michaël PerreaultFoundations for Mathematical Reality

Mathematical Table of Generators

Let us summarize the above results in a clear table.

Group	Generators Exist?	Explanation
$\mathbb{Z}_p$	Yes, all $k$ with $1 \leq k < p$	$p$ is prime, so $\gcd(k, p) = 1$ for all nonzero $k$
$\mathbb{Z}_n$	Yes, for $k$ with $\gcd(k, n) = 1$	$n$ not prime, only some $k$ are coprime to $n$

Here, $\gcd(k, n)$ stands for the greatest common divisor of $k$ and $n$ .