Every cyclic group with order > 2 has at least 2 distinct generators
Here's what I've got so far:
Either order of our group, $G$, is finite or infinite.
Suppose infinite: then our group is isomorphic to $\mathbb Z$ under addition. This group has two distinct generators, therefore so does $G$.
Suppose finite: Then $G$ is isomorphic to $\mathbb Z_n$ under addition modulo $n$. We know order of $G$ is $\geqslant 3$. If order of $G$ is odd, both 1 and 2 are generators of $\mathbb Z_n$ under addition modulo $n$, so $G$ has at least two generators.
If order of $G$ is even, then both 1 and some odd number between 1 and $n$ are generators. How do we determine this odd number?
Does this look like I'm on the right track?