Cyclicity Condition for Units of Ring of Integers Modulo m

Theorem
Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $\left({\Z / n \Z, +, \times}\right)$ be the ring of integers modulo $n$.

Let $U = \left({ \left({ \Z / n \Z }\right)^\times, \times}\right)$ denote the group of units of $\left({\Z / n \Z, +, \times}\right)$.

Then $U$ is cyclic either:
 * $n = p^\alpha$

or:
 * $n = 2 p^\alpha$

where $p \ge 3$ is prime and $\alpha \ge 0$.

Sufficient condition
Let $U$ be cyclic.

Let $n \ge 0$ be an integer.

Let $n = p_1^{e_1} \cdots p_r^{e_r}$, be the decomposition of $n$ into distinct prime powers given by the Fundamental Theorem of Arithmetic.

Then by the corollary to the Chinese remainder theorem we have an isomorphism:
 * $\Z / n \Z \simeq \Z / p_1 \Z \times \cdots \times \Z / p_r \Z$

By Units of Direct Product are Direct Product of Units we have:
 * $\left({\Z / n \Z}\right)^\times \simeq \left({\Z / p_1 \Z}\right)^\times \times \cdots \times \left({\Z / p_r \Z}\right)^\times$

Suppose that $r \ge 2$, and choose $i, j \in \left\{{1, \ldots, r}\right\}$ such that $i \ne j$.

If $\left({\Z / p_i \Z}\right)^\times$ or $\left({\Z / p_j \Z}\right)^\times$ is not cyclic, then $\left({\Z / n \Z}\right)^\times$ cannot be cyclic.

Therefore suppose that $\left({\Z / p_i \Z}\right)^\times$ and $\left({\Z / p_j \Z}\right)^\times$ are cyclic.

By Order of Group of Units of Integers Modulo m these groups have orders:
 * $\phi \left({ p_i^{e_i} }\right)$

and:
 * $\phi \left({ p_j^{e_j} }\right)$

respectively, where $\phi$ is the Euler $\phi$ function.

By Euler Phi Function of Integer we have:
 * $\phi \left({ p_i^{e_i} }\right) = p_i^{e_i - 1} \left({ p_i - 1 }\right)$

and
 * $\phi \left({ p_j^{e_j} }\right) = p_j^{e_j - 1} \left({ p_j - 1 }\right)$

If $p_i, p_j$ are odd, $2$ divides $p_i - 1$ and $p_j - 1$.

Therefore $2$ divides $\phi \left({ p_i^{e_i} }\right)$ and $\phi \left({ p_j^{e_j} }\right)$.

In particular, $\phi \left({ p_i^{e_i} }\right)$ and $\phi \left({ p_j^{e_j} }\right)$ are not coprime.

Now by Group Direct Product of Cyclic Groups, $\left({\Z / n \Z}\right)^\times$ is not cyclic.

If $p_i$ or $p_j$ is even, WLOG we can assume $p_i = 2$.

Then:
 * $\phi \left({ p_i^{e_i} }\right) = \phi\left({ 2^{e_i} }\right) = p_i^{e_i - 1}\left({ p_i - 1 }\right)$

So if $e_i \ge 2$, then $2$ divides $\phi \left({ p_i^{e_i} }\right)$ and $ \phi\left({ p_j^{e_j} }\right)$.

In particular $\phi \left({ p_i^{e_i} }\right)$ and $\phi \left({ p_j^{e_j} }\right)$ are not coprime.

Again by Group Direct Product of Cyclic Groups, $\left({\Z / n \Z}\right)^\times$ is not cyclic.

Thus if $\left({\Z / n \Z}\right)^\times$ is cyclic, then $n = 2^e \times p^\alpha$ with $e = 0$ or $e = 1$, $\alpha \ge 0$ and $p \ge 3$ prime.