For which positive integers $m$ does there exist an infinite sequence in $\mathbb{Z} / m \mathbb{Z}$ which is both an arithmetic progression and a geometric progression, but is nonconstant?
Answer: $m$ must not be squarefree.
The problem is essentially asking when there exists a nonconstant arithmetic progression in $\mathbb{Z} / m \mathbb{Z}$ which is also a geometric progression. Now,
- If $m$ is squarefree, then consider three $(s-d, d, s+d)$ in arithmetic progression. It's geometric if and only if $d^{2}=(s-d)(s+d)(\bmod m)$, meaning $d^{2} \equiv 0(\bmod m)$. Then $d \equiv 0(\bmod m)$. So any arithmetic progression which is also geometric is constant in this case.
- Conversely if $p^{2} \mid m$ for some prime $p$, then any arithmetic progression with common difference $m / p$ is geometric by the same calculation.