Problem: For which positive integers $m$ does there exist an infinite sequence in $\ZZ/m\ZZ$
which is both an arithmetic progression and a geometric progression,
but is nonconstant?

Solution: Answer: $m$ must \emph{not} be squarefree.

The problem is essentially asking when there exists
a nonconstant arithmetic progression in $\ZZ/m\ZZ$
which is also a geometric progression.
Now,
\begin{itemize}
  \ii If $m$ is squarefree, then consider three $(s-d, d, s+d)$ in arithmetic progression.
  It's geometric if and only if $d^2 = (s-d)(s+d) \pmod m$, meaning $d^2 \equiv 0 \pmod m$.
  Then $d \equiv 0 \pmod m$.
  So any arithmetic progression which is also geometric is constant in this case.

  \ii 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.
\end{itemize}