19.4.3 ** For any positive integer $q_{0}$, consider the sequence $q_{1}, q_{2}, \cdots, q_{n}$ defined by $q_{i}=\left(q_{i-1}-1\right)^{3}+3, i=1,2, \cdots, n$. If each $q_{i}(i=1,2, \cdots, n)$ is a power of a prime, find the largest possible value of $n$.
Since $m^{3}-m=(m-1) m(m+1)$ is divisible by 3, we have $m^{3} \equiv m(\bmod 3)$. Therefore, $q_{i}-\left(q_{i-1}-1\right)^{3}+3 \equiv q_{i-1}-1(\bmod 3)$. Thus, one of $q_{1}, q_{2}, q_{3}$ must be divisible by 3, making it a power of 3. However, $\left(q_{i}-1\right)^{3}+3$ is a power of 3 only when $q_{i}=1$, so $q_{0}=1, q_{1}=3, q_{2}=11$, $q_{3}=17 \times 59$,
the maximum value of $n$ is 2.