(6) If $7n+1$ and $8n+1$ can both be expressed as the sum of three distinct positive integers in a geometric progression, then the minimum value of the positive integer $n$ is $\qquad$.
6 6 Hint: First, when $n=6$, $7 n+1=43=1+6+6^{2}, 8 n+1=$ $49=3^{2}+3 \cdot 5+5^{2}$ are both the sums of three positive integers in geometric progression. When $n \leqslant 5$, it is easy to see that
$$
\begin{array}{l}
7 \cdot 1+1=8,8 \cdot 2+1=17,8 \cdot 3+1=25, \\
7 \cdot 4+1=29,8 \cdot 5+1=41
\end{array}
$$

cannot be expressed as the sum of three distinct positive integers in geometric progression. Therefore, the minimum value of the positive integer $n$ is 6.