Three, (25 points) Given that $m$ and $n$ ($m>n$) are positive integers, and the last three digits of $3^{m}$ and $3^{n}$ are the same. Find the minimum value of $m-n$.

保留源文本的换行和格式，翻译结果如下：

```
Three, (25 points) Given that $m$ and $n$ ($m>n$) are positive integers, and the last three digits of $3^{m}$ and $3^{n}$ are the same. Find the minimum value of $m-n$.
```
Three, from the given information, $3^{m}-3^{n}$ is a multiple of 1000, i.e.,
$$
3^{m}-3^{n}=3^{n}\left(3^{m-n}-1\right)
$$

is a multiple of 1000.
Also, $\left(3^{n}, 1000\right)=1$, hence $3^{m-n}-1$ is a multiple of 1000.
Let $s=m-n$, then $3^{s}-1$ is a multiple of 1000.
We only need to find the smallest $s$ such that the last three digits of $3^{s}$ are 001.
From Table 1, we know that $s$ is a multiple of 4.
Table 1
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline$s$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & $\cdots$ \\
\hline The last digit of $3^{s}$ & 3 & 9 & 7 & 1 & 3 & 9 & 7 & 1 & $\cdots$ \\
\hline
\end{tabular}

From Table 2, we know that $s$ is a multiple of 20.
Table 2
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline$s$ & 4 & 8 & 12 & 16 & 20 & 24 & 28 & 32 & 36 & 40 & $\cdots$ \\
\hline \begin{tabular}{l}
The last two \\
digits of $3^{s}$
\end{tabular} & 81 & 61 & 41 & 21 & 01 & 81 & 61 & 41 & 21 & 01 & $\cdots$ \\
\hline
\end{tabular}

From Table 3, we know that the smallest value of $s=m-n$ is 100.
Table 3
\begin{tabular}{|c|r|r|r|r|r|r|}
\hline$s$ & 20 & 40 & 60 & 80 & 100 & $\cdots$ \\
\hline The last three digits of $3^{s}$ & 401 & 801 & 201 & 601 & 001 & $\cdots$ \\
\hline
\end{tabular}