$9 \cdot 51$ Let $m, n \in N$, find the minimum value of $\left|12^{m}-5^{n}\right|$.
[Solution] Clearly, $|12^{m}-5^{n}|$ is an odd number, and it is not divisible by 3 or 5. From $12^{m}-5^{n} \equiv -1 \pmod{4}$, we know that $12^{m}-5^{n}$ cannot be 1.
If $5^{n}-12^{m}=1$, then by
$$5^{n}-12^{m} \equiv 5^{n} \equiv (-1)^{n} \pmod{3}$$

we can deduce that $n$ is even, let $n=2k$. Also, since
$$5^{n}-12^{m} \equiv -2^{m} \equiv 2^{m+2} \pmod{5}$$

it follows from $5^{n}-12^{m}=1$ that $4 \mid m+2$, i.e., $m=4l-2$. Therefore,
$$\begin{aligned}
5^{n}-12^{m} & =5^{2k}-12^{4l-2}=\left(5^{k}+12^{2k-1}\right)\left(5^{k}-12^{2l-1}\right) \\
& \geqslant 5^{k}+12^{2l-1}>1
\end{aligned}$$

which is a contradiction! Therefore, $5^{n}-12^{m}$ cannot be 1 either.
In summary, we have $|12^{m}-5^{n}| \geqslant 7$, and since $12-5=7$, the minimum value sought is 7.