4. Find all positive integers $m, n$, such that $\left|12^{m}-5^{n}\right|=7$.


4. Solution: Considering modulo 4, we know that the equation $12^{m}-5^{n}=-7$ has no positive integer solutions. The equation
$$
12^{m}-5^{n}=7
$$

obviously has the solution $m=n=1$. Below we prove that when $m>1$, it has no positive integer solutions. Taking (1) modulo 3, we get $-(-1)^{n} \equiv 1(\bmod 3)$, so $n$ is odd, thus $5^{n} \equiv 5(\bmod 8)$. Since $m \geqslant 2$, we have $8 \mid 12^{m}$. Taking (1) modulo 8, we get $-5 \equiv 7(\bmod 8)$, which is impossible, so $m=1$, hence $n=1$.