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


2. Mod 4 reveals 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) mod 3 yields $-(-1)^{n} \equiv 1(\bmod 3)$, so $n$ is odd, thus $5^{n} \equiv 5(\bmod 8)$. Also, $m \geqslant 2$, so $8 \mid 12^{m}$. Taking (1) mod 8 yields $-5 \equiv 7(\bmod 8)$, which is impossible. Therefore, $m=1$, hence $n=1$.