14.2.11 ** Find two coprime four-digit positive integers $A$ and $B$, such that for any positive integers $m$ and $n$, the numbers $A^{m}$ and $B^{n}$ differ by at least 4000,
Take $A=8001, B=4001$. For any positive integers $m$ and $n, A^{m}, B^{n}$ when divided by 4000 both leave a remainder of 1, so $A^{m}-B^{n}$ are all multiples of 4000. Since $A^{m} \neq B^{n}$, it follows that $\left|A^{m}-B^{n}\right| \geqslant 4000$.