18.4.9 $\star$ Find all positive integers $m, n$, such that $(m+n)^{m}=n^{m}+1413$.
From $(m+n)^{m} \geqslant m^{m}+n^{m}$, we know that $m^{m} \leqslant 1413$. Therefore, $m \leqslant 4$.
Since $m=(m+n)-n$ divides $(m+n)^{m}-n^{m}=1413$, we have $m=1$ or 3. When $m=1$, $(m+n)^{m}=n^{m}+1413$ does not hold; when $m=3$, $n=11$. Therefore, the positive integers sought are $m=3$, $n=11$.