G3.1 If $m$ and $n$ are positive integers with $m^{2}-n^{2}=43$, find the value of $m^{3}-n^{3}$.
$(m+n)(m-n)=43$, which is a prime number.
$\left\{\begin{array}{l}m+n=43 \\ m-n=1\end{array} \Rightarrow m=22, n=21\right.$
$$
m^{3}-n^{3}=(m-n)\left(m^{2}+m n+n^{2}\right)=1 \times\left[(m+n)^{2}-m n\right]=43^{2}-22 \times 21=1849-462=1387
$$