G6.4 If $r, s, t, u$ are positive integers and $r^{5}=s^{4}, t^{3}=u^{2}, t-r=19$ and $d=u-s$, find the value of $d$.
Let $w=u^{\frac{1}{15}}, v=s^{\frac{1}{15}}$, then $t=u^{\frac{2}{3}}=u^{\frac{10}{15}}=w^{10}, r=s^{\frac{4}{5}}=s^{\frac{12}{15}}=v^{12}$
$$
\begin{array}{l}
t-r=19 \Rightarrow w^{10}-v^{12}=19 \\
\Rightarrow\left(w^{5}+v^{6}\right)\left(w^{5}-v^{6}\right)=19 \times 1
\end{array}
$$
$\because 19$ is a prime number, $w^{5}+v^{6}=19, w^{5}-v^{6}=1$
Solving these equations give $w^{5}=10, v^{6}=9 \Rightarrow w^{5}=10, v^{3}=3$
$$
\begin{array}{l}
u=w^{15}=1000, s=v^{15}=3^{5}=729 \\
d=u-s=1000-243=757
\end{array}
$$