4.58 Find the integers $m$ and $n$ that satisfy the equation $(5+3 \sqrt{2})^{m}=(3+5 \sqrt{2})^{n}$.
[Solution] Let $(5+3 \sqrt{2})^{m}=(3+5 \sqrt{2})^{n}=A+B \sqrt{2}$, where $A, B$ are rational numbers.
If $B=0$, then it must be that $m=n=0$.
If $B \neq 0$, then
$$
(5-3 \sqrt{2})^{m}=(3-5 \sqrt{2})^{n}=A-B \sqrt{2}
$$

On the other hand, since $0<5-3\sqrt{2}<1$ and $0<3-5\sqrt{2}<1$, it follows that $(5-3\sqrt{2})^m < 1$ and $(3-5\sqrt{2})^n < 1$. Therefore, (1) cannot hold. This is a contradiction.
Hence, the equation holds only when $m=n=0$.