Question 31: Let the function $\mathrm{f}(\mathrm{x})=\ln \mathrm{x}$ have the domain $(\mathrm{m},+\infty)$, and $M>0$. If for any $a, b, c \in (M,+\infty)$, $a, b, c$ are the three sides of a right triangle, then $f(a), f(b), f(c)$ can also be the three sides of a triangle. Find the minimum value of M.
Question 31, Answer: When $M2$. Then
$f(a)+f(b)>f(c) \Leftrightarrow \ln a+\ln b>\ln c \Leftrightarrow ab>c \Leftrightarrow a^2 b^2>c^2 \Leftrightarrow a^2 b^2-a^2-b^2>0 \Leftrightarrow$ $\left(a^2-1\right)\left(b^2-1\right)>1$.

And $a^2-1>1, b^2-1>1$, so $\left(a^2-1\right)\left(b^2-1\right)>1$, so $f(a)+f(b)>f(c)$, so $f(a)$, $f(b)$, and $f(c)$ can also be the three sides of a triangle.
In summary, the minimum value of $M$ is $\sqrt{2}$.