70 Let the set of real numbers $A_{n}=\left\{x \mid n<x^{n}<n+1, n \in \mathbf{N}\right\}$, if $A_{1} \cap A_{2} \cap \cdots \cap A_{n} \neq \emptyset$, then the maximum value of $n$ is $\qquad$.
$70 \quad 4$. Note
$$
\begin{aligned}
A_{n}^{\prime} & =\left\{x \mid n0\right\} \\
& =\{x \mid \sqrt[n]{n}\sqrt{3}>\sqrt[3]{4}>\sqrt[4]{5}>\sqrt[5]{6}$. Also, $\sqrt[3]{3}<\sqrt[4]{5}, \sqrt[5]{6}<\sqrt[3]{3}$, so
$$
\begin{array}{c}
A_{1} \cap A_{2}^{\prime} \cap A_{3}^{\prime} \cap A_{4}^{\prime} \neq \emptyset, \\
A_{1}^{\prime} \cap A_{2}^{\prime} \cap A_{3}^{\prime} \cap A_{4}^{\prime} \cap A_{5}^{\prime}=\emptyset .
\end{array}
$$

That is, the maximum value of $n$ is 4.