3.1. (Netherlands, 82). What is larger: $(17091982!)^{2}$ or $17091982^{1701982} ?$
3.1. We will prove that for any value of $n \in \mathbf{N}$, greater than 2, the inequality

$$
(n!)^{2}>n^{n}
$$

holds. Indeed, we have

$$
n' \cdot n!=(1 \cdot 2 \ldots n)(n(n-1) \ldots .1)=(1 \cdot n)(2(n-1)) \ldots(n \cdot 1)>n^{n}
$$

since

$$
1 \cdot n=n \cdot 1=n \text { and } k(n-k+1)=(n-k)(k-1)+n>n
$$

for $k=2, \ldots, n-1$. By substituting in the proven inequality

$$
n=17091982,
$$

we obtain the answer:

$$
(17091982!)^{2}>17091982^{17081982}
$$