4. $1991^{2000}$ divided by $10^{6}$, the remainder is
880001
4.【Analysis and Solution】
$$
\begin{aligned}
1991^{2000} & =(1+1990)^{2000} \\
& =1+2000 \times 1990+\frac{1}{2} \times 2000 \times 1999 \times 1990^{2}+\cdots+1990^{2000} .
\end{aligned}
$$

According to the above expansion, we only need to consider the sum of the first three terms modulo \(10^{6}\), so the remainder is 880001.