  We study the probability density function (PDF) of the cover time $t_c$ of a
finite interval of size $L$, by $N$ independent one-dimensional Brownian
motions, each with diffusion constant $D$. The cover time $t_c$ is the minimum
time needed such that each point of the entire interval is visited by at least
one of the $N$ walkers. We derive exact results for the full PDF of $t_c$ for
arbitrary $N \geq 1$, for both reflecting and periodic boundary conditions. The
PDFs depend explicitly on $N$ and on the boundary conditions. In the limit of
large $N$, we show that $t_c$ approaches its average value $\langle t_c \rangle
\approx L^2/(16\, D \, \ln N)$, with fluctuations vanishing as $1/(\ln N)^2$.
We also compute the centered and scaled limiting distributions for large $N$
for both boundary conditions and show that they are given by nontrivial
$N$-independent scaling functions.
