  This short but systematic work demonstrates a link between Chebyshev's
theorem and the explicit integration in cosmological time $t$ and conformal
time $\eta$ of the Friedmann equations in all dimensions and with an arbitrary
cosmological constant $\Lambda$. More precisely, it is shown that for spatially
flat universes an explicit integration in $t$ may always be carried out, and
that, in the non-flat situation and when $\Lambda$ is zero and the ratio $w$ of
the pressure and energy density in the barotropic equation of state of the
perfect-fluid universe is rational, an explicit integration may be carried out
if and only if the dimension $n$ of space and $w$ obey some specific relations
among an infinite family. The situation for explicit integration in $\eta$ is
complementary to that in $t$. More precisely, it is shown in the flat-universe
case with $\Lambda\neq0$ that an explicit integration in $\eta$ can be carried
out if and only if $w$ and $n$ obey similar relations among a well-defined
family which we specify, and that, when $\Lambda=0$, an explicit integration
can always be carried out whether the space is flat, closed, or open. We also
show that our method may be used to study more realistic cosmological
situations when the equation of state is nonlinear.
