  We derive the general form of the cosmological scalar field potential which
is compatible both with the existence of black holes and p-branes related to
string/M theory and with multidimensional inflationary cosmology. It is shown
that the scalar potential alters non-trivially from dimension to dimension yet
always obeys one single equation where the number of spacetime dimensions is a
free parameter. Using this equation we formulate an eigenvalue problem for the
dimensionality parameter. It turns out that in the low-energy regime of
sub-Planckian values of the inflaton field, i.e., when the Universe has cooled
and expanded sufficiently enough, the value four arises as the largest
admissible (eigen)value of this parameter.
