  We investigate the many-body state and the static and the dynamic behaviour
of the pair-correlation function of a Bose-Einstein condensate with a finite
atom number, which is confined in a quasi-one-dimensional toroidal/annular
potential, both for repulsive, and for attractive interactions. We link the
dynamic pair-correlation function that we evaluate with the problem of quantum
time crystals. For weak repulsive interatomic interactions and a finite number
of atoms the pair-correlation function shows a periodic temporal behaviour,
which disappears in the limit of a large atom number, in agreement with general
arguments. Finally we provide some insight into older results of attractive
interactions, where the time-crystalline behaviour exists only in the limit of
a large atom number.
