  We derive exact thermodynamic identities relating the average number of
condensed atoms and the root-mean-square fluctuations determined in different
statistical ensembles for the weakly interacting Bose gas confined in a box.
This is achieved by introducing the concept of {\it auxiliary partition
functions} for model Hamiltonians that do conserve the total number of
particles. Exploiting such thermodynamic identities, we provide the first,
completely analytical prediction of the microcanonical particle number
fluctuations in the weakly interacting Bose gas. Such fluctuations, as a
function of the volume V of the box are found to behave normally, at variance
with the anomalous scaling behavior V^{4/3} of the fluctuations in the ideal
Bose gas.
