  We study two-body correlations for $N$ identical bosons by use of the
hyperspherical adiabatic expansion method. We use the zero-range interaction
and derive a transcendental equation determining the key ingredient of the
hyperradial potential. The necessary renormalization is for both repulsive and
attractive interactions achieved with an effective range expansion of the
two-body phase-shifts. Our solutions including correlations provide the
properties of Bose-Einstein condensates exemplified by stability conditions as
established by mean-field Gross-Pitaevskii calculations. The many-body Efimov
states are unavoidable for large scattering lengths.
