  We show that the contact parameter of N harmonically-trapped interacting 1D
bosons at zero temperature can be analytically and accurately obtained by a
simple rescaling of the exact two-boson solution, and that N-body effects can
be almost factorized. The small deviations observed between our analytical
results and DMRG calculations are more pronounced when the interaction energy
is maximal (i.e. at intermediate interaction strengths) but they remain bounded
by the large-N local-density approximation obtained from the Lieb-Liniger
equation of state stemming from the Bethe Ansatz. The rescaled two-body
solution is so close to the exact ones, that is possible, within a simple
expression interpolating the rescaled two-boson result to the local-density, to
obtain N-boson contact and ground state energy functions in very good agreement
with DMRG calculations. Our results suggest a change of paradigm in the study
of interacting quantum systems, giving to the contact parameter a more
fundamental role than energy.
