  We consider a system of $N$ pairwise interacting particles described by the
Hamiltonian $H$, where $\sigma_{ess} (H) = [0,\infty)$ and none of the particle
pairs has a zero energy resonance. The pair potentials are allowed to take both
signs and obey certain restrictions regarding the fall off. It is proved that
if $N \geq 4$ and none of the Hamiltonians corresponding to the subsystems
containing $N-2$ or less particles has an eigenvalue equal to zero then $H$ has
a finite number of negative energy bound states. This result provides a
positive proof to a long--standing conjecture of Amado and Greenwood stating
that four bosons with an empty negative continuous spectrum have at most a
finite number of negative energy bound states. Additionally, we give a short
proof to the theorem of Vugal'ter and Zhislin on the finiteness of the discrete
spectrum and pose a conjecture regarding the existence of the "true" four--body
Efimov effect.
