  We study the original $\alpha$-Fermi-Pasta-Ulam (FPU) system with $N=16,32$
and $64$ masses connected by a nonlinear quadratic spring. Our approach is
based on resonant wave-wave interaction theory, i.e. we assume that, in the
weakly nonlinear regime (the one in which Fermi was originally interested), the
large time dynamics is ruled by exact resonances. After a detailed analysis of
the $\alpha$-FPU equation of motion, we find that the first non trivial
resonances correspond to six-wave interactions. Those are precisely the
interactions responsible for the thermalization of the energy in the spectrum.
We predict that for small amplitude random waves the time scale of such
interactions is extremely large and it is of the order of $1/\epsilon^8$, where
$\epsilon$ is the small parameter in the system. The wave-wave interaction
theory is not based on any threshold: equipartition is predicted for arbitrary
small nonlinearity. Our results are supported by extensive numerical
simulations. A key role in our finding is played by the {\it Umklapp} (flip
over) resonant interactions, typical of discrete systems. The thermodynamic
limit is also briefly discussed.
