  We show interaction between high- and low-frequency modes in periodic
$\alpha$-FPU chains with alternating large masses. The treatment discusses the
difficult case where the number of particles $N=2p$ involves $p$ prime. A key
role is played by identifying symmetric invariant manifolds, thus reducing the
dimension of the problems drastically, and a Mathematica programme focused on
these systems. We could show explicitly interaction for systems up to 100
particles with in addition strong arguments for interactions in arbitrary large
chains.
