In this paper we study arithmetically Cohen-Macaulay (ACM for short)

vector bundles E of rank k  3 on hypersurfaces Xr  P4 of degree r  1.

We consider here mainly the case of degree r = 4, which is the first unknown

case in literature. Under some natural conditions for the bundle E

we derive a list of possible Chern classes (c1, c2, c3) which may arise in the

cases of rank k = 3 and k = 4, when r = 4 and we give several examples