The Sequence n! (mod p)

We investigate a problem raised by R. K. Guy concerning the distribution modulo a large prime number p of the sequence 1!, 2!,... ,p!. Numerical computations indicate that about E residue classes modulo p are missed by this sequence. We show that this is the general behavior of a randomly chosen sequence of p elements modulo p. We also show that for p large enough, the map n -> n! (mod p) is far from being a permutation of at least one of the sets {1,2,... ,p - 1} or {2,4, • • •, [p-1/4]}, in the sense that a positive proportion of residues are missed in at least one of the above sets.