Products of Factorials Modulo p

The problem that we investigate in this note is the following: given p, find sufficient conditions that the parameters s and t should satisfy such as to ensure that Ps,t(p) contains the entire Zp. Let ε > 0 be any small number. Throughout this paper, we denote by c1, c2, . . . computable positive constants which are either absolute or depend on ε. From the way we formulated the above problem, we see that its answer is easily decidable if either both s and t are very small (with respect to p) or very large with respect to p. For example, if s < c1(log(p)) with some suitable constant c1, then it is clear that Ps,t(p), or even the union of all Ps,t(p) for all allowable values of t, cannot possibly contain the entire Zp when p is large. Indeed, the reason here is that the cardinality of the union of all Ps,t(p) for all allowable values of t is at most p(s) = O(exp(c2 √ s)) and this is much smaller than p when p is large if c1 is chosen such that c1 > c2. Here, we denoted by p(s) the number of unrestricted partitions of s, and the constant c2 can be chosen to be equal to π √ 2 3 . It is also obvious that Ps,t(p) does not generate the entire Zp (for any s) when t = 2. Moreover, the fact that there exist infinitely many prime numbers p for which the smallest nonquadratic residue modulo p is at least c3 log(p), shows that if one wants to generate ∗Also Associated to the Institute of Mathematics of Romanian Academy, Bucharest, Romania