An Extension of Wilson’s Theorem

Let N[k] be the multiset containing the (n−1k) products of k-subsets of N={1,…,n−1}. We show that if n≥(2c+3)2, then ((−1)c+∑M∈N[n−1−c]M)·(c+1)≡0 (modn), if and only if n=(c+1)p, where p is prime. This provides a combinatorial extension of Wilson’s theorem, which is the special case where c=0.