cyclic group of order p acts on the set A, and by extension it acts on the set of subsets of size bp. Every orbit of this group action has pk elements, where k is the number of incomplete rings, i.e., if there are k rings that only partly intersect a subset B in the orbit. There are orbits of size 1 and there are no orbits of size p. Thus we first obtain Babbage's theorem Examining the orbits of size p2, we also obtain Among other consequences, this equation tells us that the case a=2 and b=1 implies the general case of the second form of Wolstenholme's theorem. Switching from combinatorics to algebra, both sides of this congruence are polynomials in a for each fixed value of b. The congruence therefore holds when a is any integer, positive or negative, provided that b is a fixed positive integer. In particular, if a=-1 and b=1, the congruence becomes This congruence becomes an equation for using the relation When p is odd, the relation is When p≠3, we can divide both sides by 3 to complete the argument. A similar derivation modulo p4 establishes that for all positive a and b if and only if it holds when a=2 and b=1, i.e., if and only if p is a Wolstenholme prime. The converse as a conjecture It is conjectured that if when k=3, then n is prime. The conjecture can be understood by considering k = 1 and 2 as well as 3. When k = 1, Babbage's theorem implies that it holds for n = p2 for p an odd prime, while Wolstenholme's theorem implies that it holds for n = p3 for p > 3, and it holds for n = p4 if p is a Wolstenholme prime. When k = 2, it holds for n = p2 if p is a Wolstenholme prime. These three numbers, 4 = 22, 8 = 23, and 27 = 33 are not held for () with k = 1, but all other prime square and prime cube are held for () with k = 1. Only 5 other composite values (neither prime square nor prime cube) of n are known to hold for () with k = 1, they are called Wolstenholme pseudoprimes, they are 27173, 2001341, 16024189487, 80478114820849201, 20378551049298456998947681, ... The first three are not prime powers , the last two