In mathematics, Wolstenholme's theorem states that for a prime number , the congruence
holds, where the parentheses denote a binomial coefficient. For example, with p = 7, this says that 1716 is one more than a multiple of 343. The theorem was first proved by Joseph Wolstenholme in 1862. In 1819, Charles Babbage showed the same congruence modulo p^{2}, which holds for . An equivalent formulation is the congruence
for , which is due to Wilhelm Ljunggren^{[1]} (and, in the special case , to J. W. L. Glaisher^{[citation needed]}) and is inspired by Lucas' theorem.
No known composite numbers satisfy Wolstenholme's theorem and it is conjectured that there are none (see below). A prime that satisfies the congruence modulo p^{4} is called a Wolstenholme prime (see below).
As Wolstenholme himself established, his theorem can also be expressed as a pair of congruences for (generalized) harmonic numbers:
(Congruences with fractions make sense, provided that the denominators are coprime to the modulus.) For example, with p=7, the first of these says that the numerator of 49/20 is a multiple of 49, while the second says the numerator of 5369/3600 is a multiple of 7.
A prime p is called a Wolstenholme prime iff the following condition holds:
If p is a Wolstenholme prime, then Glaisher's theorem holds modulo p^{4}. The only known Wolstenholme primes so far are 16843 and 2124679 (sequence A088164 in the OEIS); any other Wolstenholme prime must be greater than 10^{9}.^{[2]} This result is consistent with the heuristic argument that the residue modulo p^{4} is a pseudorandom multiple of p^{3}. This heuristic predicts that the number of Wolstenholme primes between K and N is roughly ln ln N − ln ln K. The Wolstenholme condition has been checked up to 10^{9}, and the heuristic says that there should be roughly one Wolstenholme prime between 10^{9} and 10^{24}. A similar heuristic predicts that there are no "doubly Wolstenholme" primes, for which the congruence would hold modulo p^{5}.
There is more than one way to prove Wolstenholme's theorem. Here is a proof that directly establishes Glaisher's version using both combinatorics and algebra.
For the moment let p be any prime, and let a and b be any nonnegative integers. Then a set A with ap elements can be divided into a rings of length p, and the rings can be rotated separately. Thus, the afold direct sum of the 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 p^{k} 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 p^{2}, 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 p^{4} 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.
It is conjectured that if

(1) 
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 = p^{2} for p an odd prime, while Wolstenholme's theorem implies that it holds for n = p^{3} for p > 3, and it holds for n = p^{4} if p is a Wolstenholme prime. When k = 2, it holds for n = p^{2} if p is a Wolstenholme prime. These three numbers, 4 = 2^{2}, 8 = 2^{3}, and 27 = 3^{3} are not held for (1) with k = 1, but all other prime square and prime cube are held for (1) with k = 1. Only 5 other composite values (neither prime square nor prime cube) of n are known to hold for (1) with k = 1, they are called Wolstenholme pseudoprimes, they are
The first three are not prime powers (sequence A228562 in the OEIS), the last two are 16843^{4} and 2124679^{4}, 16843 and 2124679 are Wolstenholme primes (sequence A088164 in the OEIS). Besides, with an exception of 16843^{2} and 2124679^{2}, no composites are known to hold for (1) with k = 2, much less k = 3. Thus the conjecture is considered likely because Wolstenholme's congruence seems overconstrained and artificial for composite numbers. Moreover, if the congruence does hold for any particular n other than a prime or prime power, and any particular k, it does not imply that
The number of Wolstenholme pseudoprimes up to is , so the sum of reciprocals of those numbers converges. The constant follows from the existence of only three Wolstenholme pseudoprimes up to . The number of Wolstenholme pseudoprimes up to should be at least 7 if the sum of its reciprocals diverged, and since this is not satisfied, the counting function of these pseudoprimes is at most for some efficiently computable constant ; we can take as 499712.
Leudesdorf has proved that for a positive integer n coprime to 6, the following congruence holds:^{[3]}