In mathematics, Wolstenholme's theorem states that for a prime number p ≥ 5, the congruence holds, where the parentheses denote a binomial coefficient.
In 1819, Charles Babbage showed the same congruence modulo p2, which holds for p ≥ 3.
An equivalent formulation is the congruence for p ≥ 5, which is due to Wilhelm Ljunggren[1] (and, in the special case b = 1, to J. W. L. Glaisher[citation needed]) and is inspired by Lucas's theorem.
No known composite numbers satisfy Wolstenholme's theorem and it is conjectured that there are none (see below).
As Wolstenholme himself established, his theorem can also be expressed as a pair of congruences for (generalized) harmonic numbers: since (Congruences with fractions make sense, provided that the denominators are coprime to the modulus.)
[2] This result is consistent with the heuristic argument that the residue modulo p4 is a pseudo-random multiple of p3.
A similar heuristic predicts that there are no "doubly Wolstenholme" primes, for which the congruence would hold modulo p5.
Here is a proof that directly establishes Glaisher's version using both combinatorics and algebra.
Thus, the a-fold 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 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.
orbits of size 1 and there are no orbits of size p.[3] 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.
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.
Thus the conjecture is considered likely because Wolstenholme's congruence seems over-constrained and artificial for composite numbers.
The constant in the big O notation is also effectively computable in
Leudesdorf has proved that for a positive integer n coprime to 6, the following congruence holds:[4] In 1900, Glaisher[5][6] showed further that: for prime p > 3, Where Bn is the Bernoulli number.