沃尔斯滕霍尔姆定理

• 在数论上，沃尔斯滕霍尔姆定理说明，对于大于或等于5的质数，有 （公式略）
• ( （公式略）
• ( （公式略）
• ( （公式略）

以上四个等式是等价的。

只有少数质数符合 （公式略），现时已知有两个这样的质数，16843 及 2124679 （OEIS:A088164）。这类质数称为沃尔斯滕霍尔姆素数，下一个这样的质数如果存在，它大于109。

这定理是19世纪英国数学家约瑟夫·沃尔斯滕霍尔姆提出的。值得一提的是沃尔斯滕霍尔姆是吴尔芙的父亲的朋友，也是吴尔芙小说《灯塔行》中奥古斯图斯·卡麦可的原形。

In mathematics, Wolstenholme's theorem states that for a prime number p ≥ 5, the congruence （公式略）

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 p2, which holds for p ≥ 3. An equivalent formulation is the congruence （公式略）

for p ≥ 5, which is due to Wilhelm Ljunggren (and, in the special case b = 1, to J. W. L. Glaisher) 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). A prime that satisfies the congruence modulo p4 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: （公式略）

（公式略）

since （公式略）

(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.

Wolstenholme primes

A prime p is called a Wolstenholme prime iff the following condition holds: （公式略）

The only known Wolstenholme primes so far are 16843 and 2124679 (sequence A088164 in the OEIS); any other Wolstenholme prime must be greater than 1011. This result is consistent with the heuristic argument that the residue modulo p4 is a pseudo-random multiple of p3. 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 1011, and the heuristic says that there should be roughly one Wolstenholme prime between 1011 and 1024. A similar heuristic predicts that there are no "doubly Wolstenholme" primes, for which the congruence would hold modulo p5.

A proof of the theorem

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 non-negative 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 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. There are （公式略） orbits of size 1 and there are no orbits of size p. Thus we first obtain Babbage's theorem （公式略）

Examining the orbits of size p2, we also obtain （公式略）

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 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 (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

27173, 2001341, 16024189487, 80478114820849201, 20378551049298456998947681,... (sequence A082180 in the OEIS)

The first three are not prime powers (sequence A228562 in the OEIS), the last two are 168434 and 21246794, 16843 and 2124679 are Wolstenholme primes (sequence A088164 in the OEIS). Besides, with an exception of 168432 and 21246792, no composites are known to hold for (1) with k = 2, much less k = 3.

Generalizations

Leudesdorf has proved that for a positive integer n coprime to 6, the following congruence holds: （公式略）

In 1900, Glaisher showed further that: for prime p > 3, （公式略）

Where Bn is the Bernoulli number.