Wolstenholme prime

Template:Short description Template:Distinguish Template:Use dmy dates Template:Infobox integer sequence

In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater than 3. Wolstenholme primes are named after mathematician Joseph Wolstenholme, who first described this theorem in the 19th century.

Interest in these primes first arose due to their connection with Fermat's Last Theorem. Wolstenholme primes are also related to other special classes of numbers, studied in the hope to be able to generalize a proof for the truth of the theorem to all positive integers greater than two.

The only two known Wolstenholme primes are 16843 and 2124679 Template:OEIS. There are no other Wolstenholme primes less than 10¹¹.^[1]

Definition

Template:Unsolved

Wolstenholme prime can be defined in a number of equivalent ways.

Definition via binomial coefficients

A Wolstenholme prime is a prime number p > 7 that satisfies the congruence

(\binom{2 p - 1}{p - 1}) \equiv 1 (\mod p^{4}),

where the expression in left-hand side denotes a binomial coefficient.^[2] In comparison, Wolstenholme's theorem states that for every prime p > 3 the following congruence holds:

(\binom{2 p - 1}{p - 1}) \equiv 1 (\mod p^{3}) .

Definition via Bernoulli numbers

A Wolstenholme prime is a prime p that divides the numerator of the Bernoulli number B_p−3.Template:Sfn Template:Sfn Template:Sfn The Wolstenholme primes therefore form a subset of the irregular primes.

Definition via irregular pairs

Template:Main A Wolstenholme prime is a prime p such that (p, p–3) is an irregular pair.Template:Sfn Template:Sfn

Definition via harmonic numbers

A Wolstenholme prime is a prime p such thatTemplate:Sfn

H_{p - 1} \equiv 0 (\mod p^{3}),

i.e. the numerator of the harmonic number $H_{p - 1}$ expressed in lowest terms is divisible by p³.

Search and current status

The search for Wolstenholme primes began in the 1960s and continued over the following decades, with the latest results published in 2022. The first Wolstenholme prime 16843 was found in 1964, although it was not explicitly reported at that time.^[3] The 1964 discovery was later independently confirmed in the 1970s. This remained the only known example of such a prime for almost 20 years, until the discovery announcement of the second Wolstenholme prime 2124679 in 1993.Template:Sfn Up to 1.2Template:E, no further Wolstenholme primes were found.Template:Sfn This was later extended to 2Template:E by McIntosh in 1995 Template:Sfn and Trevisan & Weber were able to reach 2.5Template:E.Template:Sfn The latest result as of 2022 is that there are only those two Wolstenholme primes up to 10¹¹.^[4]

Expected number of Wolstenholme primes

It is conjectured that infinitely many Wolstenholme primes exist. It is conjectured that the number of Wolstenholme primes ≤ x is about ln ln x, where ln denotes the natural logarithm. For each prime p ≥ 5, the Wolstenholme quotient is defined as

W_{p} = \frac{(\binom{2 p - 1}{p - 1}) - 1}{p^{3}} .

Clearly, p is a Wolstenholme prime if and only if W_p ≡ 0 (mod p). Empirically one may assume that the remainders of W_p modulo p are uniformly distributed in the set {0, 1, ..., p–1}. By this reasoning, the probability that the remainder takes on a particular value (e.g., 0) is about 1/p.Template:Sfn

Notes

Template:Reflist

References

Template:Refbegin

Template:Refend

External links

Caldwell, Chris K. Wolstenholme prime from The Prime Glossary
McIntosh, R. J. Wolstenholme Search Status as of March 2004 e-mail to Paul Zimmermann
Bruck, R. Wolstenholme's Theorem, Stirling Numbers, and Binomial Coefficients
Conrad, K. The p-adic Growth of Harmonic Sums interesting observation involving the two Wolstenholme primes

Template:Prime number classes

↑ Template:MathWorld
↑ Template:Citation
↑ Selfridge and Pollack published the first Wolstenholme prime in Template:Harvnb (see Template:Harvnb).
↑ Template:Cite journal

[1] Template:MathWorld

[2] Template:Citation

[3] Selfridge and Pollack published the first Wolstenholme prime in Template:Harvnb (see Template:Harvnb).

[4] Template:Cite journal

[1]

[2]

[3]

[4]