Wolstenholme prime
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In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, a Wolstenholme prime is a special type of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
satisfying a stronger version of
Wolstenholme's theorem In mathematics, Wolstenholme's theorem states that for a prime number p \geq 5, the congruence : \equiv 1 \pmod holds, where the parentheses denote a binomial coefficient. For example, with ''p'' = 7, this says that 1716 is one more than a multiple ...
. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater than 3. Wolstenholme primes are named after mathematician
Joseph Wolstenholme Joseph Wolstenholme (30 September 1829 – 18 November 1891) was an English mathematician. Wolstenholme was born in Eccles near Salford, Lancashire, England, the son of a Methodist minister, Joseph Wolstenholme, and his wife, Elizabeth, ''née' ...
, who first described this theorem in the 19th century. Interest in these primes first arose due to their connection with
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...
. 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 . There are no other Wolstenholme primes less than 109.


Definition

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 : \equiv 1 \pmod, where the expression in left-hand side denotes a binomial coefficient. In comparison,
Wolstenholme's theorem In mathematics, Wolstenholme's theorem states that for a prime number p \geq 5, the congruence : \equiv 1 \pmod holds, where the parentheses denote a binomial coefficient. For example, with ''p'' = 7, this says that 1716 is one more than a multiple ...
states that for every prime ''p'' > 3 the following congruence holds: : \equiv 1 \pmod.


Definition via Bernoulli numbers

A Wolstenholme prime is a prime ''p'' that divides the numerator of the
Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
''B''''p''−3. The Wolstenholme primes therefore form a subset of the
irregular prime In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli num ...
s.


Definition via irregular pairs

A Wolstenholme prime is a prime ''p'' such that (''p'', ''p''–3) is an irregular pair.


Definition via harmonic numbers

A Wolstenholme prime is a prime ''p'' such that :H_ \equiv 0 \pmod\, , i.e. the numerator of the harmonic number H_ expressed in lowest terms is divisible by ''p''3.


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 2007. The first Wolstenholme prime 16843 was found in 1964, although it was not explicitly reported at that time.Selfridge and Pollack published the first Wolstenholme prime in (see ). 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. Up to 1.2, no further Wolstenholme primes were found. This was later extended to 2 by McIntosh in 1995 and Trevisan & Weber were able to reach 2.5. The latest result as of 2007 is that there are only those two Wolstenholme primes up to .


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. Clearly, ''p'' is a Wolstenholme prime if and only if ''W''''p'' ≡ 0 (mod ''p'').
Empirically In philosophy, empiricism is an epistemological theory that holds that knowledge or justification comes only or primarily from sensory experience. It is one of several views within epistemology, along with rationalism and skepticism. Empir ...
one may assume that the remainders of ''W''''p'' modulo ''p'' are uniformly distributed in the set . By this reasoning, the probability that the remainder takes on a particular value (e.g., 0) is about 1/''p''.


See also

*
Wieferich prime In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime ''p'' divides . Wieferich primes were first described by Ar ...
* Wall–Sun–Sun prime * Wilson prime * Table of congruences


Notes


References

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Further reading

* * *


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 {{Prime number classes, state=collapsed Classes of prime numbers Unsolved problems in number theory