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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, a Wilson prime is a
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 ...
such that
divides
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
, where "
" denotes the
factorial function; compare this with
Wilson's theorem
In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations of m ...
, which states that every prime
divides
. Both are named for 18th-century
English
English usually refers to:
* English language
* English people
English may also refer to:
Peoples, culture, and language
* ''English'', an adjective for something of, from, or related to England
** English national ide ...
mathematician
John Wilson; in 1770,
Edward Waring
Edward Waring (15 August 1798) was a British mathematician. He entered Magdalene College, Cambridge as a sizar and became Senior wrangler in 1757. He was elected a Fellow of Magdalene and in 1760 Lucasian Professor of Mathematics, holding the ...
credited the theorem to Wilson, although it had been stated centuries earlier by
Ibn al-Haytham
Ḥasan Ibn al-Haytham, Latinized as Alhazen (; full name ; ), was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq.For the description of his main fields, see e.g. ("He is one of the prin ...
.
The only known Wilson primes are
5,
13, and
563 . Costa et al. Write that "the case
is trivial", and credit the observation that 13 is a Wilson prime to .
Early work on these numbers included searches by
N. G. W. H. Beeger
Nicolaas George Wijnand Henri Beeger (1884, in Utrecht – 1965, in Amsterdam) was a Dutch mathematician. His 1916 doctorate was on Dirichlet series. He worked for most of his life as a teacher, working on mathematics papers in his spare evenin ...
and
Emma Lehmer
Emma Markovna Lehmer ('' née'' Trotskaia) (November 6, 1906 – May 7, 2007) was a mathematician known for her work on reciprocity laws in algebraic number theory. She preferred to deal with complex number fields and integers, rather than ...
,
but 563 was not discovered until the early 1950s, when computer searches could be applied to the problem.
If any others exist, they must be greater than 2 × 10
13.
It has been
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
d that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval