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Wilson prime

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Template:Short description Template:Infobox integer sequence In number theory, a Wilson prime is a prime number <math>p</math> such that <math>p^2</math> divides <math>(p-1)!+1</math>, where "<math>!</math>" denotes the factorial function; compare this with Wilson's theorem, which states that every prime <math>p</math> divides <math>(p-1)!+1</math>. Both are named for 18th-century English mathematician John Wilson; in 1770, Edward Waring credited the theorem to Wilson,<ref>Edward Waring, Meditationes Algebraicae (Cambridge, England: 1770), page 218 (in Latin). In the third (1782) edition of Waring's Meditationes Algebraicae, Wilson's theorem appears as problem 5 on page 380. On that page, Waring states: "Hanc maxime elegantem primorum numerorum proprietatem invenit vir clarissimus, rerumque mathematicarum peritissimus Joannes Wilson Armiger." (A man most illustrious and most skilled in mathematics, Squire John Wilson, found this most elegant property of prime numbers.)</ref> although it had been stated centuries earlier by Ibn al-Haytham.<ref>Template:MacTutor Biography</ref>

The only known Wilson primes are 5, 13, and 563 (sequence A007540 in the OEIS). Costa et al. write that "the case <math>p=5</math> is trivial", and credit the observation that 13 is a Wilson prime to Template:Harvtxt.<ref name="Search"/><ref>Template:Cite book</ref> Early work on these numbers included searches by N. G. W. H. Beeger and Emma Lehmer,<ref name=lehmer>Template:Cite journal</ref><ref name="Search"/><ref>Template:Cite journal</ref> but 563 was not discovered until the early 1950s, when computer searches could be applied to the problem.<ref name="Search"/><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> If any others exist, they must be greater than 2 × 1013.<ref name="Search">Template:Cite journal</ref> It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval <math>[x,y]</math> is about <math>\log\log_x y</math>.<ref>The Prime Glossary: Wilson prime</ref>

Several computer searches have been done in the hope of finding new Wilson primes.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite journal See p. 443.</ref><ref>Template:Cite book</ref> The Ibercivis distributed computing project includes a search for Wilson primes.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Another search was coordinated at the Great Internet Mersenne Prime Search forum.<ref>Distributed search for Wilson primes (at mersenneforum.org)</ref>

GeneralizationsEdit

Wilson primes of order Template:MvarEdit

Wilson's theorem can be expressed in general as <math>(n-1)!(p-n)!\equiv(-1)^n\ \bmod p</math> for every integer <math>n \ge 1</math> and prime <math>p \ge n</math>. Generalized Wilson primes of order Template:Mvar are the primes Template:Mvar such that <math>p^2</math> divides <math>(n-1)!(p-n)! - (-1)^n</math>.

It was conjectured that for every natural number Template:Mvar, there are infinitely many Wilson primes of order Template:Mvar.

The smallest generalized Wilson primes of order <math>n</math> are: Template:Bi

Near-Wilson primesEdit

A prime <math>p</math> satisfying the congruence <math>(p-1)!\equiv -1+Bp\ (\operatorname{mod}{p^2})</math> with small <math>|B|</math> can be called a near-Wilson prime. Near-Wilson primes with <math>B=0</math> are bona fide Wilson primes. The table on the right lists all such primes with <math>|B|\le 100</math> from Template:10^ up to 4Template:E.<ref name="Search"/>

Wilson numbersEdit

A Wilson number is a natural number <math>n</math> such that <math>W(n)\equiv 0\ (\operatorname{mod}{n^2})</math>, where <math display=block>W(n) = \pm1+\prod_\stackrel{1 \le k \le n}{\gcd(k,n)=1}{k},</math>and where the <math>\pm1</math> term is positive if and only if <math>n</math> has a primitive root and negative otherwise.<ref>see Gauss's generalization of Wilson's theorem</ref> For every natural number <math>n</math>, <math>W(n)</math> is divisible by <math>n</math>, and the quotients (called generalized Wilson quotients) are listed in Template:Oeis. The Wilson numbers are Template:Bi

If a Wilson number <math>n</math> is prime, then <math>n</math> is a Wilson prime. There are 13 Wilson numbers up to 5Template:E.<ref>Template:Cite journal</ref>

See alsoEdit

ReferencesEdit

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Further readingEdit

External linksEdit

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:WilsonPrime%7CWilsonPrime.html}} |title = Wilson prime |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}


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