Wilson prime

 ( Classes Of Prime Numbers ) 

In number theory, a Wilson prime is a prime number $p$ such that $p^2$ divisor $(p-1)!+1$, where "$!$" denotes the factorial function; compare this with Wilson's theorem, which states that every prime $p$ divides $(p-1)!+1$. Both are named for 18th-century English people mathematician John Wilson; in 1770, Edward Waring credited the theorem to Wilson,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.) although it had been stated centuries earlier by Ibn al-Haytham.

The only known Wilson primes are 5, 13, and 563 . Costa et al. write that "the case $p=5$ 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 and Emma Lehmer, 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 × 10¹³. It has been that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval $x,y$ is about $\backslash log\backslash log\_x\; y$. The Prime Glossary: Wilson prime

Several computer searches have been done in the hope of finding new Wilson primes. See p. 443.

The Ibercivis distributed computing project includes a search for Wilson primes. Another search was coordinated at the Great Internet Mersenne Prime Search forum. Distributed search for Wilson primes (at mersenneforum.org)

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Generalizations

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Wilson primes of order Wilson's theorem can be expressed in general as $(n-1)!(p-n)!\backslash equiv(-1)^n\backslash \; \backslash bmod\; p$ for every integer $n\; \backslash ge\; 1$ and prime $p\; \backslash ge\; n$. Generalized Wilson primes of order are the primes such that $p^2$ divides $(n-1)!(p-n)!\; -\; (-1)^n$.

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

The smallest generalized Wilson primes of order $n$ are:

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Near-Wilson primes

+20

−30

−55

−32

−45

−88

+14

+99

+34

−50

−18

+21

+15

−73

+69

−32

−99

+75

+39

+40

+91

+45

+40

+3

−85

−75

+27

+23

−7

−62

−26

−7

−27

−53

+2

+95

+58

+63

−93

+75

+25

+58

−90

+23

−51

+21

+9

+13

+24

+60

+12

−7

−63

+4

−54

−1

−66

+13

−27

−2

−32

+94

−30

−3

−69

−70

+22

−43

+56

−92

−36

+25

+24

−62

−84

−50

−28

+60

−81

−51

−19

+64

+18

+8

−65

+5

−34

−78

+94

−74

−5

+83

+41

+51

+50

+76

−18

−97

−45

−1

+86

+52

+27

+93

+97

+46

−58

+26

+55

+10

+39

−48

−21

+91

+9

+46

−20

+87

+70

+41

+4

−52

−78

+82

A prime $p$ satisfying the congruence $(p-1)!\backslash equiv\; -1+Bp\backslash \; (\backslash operatorname\{mod\}\{p^2\})$ with small $|B|$ can be called a near-Wilson prime. Near-Wilson primes with $B=0$ are bona fide Wilson primes. The table on the right lists all such primes with $|B|\backslash le\; 100$ from up to 4.

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Wilson numbers

A Wilson number is a natural number $n$ such that $W(n)\backslash equiv\; 0\backslash \; (\backslash operatorname\{mod\}\{n^2\})$, where $$W(n)\; =\; \backslash pm1+\backslash prod\_\backslash stackrel\{1\; \backslash le\; k\; \backslash le\; n\}\{\backslash gcd(k,n)=1\}\{k\},$$and where the $\backslash pm1$ term is positive if and only if $n$ has a primitive root and negative otherwise.see Gauss's generalization of Wilson's theorem For every natural number $n$, $W(n)$ is divisible by $n$, and the quotients (called generalized ) are listed in . The Wilson numbers are

If a Wilson number $n$ is prime, then $n$ is a Wilson prime. There are 13 Wilson numbers up to 5.

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See also

• PrimeGrid
• Table of congruences
• Wall–Sun–Sun prime
• Wieferich prime
• Wolstenholme prime

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Further reading

• (2025). 9780387947778, Springer-Verlag. ISBN 9780387947778

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External links

• The Prime Glossary: Wilson prime
• Status of the search for Wilson primes

Categories: Classes Of Prime Numbers, Factorial And Binomial Topics, Unsolved Problems In Number Theory

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