Wilson prime

Wilson prime

Named after	563
OEIS index	A007540 Wilson primes: primes ${\displaystyle p}$ such that ${\displaystyle (p-1)!\equiv -1\ (\operatorname {mod} {p^{2}})}$

In number theory, a Wilson prime is a prime number ${\displaystyle p}$ such that ${\displaystyle p^{2}}$ divides ${\displaystyle (p-1)!+1}$, where "${\displaystyle !}$" denotes the

, which states that every prime ${\displaystyle p}$ divides ${\displaystyle (p-1)!+1}$. Both are named for 18th-century

The only known Wilson primes are

). Costa et al. write that "the case ${\displaystyle p=5}$ is trivial", and credit the observation that 13 is a Wilson prime to Mathews (1892).^[3][4] Early work on these numbers included searches by N. G. W. H. Beeger and Emma Lehmer,^[5][3][6] but 563 was not discovered until the early 1950s, when computer searches could be applied to the problem.^[3][7][8] If any others exist, they must be greater than 2 × 10¹³.^[3] It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval ${\displaystyle [x,y]}$ is about ${\displaystyle \log \log _{x}y}$.^[9]

Several computer searches have been done in the hope of finding new Wilson primes.^[10][11][12] The Ibercivis distributed computing project includes a search for Wilson primes.^[13] Another search was coordinated at the Great Internet Mersenne Prime Search forum.^[14]

Generalizations

Wilson primes of order n

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

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

The smallest generalized Wilson primes of order ${\displaystyle n}$ are:

Near-Wilson primes

p	B
1282279	+20
1306817	−30
1308491	−55
1433813	−32
1638347	−45
1640147	−88
1647931	+14
1666403	+99
1750901	+34
1851953	−50
2031053	−18
2278343	+21
2313083	+15
2695933	−73
3640753	+69
3677071	−32
3764437	−99
3958621	+75
5062469	+39
5063803	+40
6331519	+91
6706067	+45
7392257	+40
8315831	+3
8871167	−85
9278443	−75
9615329	+27
9756727	+23
10746881	−7
11465149	−62
11512541	−26
11892977	−7
12632117	−27
12893203	−53
14296621	+2
16711069	+95
16738091	+58
17879887	+63
19344553	−93
19365641	+75
20951477	+25
20972977	+58
21561013	−90
23818681	+23
27783521	−51
27812887	+21
29085907	+9
29327513	+13
30959321	+24
33187157	+60
33968041	+12
39198017	−7
45920923	−63
51802061	+4
53188379	−54
56151923	−1
57526411	−66
64197799	+13
72818227	−27
87467099	−2
91926437	−32
92191909	+94
93445061	−30
93559087	−3
94510219	−69
101710369	−70
111310567	+22
117385529	−43
176779259	+56
212911781	−92
216331463	−36
253512533	+25
282361201	+24
327357841	−62
411237857	−84
479163953	−50
757362197	−28
824846833	+60
866006431	−81
1227886151	−51
1527857939	−19
1636804231	+64
1686290297	+18
1767839071	+8
1913042311	−65
1987272877	+5
2100839597	−34
2312420701	−78
2476913683	+94
3542985241	−74
4036677373	−5
4271431471	+83
4296847931	+41
5087988391	+51
5127702389	+50
7973760941	+76
9965682053	−18
10242692519	−97
11355061259	−45
11774118061	−1
12896325149	+86
13286279999	+52
20042556601	+27
21950810731	+93
23607097193	+97
24664241321	+46
28737804211	−58
35525054743	+26
41659815553	+55
42647052491	+10
44034466379	+39
60373446719	−48
64643245189	−21
66966581777	+91
67133912011	+9
80248324571	+46
80908082573	−20
100660783343	+87
112825721339	+70
231939720421	+41
258818504023	+4
260584487287	−52
265784418461	−78
298114694431	+82

A prime ${\displaystyle p}$ satisfying the congruence ${\displaystyle (p-1)!\equiv -1+Bp\ (\operatorname {mod} {p^{2}})}$ with small ${\displaystyle |B|}$ can be called a near-Wilson prime. Near-Wilson primes with ${\displaystyle B=0}$ are bona fide Wilson primes. The table on the right lists all such primes with ${\displaystyle |B|\leq 100}$ from 10⁶ up to 4×10¹¹.^[3]

Wilson numbers

A Wilson number is a natural number ${\displaystyle n}$ such that ${\displaystyle W(n)\equiv 0\ (\operatorname {mod} {n^{2}})}$, where

and where the ${\displaystyle \pm 1}$ term is positive if and only if ${\displaystyle n}$ has a primitive root and negative otherwise.^[15] For every natural number ${\displaystyle n}$, ${\displaystyle W(n)}$ is divisible by ${\displaystyle n}$, and the quotients (called generalized Wilson quotients) are listed in OEIS: A157249. The Wilson numbers are

If a Wilson number ${\displaystyle n}$ is prime, then ${\displaystyle n}$ is a Wilson prime. There are 13 Wilson numbers up to 5×10⁸.^[16]

See also

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

References

Further reading

• Crandall, Richard E.; Pomerance, Carl (2001). Prime Numbers: A Computational Perspective. Springer-Verlag. p. 29.
  ISBN 978-0-387-94777-8
  .
• Pearson, Erna H. (1963). "On the Congruences (p − 1)! ≡ −1 and 2^p−1 ≡ 1 (mod p²)" (PDF). Math. Comput. 17: 194–195.

External links

• The Prime Glossary: Wilson prime
• Weisstein, Eric W. "Wilson prime". MathWorld.
• Status of the search for Wilson primes