Wilson prime

Wilson prime

Named after	John Wilson
Publication year	1938
Author of publication	Emma Lehmer
Number of known terms	3
First terms	5, 13, 563
Largest known term	563
OEIS index	A007540

A Wilson prime, named after English mathematician John Wilson, is a prime number p such that p² divides (p − 1)! + 1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p − 1)! + 1.

The only known Wilson primes are 5, 13, and 563 (sequence A007540 in OEIS); if any others exist, they must be greater than 2×10¹³. It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval [x, y] is about log(log(y)/log(x)).

Several computer searches have been done in the hope of finding new Wilson primes. The Ibercivis distributed computing project includes a search for Wilson primes. Another search is coordinated at the mersenneforum.

Generalizations

Wilson primes of order n

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

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

$n$	prime $p$ such that $p^2$ divides $(n-1)!(p-n)! - (-1)^n$	OEIS sequence
1	5, 13, 563, ...	A007540
2	2, 3, 11, 107, 4931, ...	A079853
3	7, ...
4	10429, ...
5	5, 7, 47, ...
6	11, ...
7	17, ...
8	...
9	541, ...
10	11, 1109, ...
11	17, 2713, ...
12	...
13	13, ...
14	...
15	349, ...
16	31, ...
17	61, 251, 479, ...	A152413
18	13151527, ...
19	71, ...
20	59, 499, ...
21	217369, ...
22	...
23	...
24	47, 3163, ...
25	...
26	97579, ...
27	53, ...
28	347, ...
29	...
30	137, 1109, 5179, ...

Least generalized Wilson number of order n are

5, 2, 7, 10429, 5, 11, 17, ... (The next term > 1.4×10⁷) (sequence A128666 in OEIS)

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 p satisfying the congruence (p − 1)! ≡ − 1 + Bp mod p² with small | B | can be called a near-Wilson prime. Near-Wilson primes with B = 0 represent Wilson primes. The following table lists all such primes with | B | ≤ 100 from 10⁶ up to 4×10¹¹:

Wilson numbers

A Wilson number is a natural number n such that W(n) ≡ 0 (mod n²), where W(n) = $\left(\prod_{1<=k<=n,gcd(k,n)=1}{k}\right)+e$, the constant e = 1 if and only if n have a primitive root, otherwise, e = -1. For every natural number n, W(n) is divisible by n, and the quotients (called generalized Wilson quotients) are listed in  A157249. The Wilson numbers are

1, 5, 13, 563, 5971, 558771, 1964215, 8121909, 12326713, 23025711, 26921605, 341569806, 399292158, ... (sequence A157250 in OEIS)

If a Wilson number n is prime, then n is a Wilson prime. There are 13 Wilson numbers up to 5×10⁸.

See also

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

External links

• The Prime Glossary: Wilson prime
• Weisstein, Eric W., "Wilson prime", MathWorld.
• Status of the search for Wilson primes
• On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson