Wilson prime


A Wilson prime, named after English mathematician John Wilson, is a prime number p such that p2 divides ! + 1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides ! + 1.
The only known Wilson primes are 5, 13, and 563 ; if any others exist, they must be greater than 2. It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval is about log/log).
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 was coordinated at the Great Internet Mersenne Prime Search forum.

Generalizations

Wilson primes of order

Wilson's theorem can be expressed in general as for every integer and prime. Generalized Wilson primes of order are the primes such that divides.
It was conjectured that for every natural number, there are infinitely many Wilson primes of order.
prime such that divides OEIS sequence
15, 13, 563,...
22, 3, 11, 107, 4931,...
37,...
410429,...
55, 7, 47,...
611,...
717,...
8...
9541,...
1011, 1109,...
1117, 2713,...
12...
1313,...
14...
15349, 41341,...
1631,...
1761, 251, 479,...
1813151527,...
1971, 621629,...
2059, 499, 43223, 214009,...
21217369,...
22...
23...
2447, 3163,...
25...
2697579,...
2753,...
28347, 739399,...
29...
30137, 1109, 5179,...

Least generalized Wilson prime of order n are

Near-Wilson primes

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 with small can be called a near-Wilson prime. Near-Wilson primes with represent Wilson primes. The following table lists all such primes with from up to 4:

Wilson numbers

A Wilson number is a natural number n such that W ≡ 0, where, the constant e = 1 if and only if n have a primitive root, otherwise, e = -1 For every natural number n, W is divisible by n, and the quotients 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.