The only known Wilson primes are 5, 13, and 563 (sequence A007540 in the OEIS). Costa et al. write that "the case 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 is about .[9]
Wilson's theorem can be expressed in general as for every integer and prime . Generalized Wilson primes of order n are the primes p such that divides .
It was conjectured that for every natural numbern, there are infinitely many Wilson primes of order n.
The smallest generalized Wilson primes of order are:
5, 2, 7, 10429, 5, 11, 17, ... (The next term > 1.4 × 107) (sequence A128666 in the OEIS)
Near-Wilson primesedit
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 satisfying the congruence with small can be called a near-Wilson prime. Near-Wilson primes with are bona fide Wilson primes. The table on the right lists all such primes with from 106 up to 4×1011.[3]
Wilson numbersedit
A Wilson number is a natural number such that , where
and where the term is positive if and only if has a primitive root and negative otherwise.[15] For every natural number , is divisible by , and the quotients (called generalized Wilson quotients) are listed in OEIS: A157249. The Wilson numbers are
1, 5, 13, 563, 5971, 558771, 1964215, 8121909, 12326713, 23025711, 26921605, 341569806, 399292158, ... (sequence A157250 in the OEIS)
If a Wilson number is prime, then is a Wilson prime. There are 13 Wilson numbers up to 5×108.[16]
^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.)
^ abcdeCosta, Edgar; Gerbicz, Robert; Harvey, David (2014). "A search for Wilson primes". Mathematics of Computation. 83 (290): 3071–3091. arXiv:1209.3436. doi:10.1090/S0025-5718-2014-02800-7. MR 3246824. S2CID 6738476.
^Mathews, George Ballard (1892). "Example 15". Theory of Numbers, Part 1. Deighton & Bell. p. 318.
^Lehmer, Emma (April 1938). "On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson" (PDF). Annals of Mathematics. 39 (2): 350–360. doi:10.2307/1968791. JSTOR 1968791. Retrieved 8 March 2011.
^Wall, D. D. (October 1952). "Unpublished mathematical tables" (PDF). Mathematical Tables and Other Aids to Computation. 6 (40): 238. doi:10.2307/2002270. JSTOR 2002270.
^Goldberg, Karl (1953). "A table of Wilson quotients and the third Wilson prime". J. London Math. Soc.28 (2): 252–256. doi:10.1112/jlms/s1-28.2.252.
^Crandall, Richard E.; Dilcher, Karl; Pomerance, Carl (1997). "A search for Wieferich and Wilson primes". Math. Comput. 66 (217): 433–449. Bibcode:1997MaCom..66..433C. doi:10.1090/S0025-5718-97-00791-6. See p. 443.
^Ribenboim, P.; Keller, W. (2006). Die Welt der Primzahlen: Geheimnisse und Rekorde (in German). Berlin Heidelberg New York: Springer. p. 241. ISBN 978-3-540-34283-0.
^"Ibercivis site". Archived from the original on 2012-06-20. Retrieved 2011-03-10.
^Distributed search for Wilson primes (at mersenneforum.org)