Wilson prime

Wilson prime
Named after	John Wilson
Publication year	1938[1]
Author of publication	Emma Lehmer
No. of known terms	3
First terms	5, 13, 563
Largest known term	563
OEIS index	A007540; Wilson primes: primes p such that (p-1)! == -1 (mod p^2);

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 the OEIS); if any others exist, they must be greater than 2×10¹³.^[2] 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)).^[3]

Several computer searches have been done in the hope of finding new Wilson primes.^[4]^[5]^[6] The Ibercivis distributed computing project includes a search for Wilson primes.^[7] Another search is coordinated at the mersenneforum.^[8]

Generalizations

Wilson primes of order $n$

Wilson's theorem can be expressed in general as $(n-1)!(p-n)!\equiv (-1)^{n}\ {\bmod {p}}$ for every integer $n\geq 1$ and prime $p\geq n$ . 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 $(n-1)!(p-n)! - (-1)^n$ (checked up to 10000)	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 prime of order n are

5, 2, 7, 10429, 5, 11, 17, ... (The next term > 1.4×10⁷) (sequence A128666 in the 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 $(p - 1)! \equiv - 1 + Bp mod p 2$ 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 $| B | \leq 100$ from 10⁶ up to 4×10¹¹:^[2]

Wilson numbers

A Wilson number is a natural number n such that W(n) ≡ 0 (mod n²), where $W(n)=\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}{k}+e$ , the constant e = 1 if and only if n have a primitive root, otherwise, e = -1^[9] 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 the OEIS)

If a Wilson number n is prime, then n is a Wilson prime. There are 13 Wilson numbers up to 5×10⁸.^[10]

Notes

↑ 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.
1 2 A Search for Wilson primes Retrieved on November 2, 2012.
↑ The Prime Glossary: Wilson prime
↑ McIntosh, R. (9 March 2004). "WILSON STATUS (Feb. 1999)". E-Mail to Paul Zimmermann. Retrieved 6 June 2011.
↑ A search for Wieferich and Wilson primes, p 443
↑ Ribenboim, P.; Keller, W. (2006). Die Welt der Primzahlen: Geheimnisse und Rekorde (in German). Berlin Heidelberg New York: Springer. p. 241. ISBN 3-540-34283-4.
↑ Ibercivis site
↑ Distributed search for Wilson primes (at mersenneforum.org)
↑ see Gauss's generalization of Wilson's theorem
↑ Agoh, Takashi; Dilcher, Karl; Skula, Ladislav (1998). "Wilson quotients for composite moduli" (PDF). Math. Comput. 67 (222): 843–861. doi:10.1090/S0025-5718-98-00951-X.

References

Beeger, N. G. W. H. (1913–1914). "Quelques remarques sur les congruences r^p−1 ≡ 1 (mod p²) et (p − 1!) ≡ −1 (mod p²)". The Messenger of Mathematics. 43: 72–84.
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.
Ribenboim, Paulo (1996). The new book of prime number records. Springer-Verlag. p. 346. ISBN 0-387-94457-5.
Crandall, Richard E.; Dilcher, Karl; Pomerance, Carl (1997). "A search for Wieferich and Wilson primes". Math. Comput. 66 (217): 433–449. doi:10.1090/S0025-5718-97-00791-6.
Crandall, Richard E.; Pomerance, Carl (2001). Prime Numbers: A Computational Perspective. Springer-Verlag. p. 29. ISBN 0-387-94777-9.
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

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[1] 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.

[Search-2] 1 2 A Search for Wilson primes Retrieved on November 2, 2012.

[3] The Prime Glossary: Wilson prime

[4] McIntosh, R. (9 March 2004). "WILSON STATUS (Feb. 1999)". E-Mail to Paul Zimmermann. Retrieved 6 June 2011.

[5] A search for Wieferich and Wilson primes, p 443

[6] Ribenboim, P.; Keller, W. (2006). Die Welt der Primzahlen: Geheimnisse und Rekorde (in German). Berlin Heidelberg New York: Springer. p. 241. ISBN 3-540-34283-4.

[7] Ibercivis site

[8] Distributed search for Wilson primes (at mersenneforum.org)

[9] see Gauss's generalization of Wilson's theorem

[10] Agoh, Takashi; Dilcher, Karl; Skula, Ladislav (1998). "Wilson quotients for composite moduli" (PDF). Math. Comput. 67 (222): 843–861. doi:10.1090/S0025-5718-98-00951-X.

Prime number classes
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