Wilson prime

Wilson prime
Named after	John Wilson
Publication year	1938
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 was coordinated at the Great Internet Mersenne Prime Search forum.^[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 1000000)	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, 41341, ...
16	31, ...
17	61, 251, 479, ...	A152413
18	13151527, ...
19	71, 621629, ...
20	59, 499, 43223, 214009, ...
21	217369, ...
22	...
23	...
24	47, 3163, ...
25	...
26	97579, ...
27	53, ...
28	347, 739399, ...
29	...
30	137, 1109, 5179, ...

The smallest 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 $B = 0$ are bona fide Wilson primes. The table on the right 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 equals 1 if and only if n has 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 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 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.
^ ^a ^b 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 978-3-540-34283-0.
^ 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. pp. 346. ISBN 978-0-387-94457-9.
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 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

[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] 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 978-3-540-34283-0.

[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.

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v t Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient
Base-dependent	Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Delicate Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n - 1, n + 1, 2 n - 1, 2 n + 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k-tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Mega (1,000,000+ digits) Largest known
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers

Wilson prime

Contents

Generalizations

Wilson primes of order $n$

Near-Wilson primes

Wilson numbers

See also

Notes

References

External links

Wilson prime

Generalizations

Wilson primes of order n

Near-Wilson primes

Wilson numbers

See also

Notes

References

External links

Wilson primes of order $n$