Wilson prime numbers (after Sir John Wilson ) are prime numbers that are divisible by . It is a stronger form of Wilson's theorem . So far, only the Wilson primes 5, 13 and 563 are known.
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![(p-1)! + 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cfd25ee2b763467e4925821e3cfc9a365d7b5e4)
![p ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef685027b97072ee63a8c738f395cd40f63767e1)
definition
- For notation, see faculty , divisibility and congruence
Wilson's theorem says that is divisible by if and only if is a prime number. So for every prime number :
![(p-1)! + 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cfd25ee2b763467e4925821e3cfc9a365d7b5e4)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p \ mid (p-1)! + 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c9c9680b694f46f461a105c14780c04d437f79c)
This can be described as congruence as follows:
![(p-1)! \ equiv-1 \ pmod p](https://wikimedia.org/api/rest_v1/media/math/render/svg/45d18a5a5793c65b8e51e36bac25d0b9e8bfe22c)
or
![(p-1)! + 1 \ equiv 0 \ pmod p](https://wikimedia.org/api/rest_v1/media/math/render/svg/15c856d74d3be1912c36e032dade0fffb81851a9)
The integer result of the division
![{\ frac {(p-1)! + 1} {p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be8112076b480e54a23a7418fd701a142ddd7b70)
is also referred to in this context as the Wilson quotient (sequence A007619 in OEIS ).
![W (p)](https://wikimedia.org/api/rest_v1/media/math/render/svg/89391f0c76d84aa0d2460dce4d26ff6243b0e153)
A Wilson prime number is now any prime number that is also a divisor of “its” Wilson quotient (and thus fulfills Wilson's theorem almost twice ).
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
proof
Without loss of generality
-
is
has a clear solution![{\ displaystyle ax \ equiv 1 (\ mod n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d0ca9c093704c651cad902b0f27fdf09ff39b9e)
or
-
is
Adoption:
With
Contradiction: can not share
and simultaneously![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![n-1](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521)
example
The number is a factor of :
![p = 13](https://wikimedia.org/api/rest_v1/media/math/render/svg/23a2ea845f4e317ae5075db84475c3acd66d6e5b)
![(p-1)! + 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cfd25ee2b763467e4925821e3cfc9a365d7b5e4)
![\ frac {(13-1)! + 1} {13} = \ frac {479,001,600 + 1} {13} = 36,846,277](https://wikimedia.org/api/rest_v1/media/math/render/svg/43c3d2ac1bf04411cc906f41e62455a9644e3ea4)
So because of Wilson's Theorem, is prime. Since it is also a divisor of the corresponding Wilson quotient (36,846,277 13 = 2,834,329), it is even a Wilson prime number.
![p = 13](https://wikimedia.org/api/rest_v1/media/math/render/svg/23a2ea845f4e317ae5075db84475c3acd66d6e5b)
![:](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f)
The repeated division is equivalent to dividing by the square of the starting number. Analogously to Wilson's theorem, it is true that every prime number is a Wilson prime number if and only if:
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p ^ 2 \ mid (p-1)! + 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf3bc2b9833a0df3475aaa3071bce58f5fec39d1)
Or:
![(p-1)! \ equiv-1 \ pmod {p ^ 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f89ca2024376626a4fe41c4411a20dd1a323312)
or
![\ frac {(p-1)! + 1} {p} = W (p) \ equiv 0 \ pmod p](https://wikimedia.org/api/rest_v1/media/math/render/svg/b650f790a2ef660bbcf46476a55fd79192b2fdc1)
Occurrence
So far, only the Wilson primes 5, 13 and 563 are known (sequence A007540 in OEIS ). Should further Wilson prime numbers exist, they are greater than . It is assumed that there are an infinite number of Wilson prime numbers between and .
![{\ displaystyle 2 \ cdot 10 ^ {13}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72b63ec750d72e12ccb1d0afe3750a25ffffffc9)
![{\ displaystyle \ ln (\ ln (y) / \ ln (x))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ab11bca003fff29a7a4dd664d0e9e1b5eb7fe64)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d)
Generalizations
Wilson prime numbers of order n
The generalization of Wilson's theorem states that a natural number is prime if and only if it holds for all :
![{\ displaystyle p \ in \ mathbb {N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4dd27ec9da79b60215c701fa49fb4c5af302b7)
![{\ displaystyle 1 \ leq n \ leq p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/755ae14eb2a30cc881d179d45f5ed01902e5e79e)
![{\ displaystyle (n-1)! \ cdot (pn)! \ equiv (-1) ^ {n} {\ pmod {p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b65b0bdd6062f83317968a3417ef38c18bee2c17)
It is therefore a prime if is an integer.
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![{\ displaystyle {\ frac {(n-1)! \ cdot (pn)! - (- 1) ^ {n}} {p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c48cdcd068ecde922544f1672a4998ef0562fe)
A generalized Wilson prime number of order n is a prime number for which the following applies:
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
-
is divisor of with ,![{\ displaystyle (n-1)! \ cdot (pn)! - (- 1) ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8c8614e0ec11f9c8a943686533e8d535e247320)
![n \ geq 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe)
It is therefore a generalized Wilson Prime of order n if is an integer.
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![{\ displaystyle {\ frac {(n-1)! \ cdot (pn)! - (- 1) ^ {n}} {p ^ {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a002a98173d22b92ee50a27ab7fc4d1f7b1f6f50)
This can be described as congruence as follows:
![{\ displaystyle (n-1)! \ cdot (pn)! \ equiv (-1) ^ {n} {\ pmod {p ^ {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d9a72321c5aad765a883d67a2af4b55b329952)
or
![{\ displaystyle (n-1) \ cdot (pn)! - (- 1) ^ {n} \ equiv 0 {\ pmod {p ^ {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf38214d41524358b6626497249b67475cc41ca)
It is conjectured that for any natural number there are infinitely many generalized Wilson prime numbers of the order .
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
example
Let be a prime number and . The square number is a factor of :
![{\ displaystyle p = 17 \ in \ mathbb {P}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2a58a7b8f58e7ba604f9059d2f433fb646f5a81)
![n = 7](https://wikimedia.org/api/rest_v1/media/math/render/svg/a264db40fcdf1645e6dabe1885e21595de502623)
![{\ displaystyle p ^ {2} = 17 ^ {2} = 289}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1b86f661e98f8ad20ce5064fec101c559162c7)
![{\ displaystyle (n-1)! \ cdot (pn)! - (- 1) ^ {n} = 6! \ cdot (p-7)! + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0867b30a12631243fc5566e77d493966e003b05)
![{\ displaystyle {\ frac {6! \ cdot (17-7)! + 1} {17 ^ {2}}} = {\ frac {720 \ cdot 3628800 + 1} {289}} = {\ frac {2612736001 } {289}} = 9040609}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13fec3999b69b05cfa6f2db35210e016b27a0708)
So is a divisor of the corresponding generalized Wilson quotient and is therefore a generalized Wilson prime number of order .
![p = 17](https://wikimedia.org/api/rest_v1/media/math/render/svg/70c557c056621563c0c8375165eccc093a47112b)
![n = 7](https://wikimedia.org/api/rest_v1/media/math/render/svg/a264db40fcdf1645e6dabe1885e21595de502623)
The following table shows the generalized Wilson prime numbers of the order for :
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle 1 \ leq n \ leq 30}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ae24dc79337a8d10d8654fcd73346ce44df6159)
|
|
Prime , so divider of is
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36) ![p ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef685027b97072ee63a8c738f395cd40f63767e1)
|
OEIS link
|
1 |
![(p-1)! + 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cfd25ee2b763467e4925821e3cfc9a365d7b5e4) |
5, 13, 563 ... |
(Follow A007540 in OEIS )
|
2 |
![{\ displaystyle (p-2)! - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b1e56bd5c71f8571f85c31ba89160356d43465b) |
2, 3, 11, 107, 4931 ... |
(Follow A079853 in OEIS )
|
3 |
![{\ displaystyle 2! \ cdot (p-3)! + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b840c2ec2bf1271bb2e71a2978b96278a4ce1511) |
7… |
|
4th |
![{\ displaystyle 3! \ cdot (p-4)! - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/244a4d959249ee27c7e512f2b52cbcf44ce13148) |
10429 ... |
|
5 |
![{\ displaystyle 4! \ cdot (p-5)! + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6accae85a670f7fa645aae082c8400f08c8fa53) |
5, 7, 47 ... |
|
6th |
![{\ displaystyle 5! \ cdot (p-6)! - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/922ede4d3aa8c70596b54d19fcf483b45befa54e) |
11 ... |
|
7th |
![{\ displaystyle 6! \ cdot (p-7)! + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cae8f042010f4eada5c9f3fad5e4b5720d8f2fc0) |
17 ... |
|
8th |
![{\ displaystyle 7! \ cdot (p-8)! - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fef0c261cf6cc3f6902cbf038bdce0156381fdcd) |
... |
|
9 |
![{\ displaystyle 8! \ cdot (p-9)! + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bde38c457c28d76fe9c61057eb3d89f7f22917e6) |
541… |
|
10 |
![{\ displaystyle 9! \ cdot (p-10)! - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cd98d2cc474e63e85355fb5cd015dd963a46552) |
11, 1109 ... |
|
11 |
![{\ displaystyle 10! \ cdot (p-11)! + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5adadf30e16ca27d98fdfa041c48825524b593) |
17, 2713 ... |
|
12 |
![{\ displaystyle 11! \ cdot (p-12)! - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05e18c9185276a1c255129c0be3b36ea114b8358) |
... |
|
13 |
![{\ displaystyle 12! \ cdot (p-13)! + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cad56e2bea5ae00512c98cb2b441cbd7b481921c) |
13… |
|
14th |
![{\ displaystyle 13! \ cdot (p-14)! - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4b63e40f353bc0583e6c7782f09bc2d4577cd92) |
... |
|
15th |
![{\ displaystyle 14! \ cdot (p-15)! + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e884a3ec8c5c293e1c3dac5df10a7af86eeeefe) |
349 ... |
|
|
|
|
Prime , so divider of is
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36) ![p ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef685027b97072ee63a8c738f395cd40f63767e1)
|
OEIS link
|
16 |
![{\ displaystyle 15! \ cdot (p-16)! - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96567eeb4a1c6508ec892bf67d4d46d03fddfbe7) |
31 ... |
|
17th |
![{\ displaystyle 16! \ cdot (p-17)! + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97e0439849bc515712abfdfec44923d1b531e347) |
61, 251, 479 ... |
(Follow A152413 in OEIS )
|
18th |
![{\ displaystyle 17! \ cdot (p-18)! - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3977555a1148bfcc696e32698c79a56cd2279cf2) |
13151527 ... |
|
19th |
![{\ displaystyle 18! \ cdot (p-19)! + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa9e67188e2140450ad2eae0c094c7214713d2f) |
71 ... |
|
20th |
![{\ displaystyle 19! \ cdot (p-20)! - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/057b2aad7491f3ba84b8a670692332ec08223033) |
59, 499 ... |
|
21st |
![{\ displaystyle 20! \ cdot (p-21)! + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f06d5a0d12d026c8fb170f57965a38b50739720b) |
217369 ... |
|
22nd |
![{\ displaystyle 21! \ cdot (p-22)! - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25bbc39edf50b59b643e0190398f57885c7577b1) |
... |
|
23 |
![{\ displaystyle 22! \ cdot (p-23)! + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fecc1f9af8a38061203c17d16d0a76a1998b47e9) |
... |
|
24 |
![{\ displaystyle 23! \ cdot (p-24)! - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea3d560a0dbd8c0118f7562e68b752ba60a5598) |
47, 3163 ... |
|
25th |
![{\ displaystyle 24! \ cdot (p-25)! + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f523d648766039640b9451416f0250e5208edd0d) |
... |
|
26th |
![{\ displaystyle 25! \ cdot (p-26)! - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41059a777fc5a4e294c710d39969c0626ecbd547) |
97579 ... |
|
27 |
![{\ displaystyle 26! \ cdot (p-27)! + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b2a013645d6a12f7bd0d3a18045593de2d13c08) |
53… |
|
28 |
![{\ displaystyle 27! \ cdot (p-28)! - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/545e064bf0ab4e2fcf46ecc27e3a2872fd2f2456) |
347 ... |
|
29 |
![{\ displaystyle 28! \ cdot (p-29)! + 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/340ef7c3ac4ccfcb2f34fc1cbfbfac313b4720a2) |
... |
|
30th |
![{\ displaystyle 29! \ cdot (p-30)! - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91b4707259ecf6710c0e5fd11babf6e839bd829f) |
137, 1109, 5179 ... |
|
|
The smallest generalized Wilson prime numbers of the order are (with increasing ):
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![n = 1,2,3, \ ldots](https://wikimedia.org/api/rest_v1/media/math/render/svg/a508e267228dcaa193fe3a7df64c94b0f31e143e)
- 5, 2, 7, 10429, 5, 11, 17 ... (Follow A128666 in OEIS )
The next generalized Wilson prime of the order is not already known, but it must be greater than .
![n = 8](https://wikimedia.org/api/rest_v1/media/math/render/svg/1344e2c194f016e12bb978493013955fe2f445f5)
![{\ displaystyle 1.4 \ cdot 10 ^ {7}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9671e1a60829d1f7c5c0ff75cd4d8aeadcd257c4)
Almost Wilson primes
A prime number representing the congruence
![{\ displaystyle p \ in \ mathbb {P}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7891a37dcb8b0ee507f9ef2038a853e245d76657)
-
with a small amount
met, is called fast-Wilson Prime (English near-Wilson primes ).
Is , we get and get the Wilson prime numbers.
![B = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/af1d79a64f775646f76d79a452ca8e1082fb1f17)
![{\ displaystyle (p-1)! \ equiv -1 {\ pmod {p ^ {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f89ca2024376626a4fe41c4411a20dd1a323312)
The following table shows all such Fast Wilson prime numbers for with :
![{\ displaystyle | B | \ leq 100}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a2f7870d319f258443b7b0e3032de89c21cfa4f)
![{\ displaystyle 10 ^ {6} <p <4 \ cdot 10 ^ {11}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b7b659cec5eecaedba37429ede4d10beac6e847)
|
|
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
|
|
Wilson numbers
A Wilson number is a natural number for which the following applies:
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
-
, With
Here is if and only if a primitive root has, otherwise is .
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle e = -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46cb2d1d084fd9236982ddddd3c65cde7747bdc5)
For every natural number is by divisible. The quotient is called the generalized Wilson quotient . The first generalized Wilson quotients are:
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle W (n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a15a89d2e090b211d2d2960dc6bea69ea0a6b294)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle {\ frac {W (n)} {n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43ebe6d970b3e5fa1c3807b724c4e839bb2190d2)
- 2, 1, 1, 1, 5, 1, 103, 13, 249, 19, 329891, 32, 36846277, 1379, 59793, 126689, 1230752346353, 4727, 336967037143579, 436486, 2252263619, 56815333, 488695968598959812886, 154925968598959812886, 15492596859895986087, 154925968598959812886382505 Follow A157249 in OEIS )
If the generalized Wilson quotient is divisible by, you get a Wilson number. These are:
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
- 1, 5, 13, 563, 5971, 558771, 1964215, 8121909, 12326713, 23025711, 26921605, 341569806, 399292158 (follow A157250 in OEIS )
If a Wilson number is prime, then Wilson is prime. There are 13 Wilson numbers for .
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle n <5 \ cdot 10 ^ {8}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/430bc6ec46f21a128e2bafd146eedfa758764b18)
literature
Web links
-
Eric W. Weisstein : Wilson Prime . In: MathWorld (English).
- Chris K. Caldwell: Wilson prime . The Prime Glossary.
-
Here is the latest update on… - Email from Richard McIntosh to Paul Zimmermann on March 9, 2004
-
Emma Lehmer : On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson. Annals of Mathematics 39 (2), April 1938, pp. 350-360 , accessed February 3, 2020 .
-
Distributed search for Wilson primes. mersenneforum.org, accessed February 3, 2020 .
-
Erna H. Pearson : On the Congruences ( p - 1)! ≡ −1 and 2 p −1 ≡ 1 (mod p 2 ). Mathematics of Computation 17 , April 6, 1962, pp. 194-195 , accessed February 3, 2020 .
Individual evidence
-
↑ Eric W. Weisstein : Wilson Quotient . In: MathWorld (English).
-
^ Karl Goldberg: A table of Wilson quotients and the third Wilson prime . In: Journal of the London Mathematical Society , April 28, 1953, pp. 252-256 (English)
-
^ A b Edgar Costa, Robert Gerbicz, David Harvey : A search for Wilson primes. October 27, 2012, pp. 1–25 , accessed February 1, 2020 .
-
^ Richard Crandall , Karl Dilcher, Carl Pomerance : A search for Wieferich and Wilson primes . Mathematics of Computation 66 , January 1997, pp. 433–449 (English)
-
↑ Chris K. Caldwell: Wilson prime . The Prime Glossary.
-
↑ Takashi Agoh, Karl Dilcher, Ladislav Skula: Wilson Quotients for composite moduli. Mathematics of Computation 67 (222), April 1998, pp. 843-861 , accessed February 2, 2020 .
|
|
formula based
|
Carol ((2 n - 1) 2 - 2) |
Cullen ( n ⋅2 n + 1) |
Double Mersenne (2 2 p - 1 - 1) |
Euclid ( p n # + 1) |
Factorial ( n! ± 1) |
Fermat (2 2 n + 1) |
Cubic ( x 3 - y 3 ) / ( x - y ) |
Kynea ((2 n + 1) 2 - 2) |
Leyland ( x y + y x ) |
Mersenne (2 p - 1) |
Mills ( A 3 n ) |
Pierpont (2 u ⋅3 v + 1) |
Primorial ( p n # ± 1) |
Proth ( k ⋅2 n + 1) |
Pythagorean (4 n + 1) |
Quartic ( x 4 + y 4 ) |
Thabit (3⋅2 n - 1) |
Wagstaff ((2 p + 1) / 3) |
Williams (( b-1 ) ⋅ b n - 1)
Woodall ( n ⋅2 n - 1)
|
Prime number follow
|
Bell |
Fibonacci |
Lucas |
Motzkin |
Pell |
Perrin
|
property-based
|
Elitist |
Fortunate |
Good |
Happy |
Higgs |
High quotient |
Isolated |
Pillai |
Ramanujan |
Regular |
Strong |
Star |
Wall – Sun – Sun |
Wieferich |
Wilson
|
base dependent
|
Belphegor |
Champernowne |
Dihedral |
Unique |
Happy |
Keith |
Long |
Minimal |
Mirp |
Permutable |
Primeval |
Palindrome |
Repunit ((10 n - 1) / 9) |
Weak |
Smarandache – Wellin |
Strictly non-palindromic |
Strobogrammatic |
Tetradic |
Trunkable |
circular
|
based on tuples
|
Balanced ( p - n , p , p + n) |
Chen |
Cousin ( p , p + 4) |
Cunningham ( p , 2 p ± 1, ...) |
Triplet ( p , p + 2 or p + 4, p + 6) |
Constellation |
Sexy ( p , p + 6) |
Safe ( p , ( p - 1) / 2) |
Sophie Germain ( p , 2 p + 1) |
Quadruplets ( p , p + 2, p + 6, p + 8) |
Twin ( p , p + 2) |
Twin bi-chain ( n ± 1, 2 n ± 1, ...)
|
according to size
|
Titanic (1,000+ digits) |
Gigantic (10,000+ digits) |
Mega (1,000,000+ digits) |
Beva (1,000,000,000+ positions)
|
Composed
|
Carmichael |
Euler's pseudo |
Almost |
Fermatsche pseudo |
Pseudo |
Semi |
Strong pseudo |
Super Euler's pseudo
|