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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 |
|
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 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 number n, there are infinitely many Wilson primes of order n.
prime such that divides (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:
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)! ≡ −1 + Bp mod p2 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| ≤ 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 , 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]
See also
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 rp−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 2p−1 ≡ 1 (mod p2)" (PDF). Math. Comput. 17: 194–195.
External links
- The Prime Glossary: Wilson prime
- Weisstein, Eric W. "Wilson prime". MathWorld.
- Status of the search for Wilson primes
- Classes of prime numbers
- Factorial and binomial topics
- Unsolved problems in number theory