Wilson prime
Encyclopedia
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 ; if any others exist, they must be greater than 4. It has been conjecture
d 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)).
Several computer searches have been done in the hope of finding new Wilson primes.
The Ibercivis
distributed computing
project includes a search for Wilson primes. Another search is coordinated at the mersenneforum.
English people
The English are a nation and ethnic group native to England, who speak English. The English identity is of early mediaeval origin, when they were known in Old English as the Anglecynn. England is now a country of the United Kingdom, and the majority of English people in England are British Citizens...
mathematician John Wilson
John Wilson (mathematician)
John Wilson was an English mathematician. The theorem, Wilson's Theorem, named after him for its discovery from Ibn al-Haytham, not its proof....
, is a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
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
13 (number)
13 is the natural number after 12 and before 14. It is the smallest number with eight letters in its name spelled out in English. It is also the first of the teens – the numbers 13 through 19 – the ages of teenagers....
, and 563 ; if any others exist, they must be greater than 4. It has been conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
d 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)).
Several computer searches have been done in the hope of finding new Wilson primes.
The Ibercivis
Ibercivis
Ibercivis is a distributed computing platform which allows internet users to participate in scientific research by donating unused computer cycles to run scientific simulations and other tasks...
distributed computing
Distributed computing
Distributed computing is a field of computer science that studies distributed systems. A distributed system consists of multiple autonomous computers that communicate through a computer network. The computers interact with each other in order to achieve a common goal...
project includes a search for Wilson primes. Another search is coordinated at the mersenneforum.
Near-Wilson primes
- 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 represent Wilson primes. The following table lists all such primes with |B| ≤ 100 from 106 up to 4:
| 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 |