Wilson prime
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A Wilson prime is a special kind of prime number. A prime number p is a Wilson prime if (and only if [ iff ])
[math]\displaystyle{ \frac{\left(p-1\right)! + 1}{p^2} = n\,\! }[/math]
where n is a positive integer (sometimes called natural number). Wilson primes were first described by Emma Lehmer.[1]
The only known Wilson primes are 5, 13, and 563 Template:OEIS; if any others exist, they must be greater than 5Template:E.[2] It has been conjectured[3] that there are an infinite number of Wilson primes, and that the number of Wilson primes in an interval Template:Math is about
[math]\displaystyle{ \frac{\log \left ( \log y \right )}{\log x} }[/math].
Compare this with Wilson's theorem, which states that every prime p divides (p − 1)! + 1.
Related pages
Notes
- ↑ On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of. Math. 39(1938), 350-360.
- ↑ Status of the search for Wilson primes, also see Crandall et al. 1997
- ↑ The Prime Glossary: Wilson prime
References
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