Wilson prime

A Wilson prime is a special kind of prime number. A prime number p is a Wilson prime if (and only if [ ${\displaystyle \iff }$ ])

${\displaystyle {\frac {(p-1)!+1}{p^{2}}}=n}$

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 (sequence A007540 in the OEIS); if any others exist, they must be greater than 5×10⁸.^[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 ${\displaystyle [x,y]}$ is about

${\displaystyle {\frac {\log \left(\log y\right)}{\log x}}}$.

Compare this with Wilson's theorem, which states that every prime p divides (p − 1)! + 1.

Related pages

• Wall-Sun-Sun prime
• Wieferich prime
• Wolstenholme prime