Wilson prime facts for kids

A Wilson prime is a special kind of prime number. A prime number p is a Wilson prime if (and only if [ iff ])

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

where n is a positive integer (sometimes called natural number). Wilson primes were first described by Emma Lehmer.

The only known Wilson primes are 5, 13, and 563 (sequence A007540 in OEIS); if any others exist, they must be greater than 5×10⁸. It has been conjectured that there are an infinite number of Wilson primes, and that the number of Wilson primes in an interval $[x, y]$ is about

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

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

Wieferich prime
Wall-Sun-Sun prime
Wolstenholme prime

Eric W. Weisstein, Wilson prime at MathWorld.

Wilson prime facts for kids

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See also