Wilson quotient

The Wilson quotient W(p) is defined as:

$W(p)={\frac {(p-1)!+1}{p}}$

If p is a prime number, the quotient is an integer by Wilson's theorem; moreover, if p is composite, the quotient is not an integer. If p divides W(p), it is called a Wilson prime. The integer values of W(p) are (sequence A007619 in the OEIS):

W(2) = 1

W(3) = 1

W(5) = 5

W(7) = 103

W(11) = 329891

W(13) = 36846277

W(17) = 1230752346353

W(19) = 336967037143579

...

It is known that

$W(p)\equiv B_{2(p-1)}-B_{p-1}{\pmod {p}},$

$p-1+ptW(p)\equiv pB_{t(p-1)}{\pmod {p^{2}}},$

where $B_{k}$ is the k-th Bernoulli number. Note that the first relation comes from the second one by subtraction, after substituting $t=1$ and $t=2$.

See also

• Fermat quotient

External links

• MathWorld: Wilson Quotient