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 Template: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

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

$p-1+ptW(p)\equiv p{B}_{t(p-1)}(\mathrm{mod}{p}^2),$

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