Wilson quotient explained

The Wilson quotient W(p) is defined as:

W(p) = ((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 :

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^[1]

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

References

1. Lehmer . Emma. On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson. Annals of Mathematics . 1938 . 39 . 2. 350–360 . 10.2307/1968791. 1968791.

External links

• MathWorld: Wilson Quotient