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Wilson quotient

From Wikipedia, the free encyclopedia

The Wilson quotient W(p) is defined as:

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[1]

where is the k-th Bernoulli number. Note that the first relation comes from the second one by subtraction, after substituting and .

See also



  1. ^ Lehmer, Emma (1938). "On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson". Annals of Mathematics. 39 (2): 350–360. doi:10.2307/1968791. JSTOR 1968791.