Wilson quotient
From HandWiki
The Wilson quotient W(p) is defined as:
- [math]\displaystyle{ W(p) = \frac{(p-1)! + 1}{p} }[/math]
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]
- [math]\displaystyle{ W(p)\equiv B_{2(p-1)}-B_{p-1}\pmod{p}, }[/math]
- [math]\displaystyle{ p-1+ptW(p)\equiv pB_{t(p-1)}\pmod{p^2}, }[/math]
where [math]\displaystyle{ B_k }[/math] is the k-th Bernoulli number. Note that the first relation comes from the second one by subtraction, after substituting [math]\displaystyle{ t=1 }[/math] and [math]\displaystyle{ t=2 }[/math].
See also
References
- ↑ 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.
External links
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