The Wilson quotient ''W''(''p'') is defined as:
:
If ''p'' is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, the quotient is an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
by
Wilson's theorem
In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations of m ...
; moreover, if ''p'' is
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials
...
, the quotient is not an integer. If ''p'' divides ''W''(''p''), it is called a
Wilson prime
In number theory, a Wilson prime is a prime number p such that p^2 divides (p-1)!+1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p-1)!+1. Both are named for 18th-century E ...
. 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
:
:
where
is the ''k''-th
Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
. Note that the first relation comes from the second one by subtraction, after substituting
and
.
See also
*
Fermat quotient In number theory, the Fermat quotient of an integer ''a'' with respect to an odd prime ''p'' is defined as= 3/ref>
The smallest solutions of ''q'p''(''a'') ≡ 0 (mod ''p'') with ''a'' = ''n'' are:
:2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, ...
References
External links
MathWorld: Wilson Quotient
Integer sequences