Difference between revisions of "Wilson's Theorem"
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== See also == | == See also == | ||
* [[Number theory]] | * [[Number theory]] | ||
+ | * [[Modular arithmetic]] | ||
+ | * [[Factorial]] |
Revision as of 17:49, 17 June 2006
Statement
If and only if is a prime, then
is a multiple of
. In other words
.
Proof
Wilson's theorem is easily verifiable for 2 and 3, so let's consider . If
is composite, then its positive factors are among
. Hence,
, so
.
However if is prime, then each of the above integers are relatively prime to
. So for each of these integers a there is another
such that
. It is important to note that this
is unique modulo
, and that since
is prime,
if and only if
is
or
. Now if we omit 1 and
, then the others can be grouped into pairs whose product is congruent to one,
Finally, multiply this equality by to complete the proof.
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