Wilson's Theorem
Statement
If and only if p is a prime, then 
is a multiple of p. Written more mathematically,
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 ifis prime, then each of the above integers are relatively prime to
there is another
such that
. It is important to note that this
if and only if
or
. Now if we omit
there is another 
such that 
. It is important to note that this 
if and only if 
or 
. Now if we omit 
Finally, multiply this equality by p-1 to complete the proof.
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