Editing Wilson's theorem (section)

==Gauss's generalization==
[[Carl Friedrich Gauss|Gauss]] proved<ref>Gauss, DA, art. 78</ref><ref>{{cite journal
 | last1 = Cosgrave | first1 = John B.
 | last2 = Dilcher | first2 = Karl
 | journal = Integers
 | mr = 2472057
 | article-number = A39
 | title = Extensions of the Gauss–Wilson theorem
 | url = https://emis.de/journals/INTEGERS/papers/i39/i39.Abstract.html
 | volume = 8
 | year = 2008}}</ref> that
<math display=block>
\prod_{k = 1 \atop \gcd(k,m)=1}^{m-1} \!\!k \ \equiv
\begin{cases}
-1      \pmod{m}  & \text{if } m=4,\;p^\alpha,\;2p^\alpha \\
\;\;\,1 \pmod{m}  & \text{otherwise}
\end{cases}
</math>
where ''p'' represents an odd prime and <math>\alpha</math> a positive integer. That is, the product of the positive integers less than {{mvar|m}} and relatively prime to {{mvar|m}} is one less than a multiple of {{mvar|m}} when {{mvar|m}} is equal to 4, or a power of an odd prime, or twice a power of an odd prime; otherwise, the product is one more than a multiple of {{mvar|m}}.  The values of ''m'' for which the product is −1 are precisely the ones where there is a [[Primitive root modulo n|primitive root modulo ''m'']].