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Wilson's Theorem (Gauss's Generalization)

Let $P$ be the product of Integers less than or equal to $n$ and Relatively Prime to $n$. Then

\begin{displaymath}
P\equiv \prod_{\scriptstyle k=2\atop\scriptstyle k\notdiv n}...
..., p^\alpha, 2p^\alpha$\cr
1\ ({\rm mod\ } n) & otherwise.\cr}
\end{displaymath}

When $m=2$, this reduces to $P\equiv 1\ \left({{\rm mod\ } {2}}\right)$ which is equivalent to $P\equiv -1\ \left({{\rm mod\ } {2}}\right)$.

See also Wilson's Theorem, Wilson's Theorem Corollary




© 1996-9 Eric W. Weisstein
1999-05-26