View source for Wilson theorem

Let $p$ be a prime number. Then the number $(p-1)!+1$ is divisible by $p$. The theorem was first formulated by E. Waring (1770) and is, according to him, due to J. Wilson. It was proved by J.L. Lagrange in 1771. A primality test for integers follows from Wilson's theorem: A natural number $n>1$ will be prime if and only if
$$
(n-1)! + 1 \equiv 0 \pmod n
$$

This test is not recommended for practical use, since the factorial involved rapidly becomes very large.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Bukhshtab,   "Number theory" , Moscow  (1966)  (In Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  E. Trost,   "Primzahlen" , Birkhäuser  (1953)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  I.M. [I.M. Vinogradov] Winogradow,   "Elemente der Zahlentheorie" , R. Oldenbourg  (1956)  (Translated from Russian)</TD></TR>
</table>

====Comments====
In fact, also the converse is true (and usually also called Wilson's theorem): Let $N = (p-1)!+1$, with $p \in \mathbf{N}$. Then $N$ is divisible by $p$ if and only if $p$ is a prime number.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Shanks,   "Solved and unsolved problems in number theory" , Chelsea, reprint  (1978)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  M.R. Schroeder,   "Number theory in science and communication" , Springer  (1984)  pp. 103</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  G.H. Hardy,   E.M. Wright,   "An introduction to the theory of numbers" , Clarendon Press  (1960)  pp. 68</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top"> Amrik Singh Nimbran, ''Some Remarks on Wilson's Theorem'', 'The Mathematics Student',Indian Mathematical Society, Vol. 67, Nos. 1–4 (1998), 243–245</TD></TR>
</table>

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