Let be a prime number. Then the number is divisible by . 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 will be prime if and only if
This test is not recommended for practical use, since the factorial involved rapidly becomes very large.
|||A.A. Bukhshtab, "Number theory" , Moscow (1966) (In Russian)|
|||E. Trost, "Primzahlen" , Birkhäuser (1953)|
|||I.M. [I.M. Vinogradov] Winogradow, "Elemente der Zahlentheorie" , R. Oldenbourg (1956) (Translated from Russian)|
 Amrik Singh Nimbran, Some Remarks on Wilson's Theorem, 'The Mathematics Student',Indian Mathematical Society, Vol. 67, Nos. 1–4 (1998), 243–245
In fact, also the converse is true (and usually also called Wilson's theorem): Let , with . Then is divisible by if and only if is a prime number.
|[a1]||D. Shanks, "Solved and unsolved problems in number theory" , Chelsea, reprint (1978)|
|[a2]||M.R. Schroeder, "Number theory in science and communication" , Springer (1984) pp. 103|
|[a3]||G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Clarendon Press (1960) pp. 68|
Wilson theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Wilson_theorem&oldid=17357