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)|
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://encyclopediaofmath.org/index.php?title=Wilson_theorem&oldid=17357