# Wilson theorem

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.

#### References

[1] | A.A. Bukhshtab, "Number theory" , Moscow (1966) (In Russian) |

[2] | E. Trost, "Primzahlen" , Birkhäuser (1953) |

[3] | I.M. [I.M. Vinogradov] Winogradow, "Elemente der Zahlentheorie" , R. Oldenbourg (1956) (Translated from Russian) |

[4] Amrik Singh Nimbran, *Some Remarks on Wilson's Theorem*, 'The Mathematics Student',Indian Mathematical Society,
Vol. 67, Nos. 1–4 (1998), 243–245

#### Comments

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.

#### References

[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 |

**How to Cite This Entry:**

Wilson theorem.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Wilson_theorem&oldid=17357