# Difference between revisions of "Wilson theorem"

From Encyclopedia of Mathematics

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

====References==== | ====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) ( | + | <table> |

− | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Bukhshtab, "Number theory" , Moscow (1966) (In Russian) {{ZBL|0144.27402}}</TD></TR> | |

− | + | <TR><TD valign="top">[2]</TD> <TD valign="top"> E. Trost, "Primzahlen" , Birkhäuser (1953) {{ZBL|0053.36002}}</TD></TR> | |

+ | <TR><TD valign="top">[3]</TD> <TD valign="top"> I.M. [I.M. Vinogradov] Winogradow, "Elemente der Zahlentheorie" , R. Oldenbourg (1956) (In German: translated from Russian) {{ZBL|0070.03802}}</TD></TR> | ||

+ | </table> | ||

====Comments==== | ====Comments==== | ||

− | In fact, also the converse is true (and usually also called Wilson's theorem): Let | + | 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==== | ====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></table> | + | <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> | ||

+ | |||

+ | {{TEX|done}} |

## Latest revision as of 17:21, 18 October 2017

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

[1] | A.A. Bukhshtab, "Number theory" , Moscow (1966) (In Russian) Zbl 0144.27402 |

[2] | E. Trost, "Primzahlen" , Birkhäuser (1953) Zbl 0053.36002 |

[3] | I.M. [I.M. Vinogradov] Winogradow, "Elemente der Zahlentheorie" , R. Oldenbourg (1956) (In German: translated from Russian) Zbl 0070.03802 |

#### 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

[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 |

[a4] | Amrik Singh Nimbran, "Some Remarks on Wilson's Theorem", The Mathematics Student,Indian Mathematical Society, Vol. 67, Nos. 1–4 (1998), 243–245 |

**How to Cite This Entry:**

Wilson theorem.

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

This article was adapted from an original article by N.I. Klimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article