Namespaces
Variants
Actions

Difference between revisions of "Wilson theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (References: better)
 
(3 intermediate revisions by 2 users not shown)
Line 1: Line 1:
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098010/w0980101.png" /> be a prime number. Then the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098010/w0980102.png" /> is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098010/w0980103.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098010/w0980104.png" /> will be prime if and only if
+
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
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098010/w0980105.png" /></td> </tr></table>
+
(n-1)! + 1 \equiv 0 \pmod n
 +
$$
  
 
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)  (Translated from Russian)</TD></TR></table>
+
<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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098010/w0980106.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098010/w0980107.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098010/w0980108.png" /> is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098010/w0980109.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098010/w09801010.png" /> is a prime number.
+
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