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Wilson's theorem (Theorem)

Wilson's theorem states that

$\displaystyle (p-1)! \equiv -1 \pmod{p} $
iff $ p$ is prime (note that it must be positive).



"Wilson's theorem" is owned by CWoo. [ full author list (2) | owner history (1) ]
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See Also: an algebraic identity leading to Wilson's theorem, polynomial congruence

Other names:  Wilson-Lagrange theorem

Attachments:
proof of Wilson's theorem (Proof) by Cosmin
group theoretic proof of Wilson's theorem (Proof) by ottocolori
Wilson's theorem result (Result) by mathwizard
Wilson prime (Definition) by PrimeFan
Wilson quotient (Definition) by PrimeFan
Clement's theorem on twin primes (Theorem) by PrimeFan
converse of Wilson's theorem (Theorem) by PrimeFan
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Cross-references: positive, prime, iff
There are 6 references to this entry.

This is version 8 of Wilson's theorem, born on 2001-08-13, modified 2008-05-17.
Object id is 11, canonical name is WilsonsTheorem.
Accessed 8788 times total.

Classification:
AMS MSC: 11-00 (Number theory :: General reference works )
Pending Errata and Addenda
None.
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Negative primes by pahio on 2005-10-18 12:41:49
 I perceive that not all people here believe the negative primes (-2, -3, -5 and so on). But in number-theoretical sense, they are as good as the positive primes, since both have exactly same divisibily properties. And ideal-theoretically, p and -p generate the same principal ideal, as always do the associates. In the other algebraic number fields than Q we, in fact, can not say that some of the associated primes is "better" or "more correct" than some else.
 The (rational) prime defined in PM may be negative equally as positive.
 Jussi
[ reply | up ]
even better... by drini on 2001-08-20 21:40:26
$p$ is a prime
IF AND ONLY IF
$(p-1)!\equiv -1\pmod{p}$

now tell me, isn't that great?
 f 
G ---------> H 
 \ ^ G 
p \ /_ ----- ~ f(G)
 \ / f ker f 
 Y / 
 G/ker f 
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