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Em matemática , o teorema de Wilson diz que p é um número primo se e somente se:
(
p
−
1
)
!
≡
−
1
(
mod
p
)
{\displaystyle (p-1)!\ \equiv \ -1\ ({\mbox{mod}}\ p)}
Se P é primo então
P divide [(P-1)! + 1].
Todas as n-ésimas diferenças sucessivas da sequência
1
n
,
2
n
,
3
n
,
4
n
.
.
.
{\displaystyle 1^{n},2^{n},3^{n},4^{n}...}
são constantes e iguais a n! (n fatorial).
Resolvendo algebricamente:
2
∗
1
=
1
∗
2
2
−
2
∗
1
2
+
1
∗
0
2
{\displaystyle 2*1=1*2^{2}-2*1^{2}+1*0^{2}}
3
∗
2
∗
1
=
1
∗
3
3
−
3
∗
2
3
+
3
∗
1
3
−
1
∗
0
3
{\displaystyle 3*2*1=1*3^{3}-3*2^{3}+3*1^{3}-1*0^{3}}
4
∗
3
∗
2
∗
1
=
1
∗
4
4
−
4
∗
3
4
+
6
∗
2
4
−
4
∗
1
4
+
1
∗
0
4
{\displaystyle 4*3*2*1=1*4^{4}-4*3^{4}+6*2^{4}-4*1^{4}+1*0^{4}}
5
∗
4
∗
3
∗
2
∗
1
=
1
∗
5
5
−
5
∗
4
5
+
10
∗
3
5
−
10
∗
2
5
+
5
∗
1
5
−
1
∗
0
5
{\displaystyle 5*4*3*2*1=1*5^{5}-5*4^{5}+10*3^{5}-10*2^{5}+5*1^{5}-1*0^{5}}
6
∗
5
∗
4
∗
3
∗
2
∗
1
=
1
∗
6
6
−
6
∗
5
6
+
15
∗
4
6
−
20
∗
3
6
+
15
∗
2
6
−
6
∗
1
6
+
1
∗
0
6
.
{\displaystyle 6*5*4*3*2*1=1*6^{6}-6*5^{6}+15*4^{6}-20*3^{6}+15*2^{6}-6*1^{6}+1*0^{6}.}
O teorema foi estabelecido por Alhazen [ 1] John Wilson .[ 2]
Se n é composto mas não é o quadrado de um primo podemos escrever
n
=
a
b
{\displaystyle n=ab}
1
<
a
<
b
<
n
{\displaystyle 1<a<b<n}
a
{\displaystyle a}
b
{\displaystyle b}
(
n
−
1
)
!
{\displaystyle (n-1)!}
(
n
−
1
)
!
≡
0
(
mod
n
)
{\displaystyle (n-1)!\equiv 0{\pmod {n}}}
p
2
{\displaystyle p^{2}}
p
>
2
{\displaystyle p>2}
p
{\displaystyle p}
2
p
{\displaystyle 2p}
(
n
−
1
)
!
{\displaystyle (n-1)!}
(
n
−
1
)
!
≡
0
(
mod
n
)
{\displaystyle (n-1)!\equiv 0{\pmod {n}}}
n
≠
4
{\displaystyle n\neq 4}
(
n
−
1
)
!
≡
0
(
mod
n
)
{\displaystyle (n-1)!\equiv 0{\pmod {n}}}
n
{\displaystyle n}
(
n
−
1
)
!
≡
−
2
∗
3
∗
.
.
.
∗
(
n
−
2
)
(
mod
n
)
{\displaystyle (n-1)!\equiv -2*3*...*(n-2){\pmod {n}}}
(
n
−
1
)
!
≡
−
1
(
mod
n
)
{\displaystyle (n-1)!\equiv -1{\pmod {n}}}
n
(n-1)!
(n-1)! mod n
2
1
1
3
2
2
4
6
2
5
24
4
6
120
0
7
720
6
8
5040
0
9
40320
0
10
362880
0
11
3628800
10
12
39916800
0
13
479001600
12
14
6227020800
0
15
87178291200
0
16
1307674368000
0
17
20922789888000
16
18
355687428096000
0
19
6402373705728000
18
20
121645100408832000
0
21
2432902008176640000
0
22
51090942171709440000
0
23
1124000727777607680000
22
24
25852016738884976640000
0
25
620448401733239439360000
0
26
15511210043330985984000000
0
27
403291461126605635584000000
0
28
10888869450418352160768000000
0
29
304888344611713860501504000000
28
30
8841761993739701954543616000000
0
2
∗
1
=
1
∗
(
2
2
−
1
2
)
−
1
2
{\displaystyle 2*1=1*(2^{2}-1^{2})-1^{2}}
2
∗
1
+
1
2
=
1
∗
(
2
2
−
1
2
)
{\displaystyle 2*1+1^{2}=1*(2^{2}-1^{2})}
pelo pequeno teorema do Fermat 3 divide
(
2
∗
1
+
1
2
)
{\displaystyle (2*1+1^{2})}
4
∗
3
∗
2
∗
1
=
1
∗
(
4
4
−
3
4
)
−
3
∗
(
3
4
−
2
4
)
+
3
∗
(
2
4
−
1
4
)
−
1
4
{\displaystyle 4*3*2*1=1*(4^{4}-3^{4})-3*(3^{4}-2^{4})+3*(2^{4}-1^{4})-1^{4}}
4
∗
3
∗
2
∗
1
+
1
4
=
1
∗
(
4
4
−
3
4
)
−
3
∗
(
3
4
−
2
4
)
+
3
∗
(
2
4
−
1
4
)
{\displaystyle 4*3*2*1+1^{4}=1*(4^{4}-3^{4})-3*(3^{4}-2^{4})+3*(2^{4}-1^{4})}
pelo pequeno teorema do Fermat 5 divide
(
4
∗
3
∗
2
∗
1
+
1
4
)
{\displaystyle (4*3*2*1+1^{4})}
6
∗
5
∗
4
∗
3
∗
2
∗
1
=
1
∗
(
6
6
−
5
6
)
−
5
∗
(
5
6
−
4
6
)
+
10
∗
(
4
6
−
3
6
)
−
10
(
3
6
−
2
6
)
+
5
(
2
6
−
1
6
)
−
1
6
{\displaystyle 6*5*4*3*2*1=1*(6^{6}-5^{6})-5*(5^{6}-4^{6})+10*(4^{6}-3^{6})-10(3^{6}-2^{6})+5(2^{6}-1^{6})-1^{6}}
6
∗
5
∗
4
∗
3
∗
2
∗
1
+
1
6
=
1
∗
(
6
6
−
5
6
)
−
5
∗
(
5
6
−
4
6
)
+
10
∗
(
4
6
−
3
6
)
−
10
(
3
6
−
2
6
)
+
5
(
2
6
−
1
6
)
{\displaystyle 6*5*4*3*2*1+1^{6}=1*(6^{6}-5^{6})-5*(5^{6}-4^{6})+10*(4^{6}-3^{6})-10(3^{6}-2^{6})+5(2^{6}-1^{6})}
pelo pequeno teorema do Fermat 7 divide
(
6
∗
5
∗
4
∗
3
∗
2
∗
1
+
1
6
)
{\displaystyle (6*5*4*3*2*1+1^{6})}
O interessante é que existe uma crença de que o Teorema de Wilson percebe os pseudo-primos absolutos.
O teorema de Wilson é uma simples variação do Pequeno Teorema do Pierre de Fermat .
↑ Biografia em MacTutor (em inglês )↑ Edward Waring, Meditationes Algebraicae (Cambridge, England: 1770), page 218 (in Latin). In the third (1782) edition of Waring's Mediationes Algebraicae , Wilson's theorem appears as problem 5 on page 380 . On that page, Waring states: "Hanc maxime elegantem primorum numerorum proprietatem invenit vir clarissimus, rerumque mathematicarum peritissimus Joannes Wilson Armiger." (A man most illustrious and most skilled in mathematics, Squire John Wilson, found this most elegant property of prime numbers.)
