First known appearance in a work of Ibn al-Haytham (965-1039). Stated by Leibniz in an unpublished paper around 1670. Conjectured by John Wilson before 1770. First proved by Lagrange in 1771.
i.e. if 
is prime, divides into 
Proof: The factors 
of 
all have inverses 
, so each is cancelled by its own inverse except the factors that are inverse to themselves. These are 
and 
, and no others – because if 
then:
of all have inverses , so each is cancelled by its own inverse except the factors that are inverse to themselves. These are and , and no others – because if then:
![\[x^2-1\equiv (x-1)(x+1)\equiv 0 \pmod{p},\]](https://www.adamponting.com/wp-content/ql-cache/quicklatex.com-63c802f0722f2b75d375ed91f650993b_l3.png "Rendered by QuickLaTeX.com")
i.e. divides 
. But then divides 
or 
, by the prime divisor property, so
![\[x\equiv 1 \pmod{p} \quad \text{or} \quad x\equiv -1 \pmod{p}.\]](https://www.adamponting.com/wp-content/ql-cache/quicklatex.com-d865d253e35bcf3188ec94067c8f3abc_l3.png "Rendered by QuickLaTeX.com")
Thus 
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In 1957, F.G. Elston generalized Wilson’s theorem:
Let be prime and 
.
Then 