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 Stated by Leibniz in an unpublished paper around 1670.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:






i.e. divides
. But then
divides
or
, by the prime divisor property, so
Thus ◼
In 1957, F.G. Elston generalized Wilson’s theorem:
Let be prime and
.
Then