Editing Agoh–Giuga conjecture (section)

==Relation to Wilson's theorem==
The Agoh–Giuga conjecture bears a similarity to [[Wilson's theorem]], which has been proven to be true. Wilson's theorem states that a number ''p'' is prime if and only if

:<math>(p-1)! \equiv -1 \pmod p,</math>

which may also be written as

:<math>\prod_{i=1}^{p-1} i \equiv -1 \pmod p.</math>

For an odd prime p we have

:<math>\prod_{i=1}^{p-1} i^{p-1} \equiv (-1)^{p-1} \equiv 1 \pmod p,</math>

and for p=2 we have

:<math>\prod_{i=1}^{p-1} i^{p-1} \equiv (-1)^{p-1} \equiv 1 \pmod p.</math>

So, the truth of the Agoh–Giuga conjecture combined with Wilson's theorem would give: a number ''p'' is prime if and only if
 
:<math>\sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod p</math>

and

:<math>\prod_{i=1}^{p-1} i^{p-1} \equiv 1 \pmod p.</math>