Agoh–Giuga conjecture

In number theory the Agoh–Giuga conjecture on the Bernoulli numbers Bk postulates that p is a prime number if and only if


 * $$pB_{p-1} \equiv -1 \pmod p.$$

It is named after Takashi Agoh and Giuseppe Giuga.

Equivalent formulation
The conjecture as stated above is due to Takashi Agoh (1990); an equivalent formulation is due to Giuseppe Giuga, from 1950, to the effect that p is prime if and only if


 * $$1^{p-1}+2^{p-1}+ \cdots +(p-1)^{p-1} \equiv -1 \pmod p$$

which may also be written as


 * $$\sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod p.$$

It is trivial to show that p being prime is sufficient for the second equivalence to hold, since if p is prime, Fermat's little theorem states that


 * $$a^{p-1} \equiv 1 \pmod p$$

for $$a = 1,2,\dots,p-1$$, and the equivalence follows, since $$p-1 \equiv -1 \pmod p.$$

Status
The statement is still a conjecture since it has not yet been proven that if a number n is not prime (that is, n is composite), then the formula does not hold. It has been shown that a composite number n satisfies the formula if and only if it is both a Carmichael number and a Giuga number, and that if such a number exists, it has at least 13,800 digits (Borwein, Borwein, Borwein, Girgensohn 1996). Laerte Sorini, finally, in a work of 2001 showed that a possible counterexample should be a number n greater than 1036067 which represents the limit suggested by Bedocchi for the demonstration technique specified by Giuga to his own conjecture.

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


 * $$(p-1)! \equiv -1 \pmod p,$$

which may also be written as


 * $$\prod_{i=1}^{p-1} i \equiv -1 \pmod p.$$

For an odd prime p we have


 * $$\prod_{i=1}^{p-1} i^{p-1} \equiv (-1)^{p-1} \equiv 1 \pmod p,$$

and for p=2 we have


 * $$\prod_{i=1}^{p-1} i^{p-1} \equiv (-1)^{p-1} \equiv 1 \pmod p.$$

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

and


 * $$\prod_{i=1}^{p-1} i^{p-1} \equiv 1 \pmod p.$$