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
It is named after Takashi Agoh and Giuseppe Giuga.

Equivalent formulation

The conjecture as stated above is due to Takashi Agoh ; an equivalent formulation is due to Giuseppe Giuga, from 1950, to the effect that p is prime if and only if
which may also be written as
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
for, and the equivalence follows, since

Status

The statement is still a conjecture since it has not yet been proven that if a number n is not prime, 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. 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
which may also be written as
For an odd prime p we have
and for p=2 we have
So, the truth of the Agoh–Giuga conjecture combined with Wilson's theorem would give: a number p is prime if and only if
and