# Agoh–Giuga conjecture

# Agoh–Giuga conjecture

In number theory the **Agoh–Giuga conjecture** on the Bernoulli numbers *B**k* 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 (1990); 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 (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

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