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
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.
Laerte Sorini, 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.
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