pBp-1\equiv-1\pmodp.
It is named after Takashi Agoh and Giuseppe Giuga.
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
1p-1+2p-1+ … +(p-1)p-1\equiv-1\pmodp
which may also be written as
| p-1 | |
| \sum | |
| i=1 |
ip-1\equiv-1\pmodp.
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
ap-1\equiv1\pmodp
for
a=1,2,...,p-1
p-1\equiv-1\pmodp.
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.
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\pmodp,
which may also be written as
| p-1 | |
| \prod | |
| i=1 |
i\equiv-1\pmodp.
For an odd prime p we have
| p-1 | |
| \prod | |
| i=1 |
ip-1\equiv(-1)p-1\equiv1\pmodp,
and for p=2 we have
| p-1 | |
| \prod | |
| i=1 |
ip-1\equiv(-1)p-1\equiv1\pmodp.
So, the truth of the Agoh–Giuga conjecture combined with Wilson's theorem would give: a number p is prime if and only if
| p-1 | |
| \sum | |
| i=1 |
ip-1\equiv-1\pmodp
and
| p-1 | |
| \prod | |
| i=1 |
ip-1\equiv1\pmodp.