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.

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

1^{p-1}%2B2^{p-1}%2B \cdots %2B(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.

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 iff 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).

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 iff

(p-1)! \equiv -1 \pmod p

which may also be written as

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

or, for odd prime p

\prod_{i=1}^{p-1} i^{p-1} \equiv (-1)^{p-1} \equiv %2B1 \pmod p

and, for even prime p=2

\prod_{i=1}^{p-1} i^{p-1} \equiv (-1)^{p-1} \equiv -1 \equiv %2B1 \pmod p.

So, the truth of the Agoh-Giuga conjecture combined with Wilson's theorem would give: a number p is prime iff

\sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod p,

and

\prod_{i=1}^{p-1} i^{p-1} \equiv %2B1 \pmod p.

References