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

${\displaystyle pB_{p-1}\equiv -1{\pmod {p}}.}$

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

${\displaystyle 1^{p-1}+2^{p-1}+\cdots +(p-1)^{p-1}\equiv -1{\pmod {p}}}$

which may also be written as

${\displaystyle \sum _{i=1}^{p-1}i^{p-1}\equiv -1{\pmod {p}}.}$

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

${\displaystyle a^{p-1}\equiv 1{\pmod {p}}}$

for ${\displaystyle a=1,2,\dots ,p-1}$, and the equivalence follows, since ${\displaystyle p-1\equiv -1{\pmod {p}}.}$

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, 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

${\displaystyle (p-1)!\equiv -1{\pmod {p}},}$

which may also be written as

${\displaystyle \prod _{i=1}^{p-1}i\equiv -1{\pmod {p}}.}$

For an odd prime p we have

${\displaystyle \prod _{i=1}^{p-1}i^{p-1}\equiv (-1)^{p-1}\equiv 1{\pmod {p}},}$

and for p=2 we have

${\displaystyle \prod _{i=1}^{p-1}i^{p-1}\equiv (-1)^{p-1}\equiv 1{\pmod {p}}.}$

So, the truth of the Agoh–Giuga conjecture combined with Wilson's theorem would give: a number p is prime if and only if

${\displaystyle \sum _{i=1}^{p-1}i^{p-1}\equiv -1{\pmod {p}}}$

and

${\displaystyle \prod _{i=1}^{p-1}i^{p-1}\equiv 1{\pmod {p}}.}$

See also

• Bernoulli number § Arithmetical properties of the Bernoulli numbers