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

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

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

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

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

$\prod_{i=1}^{p-1} i^{p-1} \equiv (-1)^{p-1} \equiv +1 \pmod p$

and, for even prime p=2

$\prod_{i=1}^{p-1} i^{p-1} \equiv (-1)^{p-1} \equiv -1 \equiv +1 \pmod p.$

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

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

and

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

References

• Agoh, T, "On Giuga’s conjecture" Manuscripta Math., 87(4), 501–510 (1995).
• Borwein, D.; Borwein, J. M., Borwein, P. B., and Girgensohn, R. "Giuga's Conjecture on Primality", American Mathematical Monthly, 103, 40–50, (1996). pdf
• Giuga, G. "Su una presumibile proprietà caratteristica dei numeri primi", Ist. Lombardo Sci. Lett. Rend. A, 83, 511–528 (1950).