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

p B_{p - 1} \equiv - 1 (\mod p) .

It is named after Takashi Agoh and Giuseppe Giuga.

Equivalent formulation

The conjecture as stated above is due to Takashi Agoh; an equivalent formulation is due to Giuseppe Giuga, from 1950, to the effect that $p$ is prime if and only if

1^{p - 1} + 2^{p - 1} + \dots + (p - 1)^{p - 1} \equiv - 1 (\mod p)

,

which may also be written as

\sum_{i = 1}^{p - 1} i^{p - 1} \equiv - 1 (\mod 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 (\mod p)

for $a = 1, 2, \dots, p - 1$ , and the equivalence follows since $p - 1 \equiv - 1 (\mod p) .$

Status

The statement is still a conjecture since it has not yet been proven that if a number $n$ is not prime (that is, if $n$ 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. In 2001, Sorini showed that a possible counterexample should be 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

(p - 1)! \equiv - 1 (\mod p),

which may also be written as

\prod_{i = 1}^{p - 1} i \equiv - 1 (\mod p) .

For an odd prime $p$ we have

\prod_{i = 1}^{p - 1} i^{p - 1} \equiv (- 1)^{p - 1} \equiv 1 (\mod p),

and for $p = 2$ we have

\prod_{i = 1}^{p - 1} i^{p - 1} \equiv (- 1)^{p - 1} \equiv 1 (\mod p) .

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

\sum_{i = 1}^{p - 1} i^{p - 1} \equiv - 1 (\mod p)

and

\prod_{i = 1}^{p - 1} i^{p - 1} \equiv 1 (\mod p) .

Notes

References

Giuga, Giuseppe (1950). "Su una presumibile proprietà caratteristica dei numeri primi". Istituto Lombardo di Scienze e Lettere, Rendiconti, Classe di Scienze Matematiche e Naturali (in Italian). 14 (83): 511–518. ISSN 0375-9164. Zbl 0045.01801.

Agoh, Takashi (1995). "On Giuga's conjecture". Manuscripta Mathematica. 87 (4): 501–510. doi:10.1007/bf02570490. Zbl 0845.11004.

Borwein, D.; Borwein, J. M.; Borwein, P. B.; Girgensohn, R. (1996). "Giuga's conjecture on primality". American Mathematical Monthly. 103 (1): 40–50. doi:10.2307/2975213. JSTOR 2975213. Zbl 0860.11003.

Sorini, Laerte (2001). "Un metodo euristico per la soluzione della congettura di Giuga". Quaderni di Economia, Matematica e Statistica, DESP, Università di Urbino Carlo Bo (in Italian). 68. ISSN 1720-9668.

[FOOTNOTEAgoh1995-1] Agoh (1995).

[FOOTNOTEGiuga1950-2] Giuga (1950).

[FOOTNOTEBorweinBorweinBorweinGirgensohn1996-3] Borwein et al. (1996).

[FOOTNOTESorini2001-4] Sorini (2001).