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

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

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

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

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

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

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

which may also be written as

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

For an odd prime $p$ we have

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

and for $p=2$ we have

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

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

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

and

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

See also

• Bernoulli number § Arithmetical properties of the Bernoulli numbers

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.