Agoh–Giuga conjecture

Template:Short description In number theory, the Agoh–Giuga conjecture on the Bernoulli numbers $B_k$ postulates that $p$ is a prime number if and only if

$p{B}_{p-1}\equiv -1(\mathrm{mod}p).$

It is named after Takashi Agoh and Giuseppe Giuga.

Equivalent formulation

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

$1^{p-1}+2^{p-1}+\cdots +(p-1{)}^{p-1}\equiv -1(\mathrm{mod}p)$,

which may also be written as

$\sum _{i=1}^{p-1}{i}^{p-1}\equiv -1(\mathrm{mod}p).$

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

$a^{p-1}\equiv 1(\mathrm{mod}p)$

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

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.Template:Sfnp In 2001, Sorini showed that a possible counterexample should be greater than 10³⁶⁰⁶⁷,Template:Sfnp 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

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

which may also be written as

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

For an odd prime $p$ we have

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

and for $p=2$ we have

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

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

$\sum _{i=1}^{p-1}{i}^{p-1}\equiv -1(\mathrm{mod}p)$

and

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

See also

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References

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