Agoh–Giuga conjecture

Status

The statement is still a conjecture since it has not yet been proven that if a number ${\displaystyle n}$ is not prime (that is, if ${\displaystyle n}$ is composite), then the formula does not hold. It has been shown that a composite number ${\displaystyle n}$ 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 ${\displaystyle p}$ is prime if and only if

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

which may also be written as

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

For an odd prime ${\displaystyle p}$ we have

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

and for ${\displaystyle p=2}$ we have

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

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

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

and

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