Agoh-Giuga conjecture

Statement

Formulation in terms of Bernoulli numbers

This formulation dates to Agoh in 1990. It states that a natural number $p$ is a prime number if and only if we have:

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

where $B_{p-1}$ is the Bernoulli number corresponding to $p-1$.

Formulation in terms of power sums

This formulation dates to Giuga in the 1950s.

A natural number $p$ is a prime number if and only if we have:

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

Note that one direction is immediate: if $p$ is a prime number, then the above congruence holds true as a consequence of Fermat's little theorem. The other direction is conjectural and open.

Formulation in terms of Giuga numbers

There is no (composite) natural number that is both a Carmichael number and a Giuga number. Note that this is equivalent to the power sums formulation because composite number satisfies the power sums condition iff it is both a Carmichael number and a Giuga number.