Agoh-Giuga conjecture

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Statement

Formulation in terms of Bernoulli numbers

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

where is the Bernoulli number corresponding to .

Formulation in terms of power sums

This formulation dates to Giuga in the 1950s.

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

Note that one direction is immediate: if 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.