Giuga's 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

It is named after Takashi Agoh and Giuseppe Giuga.

The conjecture as stated above is due to Takashi Agoh (1990); an equivalent formulation is due to Giuseppe Giuga, from 1950, to the effect that p is prime if

which may also be written as

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

for ${\displaystyle a=1,2,\dots ,p-1}$, and the equivalence follows, since ${\displaystyle p-1\equiv -1{\pmod {p}}.}$

The statement is still a conjecture since it has not yet been proven that if a number n is not prime (that is, n is composite), then the formula does not hold.

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