Argoh-Giuga conjecture - Definition

In number theory the Agoh-Giuga conjecture on the Bernoulli numbers B_k postulates that p is a prime number if and only if

pB_p−1 is congruent to −1 mod p.

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

<math>1^{p-1}+2^{p-1}+ \cdots +(p-1)^{p-1} \equiv -1 \pmod p.<math>